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Covid-19-vaccination

2022-02-01T22:08:40Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic. There are 5 deaths.

===Simulations with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic in this model.
The infection rate among vaccinated individuals and unvaccinated individuals behaves similar regardless of vaccination coverage. The course of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads quickly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T22:08:08Z

Laura: /* Results */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic. There are 5 deaths.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic in this model.
The infection rate among vaccinated individuals and unvaccinated individuals behaves similar regardless of vaccination coverage. The course of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads quickly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T22:07:49Z

Laura: /* Results */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic. There are 5 deaths.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic in this model.
The infection rate among vaccinated individuals and unvaccinated individuals behaves similar regardless of vaccination coverage. The course of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads quickly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T22:07:01Z

Laura: /* Vaccination rate: 100% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic. there are 5 deaths.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic in this model.
The infection rate among vaccinated individuals and unvaccinated individuals behaves similar regardless of vaccination coverage. The course of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads quickly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

WS 2021/2022

2022-02-01T21:56:34Z

Laura:

Semestral papers from winter term 2021/2022. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2021/2022|assignment approved]]

[[Carsharing_Fleet_Optimization|Carsharing Fleet Optimization]], Sergei Shcherbinin [[User:Shcs00|shcs00]] 16:57, 20 December 2021 (CET)

[https://www.simulace.info/index.php/Optimizing_the_Process_of_Baking_Wedding_Sweets#Model_Source_Code Optimizing the Process of Baking Wedding Sweets], [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 15:25, 21 January 2022 (CET)

[https://www.simulace.info/index.php/A_b_test A/B test] By Vaso Dzhinchvelashvili

[https://www.simulace.info/index.php/Is_it_possible_to_stop_fake_news Is it possible to stop fake news], [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 14:41, 30 January 2022 (CET)

[https://www.simulace.info/index.php/Scooter_rental Scooter rental], Liudmila Kalashnikova, 31 January 2022

[[Covid-19-vaccination|Effects of COVID-19 vaccination on the spread of infection]], Laura Kundmüller, 1 February 2022

Covid-19-vaccination

2022-02-01T21:41:17Z

Laura: /* Conclusion */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic in this model.
The infection rate among vaccinated individuals and unvaccinated individuals behaves similar regardless of vaccination coverage. The course of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads quickly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Assignments WS 2021/2022

2022-02-01T21:34:15Z

Laura: /* Effects of COVID-19 vaccination on the spread of infection */

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== Spread of covid19 in closed/open area markets ==

In winter 2021 in Czechia the christmas markets were banned due to another covid19 infection wave. On the other hand people are free to go into shopping malls. It would be interesting to use existing data about covid19 virus transmission in agent based simulation to see how many people get infected and in what speed depending on whether they are in a christmas (open) market, or in a shopping mall (closed). The main goal will be to see if the simulation would backup the decision that has been made about christmas markets.

Possible research papers that contain data about covid spreading
* [https://www.kaggle.com/guidant/covid19-evolution-transmission-spatialpatterns/ Kaggle notebook - Covid19, Evolution, Transmission, Spatial Patterns]
* [https://link.springer.com/article/10.1007/s10668-020-00884-x/ Research - Understanding COVID-19 transmission, health impacts and mitigation: timely social distancing is the key]

This simulation would be realised using NetLogo.

Summary:

'''WHAT will be simulated'''
* market place, which can be both open space or closed space.
* people with or without masks, who will walk from shop to shop, with some intention and some of them will be virus carriers
* virus, which will spread in places where people go through (depending on the closed/open area, the infection rates will differ)

'''GOAL of the simulation'''
* answer the question: "Where is the virus spread more significant? At the market place, or at the shopping mall?"

'''TOOL used for the simulation'''
* NetLogo
* Agent based simulation

'''Author:''' Angel Kostov, xkosa20

: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 11:58, 10 December 2021 (CET)

== Effects of COVID-19 vaccination on the spread of infection ==

'''Problem definition'''

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 50,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures are the vaccination and masks. Vaccination can reduce the risk of infection and the likelihood of transmissibility. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic.

'''Simulation'''

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model in order to simulate the scenario in a simplified form based on existing scientific data. In addition, current COVID-19 measures are considered.

Subjects of the simulation:

Environment:
* A village with around 1.000 inhabitants
* Simplified: The village is closed that is new people cannot come in and people of the village cannot go out

Agents:
* Vaccinated persons
* Unvaccinated persons
* Simulations with different vaccination rates to compare the different infection courses

Further measures:
* All people wear masks
* Nobody wears a mask

Start:
* Few agents are unknowingly infected (e.g. 1% of the inhabitants = 10 people)
* Movement of agents intended to reflect the daily behavior of people in real life in a simplified form.
* One agent can infect another agent with a certain probability if they are close to each other.
* The probability of an infection depends on the measure (vaccination / mask)

'''Goal:'''
* Identify the infection course with different vaccination rates and measures.
* Showing the importance of vaccination.

'''Method'''
* NetLogo

'''Possible data sources'''

https://www.rki.de/SharedDocs/FAQ/COVID-Impfen/FAQ_Liste_Wirksamkeit.html

Chia, P. Y., Xiang Ong, S. W., Chiew, C. J., Ang, L. W., Chavatte, J.-M., Mak, T.-M., Cui, L., Kalimuddin, S., Chia, W. N., Tan, C. W., Ann Chai, L. Y., Tan, S. Y., Zheng, S., Pin Lin, R. T., Wang, L., Leo, Y.-S., Lee, V. J., Lye, D. C., & Young, B. E. (2021). Virological and serological kinetics of SARS-CoV-2 Delta variant vaccine-breakthrough infections: A multi-center cohort study. Clinical Microbiology and Infection, S1198743X21006388. https://doi.org/10.1016/j.cmi.2021.11.010

Eyre, D. W., Taylor, D., Purver, M., Chapman, D., Fowler, T., Pouwels, K., Walker, A. S., & Peto, T. E. (2021). The impact of SARS-CoV-2 vaccination on Alpha and Delta variant transmission [Preprint]. Infectious Diseases (except HIV/AIDS). https://doi.org/10.1101/2021.09.28.21264260

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920

'''Author:''' Laura Kundmueller

: OK, but, please, elaborate it a bit. How exactly should the simulation look like, kinds of agents, etc. And mainly: the sources of data, etc. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:00, 10 December 2021 (CET)
:: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:13, 17 December 2021 (CET)

== Simulation of genetic algorithm: Travelling Salesman Problem ==

'''Simulation'''

The topic of this simulation is an old graph problem, Travelling Salesman Problem. My approach would be based on genetic learning algorithm. A random map will be generated at the start. Salesman is travelling in a car with some gas. The gas is used as he travels, it can be recharged at gas stations but it costs money. The map contains some hills and flat roads, which have a different cost of gas when going through.

'''The goal is:'''
* to find the optimum path between the towns.

'''The parameters are:'''
* number of agents (travelling salesmen)
* gas in car
* money

* number of towns
* number of hills
* number of gas stations

'''Method'''
* NetLogo

'''Author:''' [[User:Mart13|Mart13]] ([[User talk:Mart13|talk]]) 09:57, 9 December 2021 (CET)

: Although this isn't a true agent-based simulation, we sometimes accept topics from artificial intelligence and other related fields. However, it is necessary to elaborate it in deep. How exactly the algorithm will work. What is the goal (not the goal of the agent, but the goal of this work)? Etc. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:03, 10 December 2021 (CET)

:
Algorithm:
* Basic genetic learning, gene is a path of an agent. My idea is that every patch of map has a different cost of going through. The agents must decide where to go and in which order.
* Fitness function is a score of an agent (the score of its path).
* I would like to implement mutation and crossover.

