**Abstract**

SARS-CoV-2 is a global lethal disease that is highly virulent. Its classification as a pandemic has left people distressed about when it might end or when all vaccines will be administered. There was also a concern with the reduction of people inclined to obtain immunity via vaccination, in the height of the pandemic. It is unknown when herd immunity will begin for SARS-CoV-2, but if there were to be an increase in the number of people depending on it, it can increase the length of the pandemic. From a simulation of the United States population, it was found that as you increase vaccinations against SARS-CoV-2 in a population, the length of the outbreak increases, and when the population surpasses an 80% vaccination rate, the outbreak period decreases at a significant rate. This was due to herd immunity suppressing the rate of infection so the virus cannot disseminate to susceptible individuals. The experiment’s data rejected the initial hypothesis that as vaccinations increase, the outbreak period will decrease and was proven with statistical analysis. This experiment was incapable of accurately identifying the vaccination rate of when herd immunity began to overpower virulence. Subsequent measurements would be necessary to find this information.

**Introduction**

Vaccinations are medical treatments that use weakened bacteria, viruses, or RNA to trigger an immune response to generate antibodies that can defend against the disease in the future. Vaccines against *influenza, rubella, poliomyelitis*, and *varicella-zoster* prevent global communities from contracting deadly diseases. This is why there has been a rush to develop and distribute vaccines for SARS-CoV-2, a coronavirus that causes acute respiratory illnesses, fever, and death. The virus uses the respiratory tract to spread through water droplets in sneezes, coughs, and conversations within six feet (1). SARS-CoV-2 is a variation of MERS-CoV and SARS-CoV, both coronavirus respiratory diseases that were seen in 2012 and 2002 (1). In March of 2020, the World Health Organization (WHO) declared this virus a pandemic leading to countries and businesses competing to manufacture a vaccine while citizens took shelter.

From May 2020 to October 2020 there was a 21% decrease of people willing to receive the SARS-CoV-2 vaccine (2). This decrease can be attributed to the rushed vaccine, safety concerns, and social and political ideas (3). Decreasing the willingness to get vaccinated can lead to more deaths, a longer outbreak period, and resurfacing of the virus in the future. A major reason people didn’t want to receive the vaccine was because of herd immunity, the dependence on immune and vaccinated individuals to stop the spread of infectious diseases. However, fundamental education of why individuals shouldn’t depend on herd immunity shows about a 7.3% increase in the number of people willing to be vaccinated. It is also demonstrated that there is a general increase in willingness to receive vaccinations for *influenza* and SARS-CoV-2 when stressing the prosocial effects and importance of immunization (4, 5). This is what the United States and many organizations began to do and by the end of the year, 2020 interest and willingness of inoculation increased by 10%, and disinterest decreased by 6% (3).

All viruses have different means of transportation, infection, and lethality. These variables are known as the properties of the virus and play a huge role in the severity of the illness. SARS-CoV-2 has an incubation period of 2-14 days with 95% of patients suffering symptoms before 12.5 days and on average symptoms appearing within 4-5 days. The infectious period, or the time an individual can infect others, is 10 days, and the mortality rate is about 3.4% as of June 2020 (6). Different infection and mortality rates result in organizations, such as the United States government, setting different vaccination goals for different diseases. The United States seeks to vaccinate 70% of the aggregate population yearly against influenza (2). Other diseases like rubella have around a 93% vaccination rate in schools to accomplish herd immunity (2). With numerous variables that affect the spread of disease, it is essential to interpret the effects of vaccinations and the decisions people are making in a pandemic.

This experiment will focus on the effects of vaccination rates in a population where SARS-CoV-2 is prevailing. Specifically, the relationship between vaccination rates of a population and the length of the outbreak. It is predicted that if you increase the number of people vaccinated in a population then, the length of an outbreak of SARS-CoV-2 will decrease. The hypothesis is that as you increase the number of vaccinated individuals you decrease the number of individuals capable of being infected and in turn reduce the number of individuals that can spread the virus. Determining the vaccination rate needed to reduce the time of the outbreak can help predict the duration of the pandemic. It will also be significant for future predictions of how SARS-CoV-2 might spread after vaccine rates begin to plateau and establish an equilibrium. Finding information like this will aid communities around the world in regards to a return to normalcy.

