Difference between revisions of "Covid-19-vaccination"
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| − | + | '''Title:''' Effects of COVID-19 vaccination on the spread of infection | |
| − | Currently, there is a new wave of infection in the COVID-19 pandemic with high number of infections. In Germany, for example, more than | + | '''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= | =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= | =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=== | ===Environment=== | ||
| − | The model represents an exemplary village in Germany with 1605 inhabitants where the | + | 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=== | ===Agents=== | ||
| − | The people are represented by agents, which | + | 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=== | ===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. | + | 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=== | ===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. These are wearing a mask and social distancing | + | 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=== | ===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= | =Parameter= | ||
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| − | '''Infected people at the beginning''' | + | '''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=== | ===Further measures for reducing the risk of infection=== | ||
| Line 58: | Line 68: | ||
| − | '''Masks''' | + | '''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''' [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''' [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= | =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= | =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= | =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= | =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= | =NetLogo File= | ||
| + | |||
| + | [[Media:COVID-19_vacccination_impact.nlogo]] | ||
Latest revision as of 23:08, 1 February 2022
Title: Effects of COVID-19 vaccination on the spread of infection
Author: Laura Kundmüller
Method: Agent-based model
Tool: NetLogo
Contents
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.
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%
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%
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%
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%
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%
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%
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.
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.
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