Additional parameters to those introduced earlier:
* number of population
* number of mutations
* number of crossovers

Goal of work:
* graphical interpretation of AI learning

[[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 11:49, 11 December 2021 (CET)
: '''Approved'''

== Optimizing the process of baking wedding sweets ==

'''Simulation'''
There is a wedding tradition in Czech Republic of baking wedding sweets and then handing them out to the guests of the weeding as a form of invitation. Process of baking usually takes whole day and several helpers in the kitchen are needed. Into paper baskets are usually packaged two types of sweets: several small ones with 3 different flavours and one so-called "rohový koláč". Which are then delivered by the bride to wedding guests. For the purpose of this simulation are process and needed ingredients simplified.

'''The goal is:'''
The goal is to optimize the number of helpers in the kitchen and find optimal amount of basic ingredients for specified number of guests.

'''Method:'''
Discrete simulation - SIMPROCESS

'''Entities:'''
* sweets
* paper baskets
* baking trays

'''Resources'''
* pastry-cooks
* bride
* flour
* sugar
* curd
* plum jam
* poppy seed filling

'''Process steps'''
* order for paper basket
* preparing sweets: small ones (3 different flavours), "rohove kolače" sweets (using all flavours)
* baking in the oven
* sugar coating
* packaging
* delivery

'''Data:'''
* https://www.svetsvateb.cz/2021/02/623262-svatebni-kolacky/
* https://megvkuchyni.cz/recepty/speciality/svatebni-special-jak-na-svatebni-kolacky/
* experience

'''Author:''' [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 10:16, 8 December 2021 (CET)

: OK, maybe. '''Approved'''. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:05, 10 December 2021 (CET)

== Carsharing company fleet optimization ==

'''Problem definition'''

Recently, carsharing becomes more and more popular in large cities. Short-term rental (from several minutes to 24 hours) of a car with possibility to drop it anywhere in the allowed area in the city attracts people who for some reasons do not want to use their own vehicles. However, it is not always convenient. If the fleet is relatively small, the probability that a car will be somewhere close by is also quite low. Cars also must be refueled or recharged sometimes by external staff, which would increase cost of the fleet maintenance with increasing of the fleet size.

'''Simulation'''

The proposing agent-based simulation will reproduce real situation with shared cars.
Two types of agents are planned:
* cars with different states (waiting, in rent, maintenance) and characteristics (mileage, fuel level)
* drivers - users of the service, who rent the cars and have their own behavior, including decision making on taking a car, driving style, and so on.
Some data for the model will be obtained as personal observations of two carsharing services operating in Prague, [https://anytimecar.cz/ Anytime] and [https://www.uniqway.cz/ Uniqway] (for example, number of available cars, which is visible in mobile applications). Another source of data would be statistics collected by other services abroad, for example, by operating in Russia service Yandex.Drive[https://www.statista.com/statistics/1224178/daily-use-duration-of-car-sharing-vehicles-in-moscow-by-vehicle-type/].

'''The goal is:'''
to find out optimal fleet size and structure and price policy to maximize revenue of a carsharing company.

'''Method:'''
agent-based simulation - NetLogo

'''Author:''' [[User:Shcs00|Sergei Shcherbinin (shcs00)]] ([[User talk:Shcs00|talk]]) 12:34, 8 December 2021 (CET)

: I like the idea. Please, do not hesitate to contact the companies directly for the data. They are sometimes willing to help if you require just general, non sensitive data. '''Approved'''. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:08, 10 December 2021 (CET)

== How long would it take to find love ==

'''Problem definition'''

There is such thing as Drake formula used to calculate a chanse to meet an alien https://en.wikipedia.org/wiki/Drake_equation
Also there is an article https://e-lub.net/annuals/why.htm - to sum up, it is Drakes formula used to calculate a chance to meet a love of your life in London

So What I want to do is to let user enter some parameters of what he/she thinks the perfect person is and based on the entered parameters we will see how long would it take to meet a perfect person in a city (say, Prague).

'''Simulation'''

Data about the city would be taken from official sources (might be Eurostat or some other data that is gathered by Czech statistical office)
User will enter all the parameters and then see when will he meet a person.
Of course for the sake of achievability some things will be simplified. For example I don't think I will actualy add millions of agents, Maybe hundreds would be enough.

'''The goal is:'''
To see how year passes after year and love is still not found:) Or maybe to understand that lowering the expectations could correspond to finding a love faster

'''Method:'''
agent-based simulation - NetLogo or AnyLogic

Author - Vaso Dzhinchvelashvili

: To be honest, althought it is a funny idea, the problem, in fact, is verification. You would deal with soft properties and attitudes, nothing what could be measured. Even the article wasn't apparently meant to be serious.
: Please, try finding something else. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 19:17, 17 December 2021 (CET)

<nowiki>UPD:
</nowiki>
== Sample size for a/b test ==
'''Problem definition'''

There are different approaches on how to calculate sample size for A/B test when measuring difference in means for treatment and control group:
1 Calculation through power of a test
2 Rule of thumb for power of a test: https://stats.stackexchange.com/questions/11131/sample-size-formula-for-an-f-test
3 some other formula https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2876926/
4 Formula derived from confidence interval https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_confidence_intervals/bs704_confidence_intervals5.html

And I think there should be more. They give different results

Another problem is in type 1, type 2 error: researcher cannot always detect the difference or can detect the difference where there is no difference.
More importantly, in formulas the type 2 error is not used at all, whereas obviously it should be included in the models somehow, maybe as one of the axioms...
so the goal of simmulation is to illustrate the relation of type 1, type 2 errors, sample size, difference of means and the fact of rejecting/accepting null hypothesis. It will be a tool, which gives the researcher the answer 'how many observations do you need to detect the difference in a test'.

'''Simulation'''

Simulate an A/B test: set distribution parameters for control and treatment groups, for different sample sizes. Calculate statistics (a;pha,betta, p-value) for the tests
Compare number of observations to the ones derived from formulas - this way the correct one could be picked.

Illustrate real chances of type 1 and type 2 errors.
Illustrate how enhancing the difference in means leads (or doesn't) to decrease of needed sample size/ rise of significance.

'''The goal is:'''
Pick the best method for calculation of sample size.
Have a clear understanding how to design a/b test, based on some inputs (expectations)

'''Method:'''
Python, monte-carlo
Author - Vaso Dzhinchvelashvili

: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 02:24, 28 December 2021 (CET)

== Scooter rental ==

'''Problem definition'''

Today in almost each big city you can find a huge number of scooter which can be rented. The main feature of scooter rental is that you can take it at one point in the city and return it to another when you finish your route.
If there is no scooter nearby, then obviously a potential client will not take it. In addition, people prefer to ride with friends, so the likelihood of renting a scooter is higher if there are several of them at one point.
The more scooters there are in the city, the more chances that people will use this service, however, scooter support also requires additional resources and extra costs.

'''Simulation'''

The goal of agent-based simulation is to define optimal number of scooters for specific city and its destribution within the city to maximize income of the company.
Two types of agents will be used:
*scooters with a list of states (rented, broken, available, available + near other scooters) and characteristics such as, for example, charge level.
*clients with different characteristics (for example, alone, with friend, maximal suitable distance to a scooter)

'''Method:'''
Agent-based simulation in NetLogo

'''Author:''' [[User:liudmila_kalashnikova| Liudmila Kalashnikova (liudmila_kalashnikova)]]

'''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:24, 17 December 2021 (CET)

== Is it possible to stop fake news? ==

Fake news is a global problem and it is getting worse every year. Is it possible to stop fake news? Fake news are made by people named "trolls". These trolls make fake news and send them to people. On the other side of the table are elfs, who defend people from fake news and send "fake news warnings". When people receive fake news, they have basicaly three choices: resend it as it is (basicaly become troll), resend it with "fake news warning" (become elf) or dont resend it. Can elfs beat trolls and defend the majority of system from fake news?