**Materials and Methods**

** **To properly conduct this experiment, a computer with internet was required as well as a place used to log quantitative data. Microsoft excel was used during this experiment to log data and calculate means, standard deviations, t-tests, and ANOVA. An internet connection was used to access the Virus Outbreak Simulator for the US (https://bioinformaticshome.com/online_software/virus-outbreak/US/index.html#).

Independent, dependent, and constant variables were identified and set before the experiment (Table 1). The independent variable is the vaccination rate of the simulation population. This is regulated by using a slide bar under the properties tab. The dependent variables are the length of the outbreak, and the population’s death, sick, carrier, and uninfected rates. The constants included keeping virulence at 50%, lethality at 5%, incubation period of 5 days, infection time of 9 days, no containment, and maintaining the virus as coronavirus.

Each vaccination condition was tested one at a time, with a total of ten trials per vaccination rate. To conduct a trial, the operator needed to make sure all constants were accurate. When checked, the simulation began by pressing the start button. The simulation was then let run to completion and the outbreak duration at the top of the screen was then logged into that trial. To initiate another trial, the operator needed to press the reset button and reassure all constants are the same before commencing the subsequent trial.

After all the trials were collected, the mean and standard deviation was calculated and graphed to test statistical significance between different vaccination rates in a population. The ANOVA test was conducted to determine this due to there being more than two conditions. The ANOVA test provided F-values and P-values that were then used to reject or accept the null hypothesis. If the F value is greater than the F-critical value then one can reject the null hypothesis. If the P-value is less than .05 then one can accept the alternative hypothesis.

**Results**

The sample population included 327 million uninfected individuals and 55 thousand infected individuals. The first trial conducted was a population with a 0% vaccination rate and produced an average outbreak time of 98 days (+/-9.741). This was the second shortest total outbreak time behind the population with a vaccination rate of 95% with an average time of 24.8 days (+/-13.290). A population with a vaccination rate of 80% had an average outbreak time of 248.8 days (+/-45.813) and was the longest of all vaccination rates. The closest average outbreak time was a vaccination population of 60% with 140.5 days (+/-22.950). When data was graphed, it revealed a significant decline in average outbreak duration after vaccination rates surpassed 80% (Fig. 1.).

All data was put through a single factor ANOVA test to compare the means (F-value = 100.903, F-critical = 2.386, P-value = 3.924E-26). Due to the F-value being greater than F-critical, T-tests were done to compare all vaccination populations and determine significance between each population. The calculated t-values increased as the range between vaccination rates increased. There was significance between populations: 0% and 20%, 0 and 40%, 20% and 40% (T-value: -0.401, P-two tailed: 0.694, T-value: -1.13, P-two tailed: 0.273). All other populations had statistically insignificant T-tests.

During each trial a ring pattern was observed (fig. 2), the center included green (immune) and black (deceased) while the outer-ring was led by purple (infected) followed by red (sick). As the vaccination rate increased in the population the pattern stayed very similar, however, the area of purple and red decreased and took longer to expand. The trials with a vaccination rate of 95% were the only trials to not experience a ring of purple and red and were the only trials to not have the purple and red encompass the entire simulation.

**Discussion**

It was predicted that as vaccinations increase in a population against SARS-CoV-2, the duration of the outbreak would decrease. The experiment conducted rejected this hypothesis as proven by the ANOVA test. Receiving an F-value much greater than F-critical meant that the null hypothesis was rejected and the extremely small P-value supported this statement. Instead, as vaccination rates increased, the duration of the outbreak increased until 80%, where there was a significant decline.

