I.. Introduction {#sec1}
================

Epidemics of infectious disease have influenced the human civilization for many centuries. Epidemics have been documented as early as 430-427 BCE when the Athens epidemic killed as much as half of the population of ancient Athens [@ref1]. From early times, man has tried to understand the causes and remedies to infectious diseases, epidemics that took a huge toll on the human civilization. Only after the later part of the nineteenth century did vaccination become a successful tool in the fight against infectious disease.

In recent decades, the shift in climate and global temperature has raised the concern of increasing exposure to human's infectious diseases [@ref2]. The World Health Organization (WHO) has stated that the climate change and warming of the atmosphere are likely the causes of increase in transmission of many infectious diseases [@ref3]. Similarly, the *New England Journal of Medicine* has raised the concern that the climate change will cause a significant increase in infectious disease, especially vector-borne and waterborne diseases [@ref4]. This is due to the fact that insect vectors tend to be more active at higher temperatures. Moreover, there are also suspicions that the climate change has caused certain infectious diseases to spread into geographical areas that were previously unaffected [@ref5].

Another concern raised in a report conducted by WHO is that globalization and its effect on economic, environmental, demographic, and topological change of societies has caused people in today's world to be in an increased risk of confronting infectious disease [@ref6].

All these factors reemphasize the importance of controlling the spread of infectious diseases and motivated new approaches toward modeling infectious disease transmission and the behavioral response to it.

Many studies have considered social and behavioral dynamics in impeding the spread of infectious disease since these behaviors can influence the dynamics of the spread of disease in populations [@ref7], [@ref8]. This behavioral or health belief model has traditionally considered four main factors [@ref9]: 1) the perceived susceptibility of an individual or the probability that a person become infected; 2) the perceived severity or the cost of being infected; 3) the perceived barriers to behavior adoption or the cost of prevention; and 4) the perceived benefits or the benefits of adopting a behavior.

Moreover, in the classical modeling of infectious disease spread, a population is typically classified into several categories based on the individual's disease status. These categories are susceptible individuals (S), infected individuals (I), and recovered individuals which become immune. This model is known as the SIR model and is wildly used in studying infectious disease transmission [@ref8]. [Fig. 1](#fig1){ref-type="fig"} shows the schematic illustration of the SIR model, in which the susceptible individuals can become infected by the transmission rate and per contact probability of $\documentclass[12pt]{minimal}
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}{}$\beta $
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}{}$L$
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}{}$\beta $
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When considering a behavioral response to the epidemic such as vaccination, the SIR model adds one more state to the individuals who have used the prevention technique and the model is upgraded to the following form referred to as SIRV model (V for vaccinated) as shown in [Fig. 2](#fig2){ref-type="fig"}. In this model, states R and V are the absorbing states. Fig. 2.Schematic illustration of the SIRV model. The parameter $\documentclass[12pt]{minimal}
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The spread of infectious disease can be studied at the population level or based on a social interacting network. The network-based model can better describe the spread of infectious diseases since many diseases are transmitted with a direct or close contact between individuals. The network-based models have been widely studied in recent research works. For example, Fu *et al.* [@ref10] modeled the role of imitation behavior in a spatial structure and showed that the spatial structure is more sensitive to changes in the cost of vaccination. Fukuda *et al.* [@ref11] applied the same model but instead of one network of interactions, they defined two networks, in which one is the disease transmission network and the other one is the information transmission network.

In spite of vaccination being a proven and potent remedy to the spread of infectious diseases, in some regions of the U.S., there is a growing trend of parents opting out of vaccinating their children [@ref12]. Moreover, vaccines are not required in order for children to attend school in most states. Some national organizations such as "Vaccination Liberation" actively promote exemptions and encouraging physicians to refuse to vaccinate [@ref13].

When studying the reasons that deter parents from vaccinating their children, the most common reasons are fear from side effects or harm to the child [@ref12]. It seems that counting on the other vaccinated children to provide protection is not among the leading reasons. However, as a matter of fact, this behavior exemplifies the "tragedy of the commons" as the behavior that these parents are choosing to subscribe to.

In order to study the behavior of individuals in response to epidemic disease, several methods have been used. The evolutionary game theory is one of the techniques that can adequately describe the choice of individuals in population of interacting players with different payoffs [@ref14]. While prevention of infectious diseases is usually achieved by increasing the number of individuals who are using prevention techniques such as vaccinating or social distances, some individuals will choose not to get vaccinated being "protected" by the vaccinated population. The idea of being protected by the population of vaccinated individuals is not always effective if there are not enough vaccinated individuals or motivators to encourage individuals to get vaccinated. To illustrate, in 2015, California faced the worst Measles epidemic in the last decades because many parents declined to vaccinate their children due to vaccine side effects and by blindly relying on the other vaccinated ones to be safe [@ref15]. However, the idea of free protection has always been partnered with voluntary vaccination.