'''Possible parameters of simulation'''
* number of trolls at the start
* possibility to become a troll (the rest to 1 is posibility to become an elf)
* troll's number of resends
* elf's number of resends
* troll's possibility to resend
* elf's possibility to resend

* and possibly many more which I cant now think about.

'''Additional idea:'''

Elfs can change to Trolls (and vice versa) in time. It would depend on how many information from who did they receive. For example, if a troll gets after some time more warnings from elfs than fake news from trolls, he become an elf too.

'''Articles about fake news spread'''
* https://www.nytimes.com/2021/11/22/world/europe/belarus-migrants-facebook-fake-news.html
* https://semantic-visions.com/resource/defending-the-covid-19-vaccination-pipeline
* https://www.cits.ucsb.edu/fake-news/spread
* https://thenextweb.com/news/ai-isnt-going-to-stop-fake-news-syndication
* https://www.bbc.co.uk/bitesize/articles/z6kxxyc
* https://ischool.syr.edu/fake-news-why-people-believe-how-it-spreads-and-what-you-can-do-about-it/
* https://futurism.com/fake-news-study-spread

'''Goals of simulation are answers to these questions:'''
* How many trolls do you need to get to the majority of people and how many resends they should perform (in average)?
* How many elfs do we need if there is X number of trolls? And how many resends should elfs do?
* Where are the boundaries to these numbers?

'''TOOL used for the simulation'''
* NetLogo
* Agent based simulation

'''Author:''' [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 00:42, 13 December 2021 (CET)
'''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:22, 17 December 2021 (CET)

== Simulation 3D print==

The simulation would focus on printing a 3D model on a 3D printer. This model would be made of multiple materials, so a 3D printer would need a material storage and a switch between them. Each material needed a different temperature and time to print parts of the model, also each material has a different financial value. At the same time, there is a need to consider providing additional cooling that slows down the print to prevent it from warping. There is also a need to consider the influence of the environment, such as ambient temperature, wind or nozzle clogging.

'''Measurement parameters'''
* time of making
* production cost
* nozzle temperature
* time between transitions
* condition of materials
* etc

''' Goals'''
* Measure how long it takes for the model to print.
* What is its financial demands if we take into account the price of the material and the time of its printing.
* If additional cooling needs to be switched on.

'''Tool'''
* Simprocess or vensim - advice?

'''Author:''' Ondřej Pišl

:: This is good example for a calculation, but not for a simulation. In case of System Dynamics it should focus on one parameter and its dynamics - for example material consumption. What all parameters it is influenced by, including complexity of the printed model and also probability should also come in (random variables) - print failure, model collapse (and its relation to model complexity). The goal of the simulation then could be a simulation, that will tell the user most probable material consumption for the input given by the user. Modify the proposal in this way and then the Vensim could be the viable way. [[User:Oleg.Svatos|Oleg.Svatos]] ([[User talk:Oleg.Svatos|talk]]) 22:05, 19 December 2021 (CET)

Assignments WS 2021/2022

2022-02-01T21:33:45Z

Laura: /* Effects of COVID-19 vaccination on the spread of infection */

{{Ambox
| text = <div>
Please, put here your assignments. Do not forget to sign them. You can use <nowiki>~~~~</nowiki> (four tildas) for an automatic signature. Use Show preview in order to check the result before your final sumbition.
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Please, strive to formulate your assignment carefully. We expect an adequate effort to formulate the assignment as it is your semestral paper. Do not forget that your main goal is a research paper. It means your simulation model must generate the results that are specific, measurable and verifiable. Think twice how you will develop your model, which entities you will use, draw a model diagram, consider what you will measure. No sooner than when you have a good idea about the model, submit your assignment. And of course, read [[How to deal with the simulation assignment|How to deal with the simulation assignment]].
</div>
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{{Ambox
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Topics on gambling, cards, etc. are not welcome.
</div>
}}

{{Ambox
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| text = <div>
In order to avoid possible confusion, please, check if you have added '''approved''' in bold somewhere in our comment under your submission. If there is no '''approved''', it means the assignment was not approved yet.
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== Spread of covid19 in closed/open area markets ==

In winter 2021 in Czechia the christmas markets were banned due to another covid19 infection wave. On the other hand people are free to go into shopping malls. It would be interesting to use existing data about covid19 virus transmission in agent based simulation to see how many people get infected and in what speed depending on whether they are in a christmas (open) market, or in a shopping mall (closed). The main goal will be to see if the simulation would backup the decision that has been made about christmas markets.

Possible research papers that contain data about covid spreading
* [https://www.kaggle.com/guidant/covid19-evolution-transmission-spatialpatterns/ Kaggle notebook - Covid19, Evolution, Transmission, Spatial Patterns]
* [https://link.springer.com/article/10.1007/s10668-020-00884-x/ Research - Understanding COVID-19 transmission, health impacts and mitigation: timely social distancing is the key]

This simulation would be realised using NetLogo.

Summary:

'''WHAT will be simulated'''
* market place, which can be both open space or closed space.
* people with or without masks, who will walk from shop to shop, with some intention and some of them will be virus carriers
* virus, which will spread in places where people go through (depending on the closed/open area, the infection rates will differ)

'''GOAL of the simulation'''
* answer the question: "Where is the virus spread more significant? At the market place, or at the shopping mall?"

'''TOOL used for the simulation'''
* NetLogo
* Agent based simulation

'''Author:''' Angel Kostov, xkosa20

: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 11:58, 10 December 2021 (CET)

== Effects of COVID-19 vaccination on the spread of infection ==

'''Problem definition'''

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 50,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures are the vaccination and masks. Vaccination can reduce the risk of infection and the likelihood of transmissibility. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic.

'''Simulation'''

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model in order to simulate the scenario in a simplified form based on existing scientific data. In addition, current COVID-19 measures are considered.

Subjects of the simulation:

Environment:
* A village with around 1.000 inhabitants
* Simplified: The village is closed that is new people cannot come in and people of the village cannot go out

Agents:
* Vaccinated persons
* Unvaccinated persons
* Simulations with different vaccination rates to compare the different infection courses

Further measures:
* All people wear masks
* Nobody wears a mask

Start:
* Few agents are unknowingly infected (e.g. 0,1% of the inhabitants = 6 people)
* Movement of agents intended to reflect the daily behavior of people in real life in a simplified form.
* One agent can infect another agent with a certain probability if they are close to each other.
* The probability of an infection depends on the measure (vaccination / mask)

'''Goal:'''
* Identify the infection course with different vaccination rates and measures.
* Showing the importance of vaccination.