Observations of the cyclical pattern in which the virus spreads can explain the reasons for an increase in the duration of outbreaks and the rapid decline at 80%. After an 80% vaccination rate, the cyclical pattern begins to disappear because of fewer individuals being able to transmit the virus. The vaccinated and immune individuals establish a barrier that limits the spread of SARS-CoV-2, which represents herd immunity. Though this is represented in this experiment and works to some degree, the virus still possesses the ability to reach a non-immune person. If more people begin to not vaccinate and depend on herd immunity viruses will spread more easily.

Herd immunity is the cause of the prolonged duration of the outbreak as vaccinations approached 80% and a sharp decline after 80%. Vaccination rates were not at a high enough rate to prevent the spread of the virus completely, only limit who can be infected. This led to an increased time of infection between individuals and an increased total time of the outbreak. After 80%, there is a point at which herd immunity starts to overpower the infectivity of SARS-CoV-2 and results in the decline of the outbreak period. This point can be found by using a similar experiment in the future. Herd immunity is also the cause of the decrease in size of the outer ring pattern. As vaccination rates increase, the time between infected persons decreases. This decreases the rate at which the virus can travel resulting in a decreased outer ring width.

The data collected supports the idea of inoculation of the SARS-CoV-2 vaccine. If more people begin to depend on herd immunity and not get vaccinated it will result in a longer outbreak period. If another 21% decrease of willingness to be vaccinated were to happen again it would result in an outbreak period of twice the time based on the data collected from this simulation. For this reason, organizations and individuals need to spread fundamental information of the prosocial and individual benefits of vaccination because that 7.3% increase in aggregate willingness to be vaccinated can push the population past the 80% threshold.

Limitations with this experiment included being unable to change the area in which people live. Rural individuals come in contact with fewer people when compared to urban and city lifestyles. The simulation didn’t state what kind of physical and social environment it was conducting in. Another limitation to this experiment is it only applies to SARS-CoV-2. Different virulence and lethality rates will result in unknown outcomes; thus, this data cannot be used as a guide for Influenza or Rubella outbreak duration. One would need to conduct a new set of trials and statistical tests to determine the outbreak duration for another disease. Finally, this simulation did not account for means of travel like planes or boats that can introduce the virus to an area of people that are not yet sick or thought to be protected from the virus. For this reason, it is predicted that the vaccination rate might have to be higher than calculated using an experiment such as this one.

Subsequent tests would be needed to determine the significance of vaccination rates and the length of outbreaks in correlation to deaths. Collecting data such as this can help determine what the most beneficial way of administering vaccination can be, like waiting to do a mass number of people at once or slowly vaccinating people as they come in.

**References**

1. Y. R. Guo* et al.*, The origin, transmission and clinical therapies on coronavirus disease 2019 (COVID-19) outbreak – an update on the status. *Mil Med Res* **7**, 11 (2020).

2. J. R. Lewis, What Is Driving the Decline in People’s Willingness to Take the COVID-19 Vaccine in the United States? *JAMA Health Forum* **1**, e201393-e201393 (2020).

3. A. S. Kimberly H. Nguyen, Hilda Razzaghi, Walter Williams, Megan C. Lindley,, N. A. Cynthia Jorgensen, James A. Singleton, COVID-19 Vaccination Intent, Perceptions, and Reasons for Not Vaccinating Among Groups Prioritized for Early Vaccination — United States, September and December 2020. *Centers for Disease Control and Prevention* (2021).

4. H. Jung, D. Albarracin, Concerns for others increases the likelihood of vaccination against influenza and COVID-19 more in sparsely rather than densely populated areas. *Proc Natl Acad Sci U S A* **118** (2021).

5. J. Logan* et al.*, ‘What have you HEARD about the HERD?’ Does education about local influenza vaccination coverage and herd immunity affect willingness to vaccinate? *Vaccine* **36**, 4118-4125 (2018).

6. S. Fulzele* et al.*, COVID-19 Virulence in Aged Patients Might Be Impacted by the Host Cellular MicroRNAs Abundance/Profile. *Aging Dis* **11**, 509-522 (2020).