Considering this "free protection" allows modeling the spread of an infectious disease as a public goods games in which some free riders get the advantage from the contribution of other individuals in the group (i.e., vaccination). Modeling a public goods game of infection versus vaccination uses a payoff function, in which a vaccinated individual has a cost equal to cost of vaccination, an infected one pays the cost of infection, and the free riders are the ones deciding not to be vaccinated and when do not get infected pay nothing [@ref10], [@ref11], [@ref16].

In this paper, we have developed a public goods game-based model, in which the payoff of each individual is calculated based on its share of perceived cost of susceptibility, severity, and barriers for the whole group where he is a member, rather than his personal cost of vaccination or infection. The spatial game theory has been used to consider the spatial network-based structure of the population of players. In this model, in every time step, all susceptible players update their strategy for getting vaccine or not synchronously and then each of them might be infected based on its strategy and the probability of becoming infected. In order to update the strategy to get vaccine or not, different methods have been used. In the classical method of updating strategy, each player updates its strategy to the strategy of the player with the highest payoff in its neighborhood [@ref14]. In many other studies, the Fermi function is used as a probability function to change to the strategy of a neighbor [@ref10], [@ref16]--[@ref17][@ref18]. In this paper, a new approach in updating the strategy is used in which each susceptible player updates its strategy not just to the strategy of the neighbor with the highest payoff but also to the strategy of the number of neighbors with the highest payoffs. Thus, if there is someone vaccinated among the neighbors with the highest payoff, the player will change its strategy to get the vaccine. The sensitivity factor is a measure for the number of neighbors with the highest payoff that a player refers to in order to update its strategy. This sensitivity factor can be considered as a surrogate form of fear factor or the effect of media on making people aware of the severity of a disease. The concept of the effect of media and fear factor on individual's decision is studied in some research trying to show how it can affect the spread of infectious diseases [@ref19]--[@ref20][@ref21][@ref22].

In [Section II](#sec2){ref-type="sec"}, the public goods game and the proposed methodology in modeling spread of infectious disease is presented. [Section III](#sec3){ref-type="sec"} presents the result of using this methodology considering different parameters and discussing the behavior of individuals facing infectious disease outbreaks. [Section IV](#sec4){ref-type="sec"} provides a summary, discussion, and future work.

II.. Methodology {#sec2}
================

In this paper, individuals' cost functions and payoffs are determined based on their contribution to the group and the group's shared payoff which can be seen as a public goods game. This means that any payoff in a group will be distributed among the members of the group. The payoff function for a typical public goods game is as follows [@ref23]:$$\documentclass[12pt]{minimal}
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}{}\begin{align*} {\mathrm {payoff}}_{i}=\begin{cases} \dfrac {rN_{c}C}{N}, &\mathrm {if}~i~\mathrm {is~not~cooperative} \\[0.5pc] \dfrac {rN_{c}C}{N}-C, &\mathrm {if}~i~\mathrm {is~cooperative} \end{cases}\tag{1}\end{align*}
\end{document}$$ In which $\documentclass[12pt]{minimal}
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}{}$N_{c}$
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In our model, we have considered a person and all its immediate neighbors as a group; so, in a lattice, the center cell is the player in question, and its cooperation group is the eight cells adjacent to it, as shown in [Fig. 3](#fig3){ref-type="fig"}, where the yellow cells show the cooperation group for player $\documentclass[12pt]{minimal}
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The payoff for being a member of this group is defined as follows:$$\documentclass[12pt]{minimal}
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}{}\begin{align*}&\hspace {-0.5pc}{\mathrm {Payoff}}_{g}=-\left ({\left ({\frac {N_{I}}{N}\mathrm {\times }C_{I} }\right)+\left ({\frac {N_{V}}{N}\times C_{V} }\right)}\right. \\&\qquad\qquad\qquad{{\displaystyle {+\,\left.{+\left ({\frac {C_{I}\times \sum \limits _{j\mathrm {\in }S} P_{\inf j}}{N} }\right)\left ({\frac {N_{R}}{N}\mathrm {\times }C_{R} }\right) }\right)} }}\tag{2}\end{align*}
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}{}$N$
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Also, in this function, $\documentclass[12pt]{minimal}
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}{}\begin{equation*} P_{\inf j}=\frac {N_{Ij}}{N}\times \beta\tag{3}\end{equation*}
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The total payoff of a player is equal to the payoff that a player can earn from participating in a group minus the cost of being infected or getting the vaccine. The following formula shows the total payoff of a player:$$\documentclass[12pt]{minimal}
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}{}\begin{align*} {\mathrm {Payoff}}_{i}=\begin{cases} {\mathrm {Payoff}}_{g}-C_{V},&\mathrm {if}~i~\mathrm {is~vaccinated} \\ {\mathrm {Payoff}}_{g}-C_{I},&\text {if}~i~\mathrm {is~infected~or~recovered} \\ {\mathrm {Payoff}}_{g},&\mathrm {if}~i~\mathrm {is~susceptible~(free~rider)}. \end{cases} \tag{4}\end{align*}
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A.. Updating Rule {#sec2a}
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The updating rule is such that in every time step the top $\documentclass[12pt]{minimal}
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}{}$s$
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To illustrate, in the following lattice, consider the sensitivity factor equal to 3. The set of the first three neighbors with the highest payoff will be $\documentclass[12pt]{minimal}
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}{}$(-12.38,V)\}$
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III.. Experimental Results {#sec3}
==========================