'''Method'''
* NetLogo

'''Possible data sources'''

https://www.rki.de/SharedDocs/FAQ/COVID-Impfen/FAQ_Liste_Wirksamkeit.html

Chia, P. Y., Xiang Ong, S. W., Chiew, C. J., Ang, L. W., Chavatte, J.-M., Mak, T.-M., Cui, L., Kalimuddin, S., Chia, W. N., Tan, C. W., Ann Chai, L. Y., Tan, S. Y., Zheng, S., Pin Lin, R. T., Wang, L., Leo, Y.-S., Lee, V. J., Lye, D. C., & Young, B. E. (2021). Virological and serological kinetics of SARS-CoV-2 Delta variant vaccine-breakthrough infections: A multi-center cohort study. Clinical Microbiology and Infection, S1198743X21006388. https://doi.org/10.1016/j.cmi.2021.11.010

Eyre, D. W., Taylor, D., Purver, M., Chapman, D., Fowler, T., Pouwels, K., Walker, A. S., & Peto, T. E. (2021). The impact of SARS-CoV-2 vaccination on Alpha and Delta variant transmission [Preprint]. Infectious Diseases (except HIV/AIDS). https://doi.org/10.1101/2021.09.28.21264260

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920

'''Author:''' Laura Kundmueller

: OK, but, please, elaborate it a bit. How exactly should the simulation look like, kinds of agents, etc. And mainly: the sources of data, etc. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:00, 10 December 2021 (CET)
:: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:13, 17 December 2021 (CET)

== Simulation of genetic algorithm: Travelling Salesman Problem ==

'''Simulation'''

The topic of this simulation is an old graph problem, Travelling Salesman Problem. My approach would be based on genetic learning algorithm. A random map will be generated at the start. Salesman is travelling in a car with some gas. The gas is used as he travels, it can be recharged at gas stations but it costs money. The map contains some hills and flat roads, which have a different cost of gas when going through.

'''The goal is:'''
* to find the optimum path between the towns.

'''The parameters are:'''
* number of agents (travelling salesmen)
* gas in car
* money

* number of towns
* number of hills
* number of gas stations

'''Method'''
* NetLogo

'''Author:''' [[User:Mart13|Mart13]] ([[User talk:Mart13|talk]]) 09:57, 9 December 2021 (CET)

: Although this isn't a true agent-based simulation, we sometimes accept topics from artificial intelligence and other related fields. However, it is necessary to elaborate it in deep. How exactly the algorithm will work. What is the goal (not the goal of the agent, but the goal of this work)? Etc. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:03, 10 December 2021 (CET)

:
Algorithm:
* Basic genetic learning, gene is a path of an agent. My idea is that every patch of map has a different cost of going through. The agents must decide where to go and in which order.
* Fitness function is a score of an agent (the score of its path).
* I would like to implement mutation and crossover.

Additional parameters to those introduced earlier:
* number of population
* number of mutations
* number of crossovers

Goal of work:
* graphical interpretation of AI learning

[[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 11:49, 11 December 2021 (CET)
: '''Approved'''

== Optimizing the process of baking wedding sweets ==

'''Simulation'''
There is a wedding tradition in Czech Republic of baking wedding sweets and then handing them out to the guests of the weeding as a form of invitation. Process of baking usually takes whole day and several helpers in the kitchen are needed. Into paper baskets are usually packaged two types of sweets: several small ones with 3 different flavours and one so-called "rohový koláč". Which are then delivered by the bride to wedding guests. For the purpose of this simulation are process and needed ingredients simplified.

'''The goal is:'''
The goal is to optimize the number of helpers in the kitchen and find optimal amount of basic ingredients for specified number of guests.

'''Method:'''
Discrete simulation - SIMPROCESS

'''Entities:'''
* sweets
* paper baskets
* baking trays

'''Resources'''
* pastry-cooks
* bride
* flour
* sugar
* curd
* plum jam
* poppy seed filling

'''Process steps'''
* order for paper basket
* preparing sweets: small ones (3 different flavours), "rohove kolače" sweets (using all flavours)
* baking in the oven
* sugar coating
* packaging
* delivery

'''Data:'''
* https://www.svetsvateb.cz/2021/02/623262-svatebni-kolacky/
* https://megvkuchyni.cz/recepty/speciality/svatebni-special-jak-na-svatebni-kolacky/
* experience

'''Author:''' [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 10:16, 8 December 2021 (CET)

: OK, maybe. '''Approved'''. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:05, 10 December 2021 (CET)

== Carsharing company fleet optimization ==

'''Problem definition'''

Recently, carsharing becomes more and more popular in large cities. Short-term rental (from several minutes to 24 hours) of a car with possibility to drop it anywhere in the allowed area in the city attracts people who for some reasons do not want to use their own vehicles. However, it is not always convenient. If the fleet is relatively small, the probability that a car will be somewhere close by is also quite low. Cars also must be refueled or recharged sometimes by external staff, which would increase cost of the fleet maintenance with increasing of the fleet size.

'''Simulation'''

The proposing agent-based simulation will reproduce real situation with shared cars.
Two types of agents are planned:
* cars with different states (waiting, in rent, maintenance) and characteristics (mileage, fuel level)
* drivers - users of the service, who rent the cars and have their own behavior, including decision making on taking a car, driving style, and so on.
Some data for the model will be obtained as personal observations of two carsharing services operating in Prague, [https://anytimecar.cz/ Anytime] and [https://www.uniqway.cz/ Uniqway] (for example, number of available cars, which is visible in mobile applications). Another source of data would be statistics collected by other services abroad, for example, by operating in Russia service Yandex.Drive[https://www.statista.com/statistics/1224178/daily-use-duration-of-car-sharing-vehicles-in-moscow-by-vehicle-type/].

'''The goal is:'''
to find out optimal fleet size and structure and price policy to maximize revenue of a carsharing company.

'''Method:'''
agent-based simulation - NetLogo

'''Author:''' [[User:Shcs00|Sergei Shcherbinin (shcs00)]] ([[User talk:Shcs00|talk]]) 12:34, 8 December 2021 (CET)

: I like the idea. Please, do not hesitate to contact the companies directly for the data. They are sometimes willing to help if you require just general, non sensitive data. '''Approved'''. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 12:08, 10 December 2021 (CET)

== How long would it take to find love ==

'''Problem definition'''

There is such thing as Drake formula used to calculate a chanse to meet an alien https://en.wikipedia.org/wiki/Drake_equation
Also there is an article https://e-lub.net/annuals/why.htm - to sum up, it is Drakes formula used to calculate a chance to meet a love of your life in London

So What I want to do is to let user enter some parameters of what he/she thinks the perfect person is and based on the entered parameters we will see how long would it take to meet a perfect person in a city (say, Prague).

'''Simulation'''

Data about the city would be taken from official sources (might be Eurostat or some other data that is gathered by Czech statistical office)
User will enter all the parameters and then see when will he meet a person.
Of course for the sake of achievability some things will be simplified. For example I don't think I will actualy add millions of agents, Maybe hundreds would be enough.

'''The goal is:'''
To see how year passes after year and love is still not found:) Or maybe to understand that lowering the expectations could correspond to finding a love faster

'''Method:'''
agent-based simulation - NetLogo or AnyLogic

Author - Vaso Dzhinchvelashvili

: To be honest, althought it is a funny idea, the problem, in fact, is verification. You would deal with soft properties and attitudes, nothing what could be measured. Even the article wasn't apparently meant to be serious.
: Please, try finding something else. [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 19:17, 17 December 2021 (CET)

<nowiki>UPD:
</nowiki>
== Sample size for a/b test ==
'''Problem definition'''

There are different approaches on how to calculate sample size for A/B test when measuring difference in means for treatment and control group:
1 Calculation through power of a test
2 Rule of thumb for power of a test: https://stats.stackexchange.com/questions/11131/sample-size-formula-for-an-f-test
3 some other formula https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2876926/
4 Formula derived from confidence interval https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_confidence_intervals/bs704_confidence_intervals5.html

And I think there should be more. They give different results

Another problem is in type 1, type 2 error: researcher cannot always detect the difference or can detect the difference where there is no difference.
More importantly, in formulas the type 2 error is not used at all, whereas obviously it should be included in the models somehow, maybe as one of the axioms...
so the goal of simmulation is to illustrate the relation of type 1, type 2 errors, sample size, difference of means and the fact of rejecting/accepting null hypothesis. It will be a tool, which gives the researcher the answer 'how many observations do you need to detect the difference in a test'.

'''Simulation'''

Simulate an A/B test: set distribution parameters for control and treatment groups, for different sample sizes. Calculate statistics (a;pha,betta, p-value) for the tests
Compare number of observations to the ones derived from formulas - this way the correct one could be picked.

Illustrate real chances of type 1 and type 2 errors.
Illustrate how enhancing the difference in means leads (or doesn't) to decrease of needed sample size/ rise of significance.

'''The goal is:'''
Pick the best method for calculation of sample size.
Have a clear understanding how to design a/b test, based on some inputs (expectations)

'''Method:'''
Python, monte-carlo
Author - Vaso Dzhinchvelashvili

: '''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 02:24, 28 December 2021 (CET)

== Scooter rental ==

'''Problem definition'''

Today in almost each big city you can find a huge number of scooter which can be rented. The main feature of scooter rental is that you can take it at one point in the city and return it to another when you finish your route.
If there is no scooter nearby, then obviously a potential client will not take it. In addition, people prefer to ride with friends, so the likelihood of renting a scooter is higher if there are several of them at one point.
The more scooters there are in the city, the more chances that people will use this service, however, scooter support also requires additional resources and extra costs.

'''Simulation'''

The goal of agent-based simulation is to define optimal number of scooters for specific city and its destribution within the city to maximize income of the company.
Two types of agents will be used:
*scooters with a list of states (rented, broken, available, available + near other scooters) and characteristics such as, for example, charge level.
*clients with different characteristics (for example, alone, with friend, maximal suitable distance to a scooter)

'''Method:'''
Agent-based simulation in NetLogo

'''Author:''' [[User:liudmila_kalashnikova| Liudmila Kalashnikova (liudmila_kalashnikova)]]

'''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:24, 17 December 2021 (CET)

== Is it possible to stop fake news? ==

Fake news is a global problem and it is getting worse every year. Is it possible to stop fake news? Fake news are made by people named "trolls". These trolls make fake news and send them to people. On the other side of the table are elfs, who defend people from fake news and send "fake news warnings". When people receive fake news, they have basicaly three choices: resend it as it is (basicaly become troll), resend it with "fake news warning" (become elf) or dont resend it. Can elfs beat trolls and defend the majority of system from fake news?

'''Possible parameters of simulation'''
* number of trolls at the start
* possibility to become a troll (the rest to 1 is posibility to become an elf)
* troll's number of resends
* elf's number of resends
* troll's possibility to resend
* elf's possibility to resend

* and possibly many more which I cant now think about.

'''Additional idea:'''

Elfs can change to Trolls (and vice versa) in time. It would depend on how many information from who did they receive. For example, if a troll gets after some time more warnings from elfs than fake news from trolls, he become an elf too.

'''Articles about fake news spread'''
* https://www.nytimes.com/2021/11/22/world/europe/belarus-migrants-facebook-fake-news.html
* https://semantic-visions.com/resource/defending-the-covid-19-vaccination-pipeline
* https://www.cits.ucsb.edu/fake-news/spread
* https://thenextweb.com/news/ai-isnt-going-to-stop-fake-news-syndication
* https://www.bbc.co.uk/bitesize/articles/z6kxxyc
* https://ischool.syr.edu/fake-news-why-people-believe-how-it-spreads-and-what-you-can-do-about-it/
* https://futurism.com/fake-news-study-spread

'''Goals of simulation are answers to these questions:'''
* How many trolls do you need to get to the majority of people and how many resends they should perform (in average)?
* How many elfs do we need if there is X number of trolls? And how many resends should elfs do?
* Where are the boundaries to these numbers?

'''TOOL used for the simulation'''
* NetLogo
* Agent based simulation

'''Author:''' [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 00:42, 13 December 2021 (CET)
'''Approved''' [[User:Tomáš|Tomáš]] ([[User talk:Tomáš|talk]]) 00:22, 17 December 2021 (CET)

== Simulation 3D print==

The simulation would focus on printing a 3D model on a 3D printer. This model would be made of multiple materials, so a 3D printer would need a material storage and a switch between them. Each material needed a different temperature and time to print parts of the model, also each material has a different financial value. At the same time, there is a need to consider providing additional cooling that slows down the print to prevent it from warping. There is also a need to consider the influence of the environment, such as ambient temperature, wind or nozzle clogging.

'''Measurement parameters'''
* time of making
* production cost
* nozzle temperature
* time between transitions
* condition of materials
* etc

''' Goals'''
* Measure how long it takes for the model to print.
* What is its financial demands if we take into account the price of the material and the time of its printing.
* If additional cooling needs to be switched on.

'''Tool'''
* Simprocess or vensim - advice?

'''Author:''' Ondřej Pišl

:: This is good example for a calculation, but not for a simulation. In case of System Dynamics it should focus on one parameter and its dynamics - for example material consumption. What all parameters it is influenced by, including complexity of the printed model and also probability should also come in (random variables) - print failure, model collapse (and its relation to model complexity). The goal of the simulation then could be a simulation, that will tell the user most probable material consumption for the input given by the user. Modify the proposal in this way and then the Vensim could be the viable way. [[User:Oleg.Svatos|Oleg.Svatos]] ([[User talk:Oleg.Svatos|talk]]) 22:05, 19 December 2021 (CET)

Covid-19-vaccination

2022-02-01T21:32:13Z

Laura: /* Conclusion */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although despite the high vaccination rate, over 90% still become infected during the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, concerning the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:25:50Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:25:41Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:24:04Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.

[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:22:59Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this simulation and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:22:03Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures masks, quarantine, and social distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:21:25Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% is assumed.
Subsequently, the individual measures Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:19:48Z

Laura: /* Results */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:19:31Z

Laura: /* Results */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:18:44Z

Laura: /* Vaccination rate: 100% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly through vaccinated people, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:14:41Z

Laura: /* Vaccination rate: 75% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals. In comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. The number of infections decreases throughout the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher (13.5% of the population). All of them were vaccinated. Also, there are only 3 deaths.

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:08:28Z

Laura: /* Vaccination rate: 50% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population is vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. The number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected people is similar to the previous simulation, flat with slight fluctuations throughout the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted.

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T21:03:44Z

Laura: /* Vaccination rate: 25% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with a 0% vaccination rate. There is a rapidly increasing number of infections at the beginning. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the number of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated, it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected. After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:47:49Z

Laura: /* Plots */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated, and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:47:12Z

Laura: /* Parameter */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 people are infected at the beginning. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader events. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since a 20% risk of infection is assumed for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:37:50Z

Laura: /* Model */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus. Thus, the virus can spread in the village. Some of the people are vaccinated. How many people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spread. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the shape "people". As mentioned above, a part of the population defined by the vaccination rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing. The parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so few people are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:27:32Z

Laura: /* Method */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

To show how COVID-19 vaccination affects the spread of the pandemic, an agent-based model is used. This method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents. Additionally, the method allows to simulate peoples' daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:22:15Z

Laura: /* Introduction and problem definition */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with a high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are various vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and show its importance. Since there are other measures, the simulation is also performed in combination with these.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T20:18:25Z

Laura: /* Introduction and problem definition */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the its importance. Since there are different other measures, the simulation is also carried out in combination with these.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T19:36:22Z

Laura: /* Essential model parameter */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting COVID-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T19:35:36Z

Laura: /* Model */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of COVID-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from COVID-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T19:34:01Z

Laura: /* Model */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the COVID-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the COVID-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T19:22:40Z

Laura: /* Simulation with further measures */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T19:13:14Z

Laura: /* Vaccination rate: 50% */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is a little lower compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:58:24Z

Laura: /* Model */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic, several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which are colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:38:35Z

Laura: /* Method */

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:11:45Z

Laura:

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Tool:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:10:56Z

Laura:

'''Title:''' Effects of COVID-19 vaccination on the spread of infection

'''Author:''' Laura Kundmüller

'''Method:''' Agent-based model

'''Program:''' NetLogo

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:02:49Z

Laura: /* Problem definition */

=Introduction and problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T18:01:06Z

Laura:

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way. The program NetLogo is used to perform the simulation.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

WS 2021/2022

2022-02-01T17:47:34Z

Laura:

Semestral papers from winter term 2021/2022. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2021/2022|assignment approved]]

[[Carsharing_Fleet_Optimization|Carsharing Fleet Optimization]], Sergei Shcherbinin [[User:Shcs00|shcs00]] 16:57, 20 December 2021 (CET)

[https://www.simulace.info/index.php/Optimizing_the_Process_of_Baking_Wedding_Sweets#Model_Source_Code Optimizing the Process of Baking Wedding Sweets], [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 15:25, 21 January 2022 (CET)

[https://www.simulace.info/index.php/A_b_test A/B test] By Vaso Dzhinchvelashvili

[https://www.simulace.info/index.php/Is_it_possible_to_stop_fake_news Is it possible to stop fake news], [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 14:41, 30 January 2022 (CET)

[https://www.simulace.info/index.php/Scooter_rental Scooter rental], Liudmila Kalashnikova, 31 January 2022

[[Covid-19-vaccination|COVID-19 vaccination]], Laura Kundmüller, 1 February 2022

WS 2021/2022

2022-02-01T17:43:58Z

Laura:

WS 2021/2022

2022-02-01T17:39:26Z

Laura:

WS 2021/2022

2022-02-01T17:32:04Z

Laura:

Semestral papers from winter term 2021/2022. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2021/2022|assignment approved]]

[[Carsharing_Fleet_Optimization|Carsharing Fleet Optimization]], Sergei Shcherbinin [[User:Shcs00|shcs00]] 16:57, 20 December 2021 (CET)

[https://www.simulace.info/index.php/Optimizing_the_Process_of_Baking_Wedding_Sweets#Model_Source_Code Optimizing the Process of Baking Wedding Sweets], [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 15:25, 21 January 2022 (CET)

[https://www.simulace.info/index.php/A_b_test A/B test] By Vaso Dzhinchvelashvili

[https://www.simulace.info/index.php/Is_it_possible_to_stop_fake_news Is it possible to stop fake news], [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 14:41, 30 January 2022 (CET)

[https://www.simulace.info/index.php/Scooter_rental Scooter rental], Liudmila Kalashnikova, 31 January 2022

[[Effects of COVID-19 vaccination on the spread of infection]], Laura Kundmüller, 1 February 2022

WS 2021/2022

2022-02-01T17:31:03Z

Laura:

Semestral papers from winter term 2021/2022. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2021/2022|assignment approved]]

[[Carsharing_Fleet_Optimization|Carsharing Fleet Optimization]], Sergei Shcherbinin [[User:Shcs00|shcs00]] 16:57, 20 December 2021 (CET)

[https://www.simulace.info/index.php/Optimizing_the_Process_of_Baking_Wedding_Sweets#Model_Source_Code Optimizing the Process of Baking Wedding Sweets], [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 15:25, 21 January 2022 (CET)

[https://www.simulace.info/index.php/A_b_test A/B test] By Vaso Dzhinchvelashvili

[https://www.simulace.info/index.php/Is_it_possible_to_stop_fake_news Is it possible to stop fake news], [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 14:41, 30 January 2022 (CET)

[https://www.simulace.info/index.php/Scooter_rental Scooter rental], Liudmila Kalashnikova, 31 January 2022

[[Covid-19-vaccination]] [https://www.simulace.info/index.php/Covid-19-vaccination], Laura Kundmüller, 31 January 2022

WS 2021/2022

2022-02-01T17:16:35Z

Laura:

WS 2021/2022

2022-02-01T17:15:27Z

Laura:

Semestral papers from winter term 2021/2022. Please, put here links to the pages with your paper. First you need to have your [[Assignments WS 2021/2022|assignment approved]]

[[Carsharing_Fleet_Optimization|Carsharing Fleet Optimization]], Sergei Shcherbinin [[User:Shcs00|shcs00]] 16:57, 20 December 2021 (CET)

[https://www.simulace.info/index.php/Optimizing_the_Process_of_Baking_Wedding_Sweets#Model_Source_Code Optimizing the Process of Baking Wedding Sweets], [[User:Cerm18|Michaela Červinková (cerm18)]] ([[User talk:Cerm18|talk]]) 15:25, 21 January 2022 (CET)

[https://www.simulace.info/index.php/A_b_test A/B test] By Vaso Dzhinchvelashvili

[https://www.simulace.info/index.php/Is_it_possible_to_stop_fake_news Is it possible to stop fake news], [[User:Mart13|Tomáš Martínek (mart13)]] ([[User talk:Mart13|talk]]) 14:41, 30 January 2022 (CET)

[https://www.simulace.info/index.php/Scooter_rental Scooter rental], Liudmila Kalashnikova, 31 January 2022

[https://www.simulace.info/index.php/Covid-19-vaccination], Laura Kundmüller, 2 February 2022

Covid-19-vaccination

2022-02-01T17:11:18Z

Laura: /* Parameter */

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [7]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [4]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [3]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [5]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [6]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T17:10:32Z

Laura: /* Problem definition */

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.[2]

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [6]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [3]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [2]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [4]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [5]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T17:10:01Z

Laura: /* Sources */

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [6]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [3]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [2]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [4]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [5]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [https://doi.org/10.1016/j.scs.2020.102390]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T17:08:50Z

Laura: /* Sources */

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [6]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [3]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [2]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [4]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [5]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [https://doi.org/10.7326/M20-3012]

Sun, C., & Zhai, Z. (2020). The efficacy of social distance and ventilation effectiveness in preventing COVID-19 transmission. Sustainable Cities and Society, 62, 102390. [6]

=NetLogo File=

[[Media:COVID-19_vacccination_impact.nlogo]]

Covid-19-vaccination

2022-02-01T17:07:07Z

Laura: /* Sources */

=Problem definition=

Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than 160,000 new infections are currently reported every day. To reduce the infection rate, a wide variety of measures have been implemented. One of these measures is the vaccination. Vaccination can reduce the risk of infection. A simulation is conducted to vividly identify the extent to which vaccination could contain the pandemic. Because not every person wants to be vaccinated, there are different vaccination rates in different regions.
The goal of the simulation is to identify the impact of the vaccination on the spread of COVID-19 infections with different vaccination rates and showing the importance of vaccination. Since there are different other measures, the simulation is also carried out in combination with other measures.

=Method=

The purpose of the simulation is to show how COVID-19 vaccination affects the spread of the pandemic. I will use an agent-based model, this method enables to reflect the real scenario at the best. Thereby people can be represented by autonomous agents and it is possible to simulate their daily behavior and thus the spread of the virus in a simplified way.

=Model=
The model shows people in a village. The inhabitants move freely in their village, reflecting their everyday behavior. To simulate the spread of Covid-19, a certain proportion of the population is already unknowingly infected at the beginning of the simulation. As the simulation progresses, they infect their fellow villagers with the virus, allowing it to spread throughout the village. Some of the people are vaccinated. How many among the people are vaccinated defines the particular vaccination rate. Vaccination in this model reduces the risk of infection of a person. To analyze the impact of vaccination on the spread of the pandemic. Several simulations with different vaccination rates are run and their results are compared.

===Environment===
The model represents an exemplary village in Germany with 1605 inhabitants where the Covid-19 virus is spreaded. For simplification, the village is closed that is new people cannot come in and people of the village cannot go out.

===Agents===
The people are represented by agents, which have colored blue and have the "people" shape. As mentioned above, a part of the population defined by the vaccination-rate is vaccinated. This property is randomly assigned to the inhabitants. Vaccinated individuals are marked in green. For simplicity, the simulations do not distinguish between different vaccination statuses (1st, 2nd, or 3rd vaccination). In the following, the vaccination rate and the corresponding numbers of the parameters refer to a "complete" vaccination status, i.e. citizens have already received 2 vaccinations.

===Movement===
People move in a certain radius ''[mobility]'' randomly in their environment, this reflects in a very simplified way the behavior of people and their encounters with other people in everyday life, for example, people's way to work, shopping, going to restaurants, etc.

===Spread of infection===
The initially infected people (marked red or orange in the model) ''[infected-people]'' can infect other healthy people with a defined probability if they are in their vicinity. The probability of being infected, i.e. the risk of infection of a person depends on whether he is vaccinated ''[infection-risk-vac]'' or unvaccinated ''[infection-risk-unvac]''. The risk of infection is significantly lower for vaccinated individuals, so the likelihood of being infected by a person is lower. Once the person is infected, their color changes. Infected vaccinated persons turn orange and infected unvaccinated persons turn red. For simplification, we do not take into account the incubation period or the time spent with an infected person. In addition, other measures can be set, which are intended to protect against the virus by reducing the risk of infection. [[File:Dashboard.png|600px|thumb|right|Interface of the model]]
These are wearing a mask and social distancing, these parameters are explained in more detail below. If a person is infected, he or she goes into quarantine with a certain probability. This is simulated in the model in a simplified way, in that the person still moves freely, but can infect other people with a lower probability ''[quarantine-rate]''. After the recovery time ''[recovery-time]'', the previously infected person is immune (indicated by the color gray) to the virus ''[immunity-possibility]''. This means that they cannot be infected again. Since infected persons can also die from Covid-19 in the worst case, this is also taken into account in the mortality rate. The mortality rate ''[mortality-rate]'' indicates how many of the infected die. Dead persons are marked with an "x".

===End of the simulation===
The simulation ends when the Covid-19 virus is eradicated from the model village. This means that no more people are infected. This can happen when everyone is immune or so many or so few are infected that the risk of infection approaches 0.

=Parameter=

===Essential model parameter===

'''Infected people at the beginning''' [infected-people]: An infection rate of 2% is assumed at baseline, corresponding to 35 infected people at baseline. [6]

'''Vaccination rate''' [vaccination-rate]: In the following simulation, the spread of the virus is compared with vaccination rates of 0%, 25%, 50%, 75%, 90% and 100%.

'''Risk of infection of unvaccinated people''' [infection-risk-unvac]: In reality, different infection risks are found depending on external circumstances. For simplicity, this model assumes that all persons meet indoors and are not at any super-spreader event. Thus, the risk of infection for unvaccinated persons is about 20%. [3]

'''Risk of infection of vaccinated people''' [infection-risk-vac]: The probability of contracting Covid-19 is up to 90% lower for vaccinated persons. Since we assume a 20% risk of infection for unvaccinated persons, the risk of infection for vaccinated persons equals 2%. [2]

'''Mobility''' [mobility]: reflects the range of motion of the agents in their environment. In the simulations, the largest possible (10) is assumed.

'''Recovery time''' [recovery-time]: determines the time it takes an infected person to recover from the disease and become immune. In the simulation, a rate of 0.15 is assumed. The lower the value, the longer the recovery time.

'''Possibility of immunity''' [immunity-possibilty]: determines whether people are immunized after infection and can no longer become infected.

'''Mortality rate''' [mortality-rate]: defines the percentage of infected people who will die from the virus. For simplicity, we assume that the mortality rate is the same for both vaccinated and unvaccinated people, regardless of external and personal circumstances. The mortality rate is 0.6%. [4]

===Further measures for reducing the risk of infection===

'''Masks''' [masks]: Wearing a mask can reduce the risk of infection to 0.4% in the best case. In the simulations, the best case is assumed, i.e., a fresh mask is worn properly. In the initial simulations no one wears a mask since this additional measure should not bias the analysis around the effectiveness of vaccination. [1]

'''Social Distancing''' [social-distancing]: describes the number of people who engage in social distancing at the beginning of the simulation. This is reflected in a simplified model by the fact that the persons do not move. Since there is currently no lockdown, we do not assume this for the time being in the simulations and set social distancing to 0. In addition, this should not distort the analysis of the effectiveness of the vaccination.

'''Quarantine''' [quarantine-rate]: determines the probability that an infected person cannot infect another person, because he/she is in quarantine. Even in reality, not every person is in quarantine because, for example, they do not show any symptoms and thus no suspicion of COVID-19 arises that could lead to quarantine measures. It is estimated that approximately 40% of those infected people do not quarantine. Thus, the quarantine rate is 60% in the simulations. The duration of quarantine corresponds to the recovery time. [5]

=Output=

===Plots===

'''Relative results''' shows the proportion of infected (in total), infected vaccinated and infected unvaccinated people in the population over time.

'''Infection rate of unvaccinated people''' shows the proportion of infected persons among the unvaccinated people over time.

'''Infection rate of vaccinated people''' shows the proportion of infected persons among the vaccinated people over time.

=Results=

===Vaccination rate: 0%===

[[File:Results vaccination-rate-0%.png|500px|thumb|right|Spread of COVID-19 infection at 0% vaccination rate]]
First, the spread of the virus was simulated with a vaccination rate of 0%. This means that none of the persons are vaccinated. The number of infected and unvaccinated persons is thus equal to the total number of infected persons. In this case, the infection numbers immediately rise very sharply and have a peek right at the beginning. At which over 15% of the population is infected. Subsequently, the numbers of infected persons decrease due to their subsequent immunization. A total of 1603 people (approx. 99.8%) become infected with the virus until it is eradicated. In the process, 5 people die. After 166 ticks, the virus is eradicated because enough people are immune.

===Vaccination rate: 25%===

[[File:Results vaccination-rate-25%.png|500px|thumb|right|Spread of COVID-19 infection at 25% vaccination rate]]
In this simulation, 25% of the population is vaccinated, which corresponds to 401 people. The curve of the total number of infections is similar to the previous one with 0% vaccination rate at the beginning a rapidly increasing number of infections. The virus spreads rapidly through the population. Here, the maximum number of infections is 181 people (11.2% of the population). The curve of infected unvaccinated persons is approaching the curve of total infection numbers, while the numbers of infected vaccinated persons is constantly low. The proportion of infected vaccinated persons to vaccinated persons is also significantly lower (with an infection rate of 2%) and flat with little fluctuation compared to the unvaccinated. Only 70% of the vaccinated are infected in the pandemic. For the unvaccinated it is 100%. In total, about 92.5% of the population is infected in the pandemic, which is eradicated after 304 ticks. Thus it lasts longer than before. At the beginning of the pandemic, mainly unvaccinated persons become infected (peek at the beginning with steadily decreasing numbers afterwards). After immunization, the proportion of vaccinated persons among those infected is correspondingly higher.

===Vaccination rate: 50%===

[[File:Results vaccination-rate-50%.png|500px|thumb|right|Spread of COVID-19 infection at 50% vaccination rate]]
In this simulation, half of the population was vaccinated. The results are similar to those of the previous vaccination rate. At the beginning of the pandemic, high infection rates (peek at 136 infected persons), where the majority of infected persons are unvaccinated. The proportion of unvaccinated is somewhat low compared to the simulation before. The numbers of infected drop slowly thereafter. Although the number of vaccinated who became infected increased, this is because the total number of vaccinated also increased. The infection rate of vaccinated infected is similarly flat with slight fluctuations over the course of the pandemic. In this scenario, a total of 90% become infected, slightly less than if the vaccination rate was 25%. Again, all of the unvaccinated become infected and 80% of the vaccinated become infected. The eradication of the pandemic takes longer here (421 ticks). Here 7 deaths are counted

===Vaccination rate: 75%===

[[File:Results vaccination-rate-75%.png|500px|thumb|right|Spread of COVID-19 infection at 75% vaccination rate]]
In this scenario, the increase in the number of infections at the beginning of the pandemic is significantly lower. The maximum number of infected persons is only 90 (5.6% of the population). This number is made up of both vaccinated and unvaccinated individuals; in comparison, the proportion of unvaccinated individuals was dominant at lower vaccination rates. Infection rates decrease over the course of the pandemic. Infection rates are similar among the unvaccinated and vaccinated, as in previous scenarios. The time (419 ticks) to eradication is similar to that for the 50% vaccination rate. However, the number of people who were not infected is higher - 86.5% were not, all of whom were vaccinated. Also the death rate here is only 3

===Vaccination rate: 90%===

[[File:Results vaccination-rate-90%.png|500px|thumb|right|Spread of COVID-19 infection at 90% vaccination rate]]
In this scenario, 90% of people are vaccinated. The curve of the total infection numbers is much flatter and approaches the curve of the vaccinated infected. The peek at the beginning of the pandemic is much lower. Here, the maximum number of persons infected at the same time is only 54 (3.4% of the population). Since most people are vaccinated, the number of vaccinated infected people is also significantly higher. However, the number of infected persons among the vaccinated and unvaccinated behaves similarly as in the scenarios before. The number of concurrently infected individuals is lower, but the duration of the pandemic to eradication is significantly longer (643 ticks). In addition, the total number of infected persons is higher, as in the scenario before. Here, 93% became infected during the pandemic and 7 people died from the virus.

===Vaccination rate: 100%===

[[File:Results vaccinationn-rate-100%.png|500px|thumb|right|Spread of COVID-19 infection at 100% vaccination rate]]
The results are similar to the simulation with a vaccination rate of 90%, except that here the total number of infected persons corresponds to the infection rate of the vaccinated persons. If we compare the number of infected people over time with the number of infected people with a low vaccination rate, we see that the course is much flatter and that there is no steep increase in the number of infected people at the beginning of the pandemic. This means that the virus spreads more slowly, but this leads to a longer-lasting pandemic.

===Simulation with further measures===
For the simulations with measures, a vaccination rate of 50% was assumed.
Subsequently, the individual measures: Masks, Quarantine and Social Distancing were applied. The results show that the measures can again reduce the infection rates. Especially the mask is an effective measure (plot 1). This prevents a steep rise in infections and thus significantly reduces infections. This also shortened the duration of the pandemic. However, it should be taken into account here that the simulation assumed the best case scenario. Another measure that works is quarantine. An increase in the quarantine rate leads to a flatter course of the infection curve (plot 2). Social distancing, on the other hand, reduces infection rates only slightly in this mode and is thus the weakest of all measures (plot 3). The measures are equally successful for both vaccinated and unvaccinated. The vaccination status does not influence the effect of the measures.
[[File:Measures.png|900px|thumb|center|Spread of COVID-19 infection with different measures]]

=Conclusion=

As expected, the results show that vaccination against COVID-19 has a positive impact on the spread of the virus. Similarly, the different vaccination rates affect the pandemic in different ways. Although vaccination prolongs the duration of the pandemic, fewer people become infected with the virus during it, even at low vaccination rates. Among the uninfected at the end of the pandemic, there are only vaccinated people. Conversely, this means that unvaccinated individuals have a 100% probability of becoming infected with COVID-19 during the pandemic.
The infection rate among vaccinated individuals and unvaccinated individuals follows a similar trajectory regardless of vaccination coverage. The trajectory of total infection rates approaches either the infection rate among vaccinated individuals or unvaccinated individuals, depending on the vaccination rate. If the vaccination rate is low, the COVID-19 virus spreads rapidly at the beginning of the pandemic, meaning that infection numbers increase rapidly and decrease as the pandemic progresses. The higher the vaccination rate, the slower the virus spreads, resulting in a flat curve of infection numbers. As a result, the number of infections is significantly lower and relatively constant at the beginning. Although, despite the high vaccination rate, over 90% still become infected during the course of the pandemic, vaccination avoids a rapid spread of the virus and thus extremely high infection numbers. This can be helpful, for example, with regard to the capacity of hospitals and intensive care units, as they do not have to care for a large number of people at once and thus reach their limits. In addition, other measures, such as masks or quarantine, help to further contain the virus. For example, the quarantine rate can be increased by regular COVID-19 testing. In addition, the risk of infection depends on the environment and COVID-19 variants. By adjusting the infection risks, the simulation can be run again with different infection probabilities and compared.

=Sources=
Bagheri, G., Thiede, B., Hejazi, B., Schlenczek, O., & Bodenschatz, E. (2021). An upper bound on one-to-one exposure to infectious human respiratory particles. Proceedings of the National Academy of Sciences, 118(49), e2110117118. [https://doi.org/10.1073/pnas.2110117118]

Coronavirus Disease 2019 (COVID-19) Daily Situation Report by the Robert Koch Institute. (2022). Robert Koch Institute. [https://www.rki.de/DE/Content/InfAZ/N/Neuartiges_Coronavirus/Situationsberichte/Feb_2022/2022-02-01-en.pdf?__blob=publicationFile]

Harder, T., Külper-Schiek, W., Reda, S., Treskova-Schwarzbach, M., Koch, J., Vygen-Bonnet, S., & Wichmann, O. (2021). Effectiveness of COVID-19 vaccines against SARS-CoV-2 infection with the Delta (B.1.617.2) variant: Second interim results of a living systematic review and meta-analysis, 1 January to 25 August 2021. Eurosurveillance, 26(41). [https://doi.org/10.2807/1560-7917.ES.2021.26.41.2100920]

Lelieveld, J., Helleis, F., Borrmann, S., Cheng, Y., Drewnick, F., Haug, G., Klimach, T., Sciare, J., Su, H., & Pöschl, U. (2020). Model Calculations of Aerosol Transmission and Infection Risk of COVID-19 in Indoor Environments. International Journal of Environmental Research and Public Health, 17(21), 8114. [https://doi.org/10.3390/ijerph17218114]

Meyerowitz-Katz, G., & Merone, L. (2020). A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates. International Journal of Infectious Diseases, 101, 138–148. [https://doi.org/10.1016/j.ijid.2020.09.1464]

Oran, D. P., & Topol, E. J. (2020). Prevalence of Asymptomatic SARS-CoV-2 Infection: A Narrative Review. Annals of Internal Medicine, 173(5), 362–367. [5]

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