In the following, the results of changing different factors that can affect the dynamics of an epidemic are studied. In the lattices, purple cells show susceptible, yellow cells show infected, red cells show recovered, and light blue cells show vaccinated individuals. The parameters which are used in the models are presented in [Table I](#table1){ref-type="table"}.TABLE IParameters of the ModelParameterMeaningValue$\documentclass[12pt]{minimal}
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A.. Effect of Changing Cost of Vaccination {#sec3a}
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One of the variables that can affect the spread of infectious disease and is under control of the health policy makers is the cost of vaccination. It is reasonable for the cost of vaccination to be less than the cost of infection and practically it should be much less than the cost of infection otherwise people will prefer not to pay the cost of vaccination. Thus, to examine the effect of $\documentclass[12pt]{minimal}
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In [Fig. 5](#fig5){ref-type="fig"}, the results of updating a lattice using different vaccination costs from 0 to a cost equal to the cost of infection is presented. Fig. 5.Number of free riders, infected, and vaccinated individuals for different vaccination costs.

[Fig. 5(a)](#fig5){ref-type="fig"} shows the number of free riders, infected, and vaccinated individuals under different vaccination costs when $\documentclass[12pt]{minimal}
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This also shows the benefit of increasing $\documentclass[12pt]{minimal}
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B.. Effect of Changing Transmission Rate {#sec3b}
----------------------------------------

The disease transmission rate is another variable in the model which depends on the characteristic of the disease. In order to examine the behavior of this model facing different diseases with different transmission rates, the number of finally vaccinated and infected individuals for different transmission rates is studied. In [Fig. 7](#fig7){ref-type="fig"}, the lattice on the left side shows the result of updating a $\documentclass[12pt]{minimal}
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}{}$\beta =0.9$
\end{document}$. It can be seen that more people get vaccinated when the transmission rate is high (especially in the lower rates of transmission) to save themselves and their community. This phenomenon can be better seen in [Fig. 8](#fig8){ref-type="fig"}, in which the number of vaccinated, infected, and free riders is plotted for different transmission rates from 0.1 to 1. [Fig. 8](#fig8){ref-type="fig"} shows that the public behavior is sensitive to the transmission rate at the lower end and is more stable at high rates of transmission. This sensitivity was experienced during the 2003 SARS episode in Hong Kong [@ref9]. Fig. 7.Distribution of players in the last lattice when $\documentclass[12pt]{minimal}
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Moreover, the epidemic tends to end sooner with a higher transmission rate, due to the faster response of individuals to the epidemic because of its high threat to the players (i.e., higher cost) as shown in [Fig. 9](#fig9){ref-type="fig"}. Fig. 9.Length of epidemic for different transmission rates.

C.. Effect of Changing Initially Vaccinated Population {#sec3c}
------------------------------------------------------

The number of initially vaccinated individuals in the population is another parameter that can be controlled by the health policy makers by forcing some individuals in the population to get vaccinated either using free-subsidy policy or partial-subsidy policy. The result of changing the percentage of initially vaccinated individuals shows that increasing the percentage of vaccinated individuals is beneficial to the society as long as the number of vaccinated is not beyond a certain percentage causing some individuals to get the benefits of living among vaccinated individuals in an immune society. As it is apparently shown in [Fig. 10](#fig10){ref-type="fig"}, the number of free riders will increase as the number of initially vaccinated increase, but it will start to decrease as the number of vaccinated increase more than 20%. These experiments are done in a $\documentclass[12pt]{minimal}
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Studying the behavior of population using different sensitivity factors and initial vaccination rate shows that the higher sensitivity will cause the number of free riders to grow faster. In [Fig. 11](#fig11){ref-type="fig"}, each line shows the number of free riders for each value of $\documentclass[12pt]{minimal}
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D.. Effect of Changing Initial Distribution of Players in the Lattice {#sec3d}
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The distribution of players in the lattice is also a factor that can be controlled to achieve better vaccine coverage. Although the health policy makers have control over the initially vaccinated individuals in society, they have no control over the number of initially infected ones. Thus, to model the effect of changing distribution of players, the initially infected individuals are randomly distributed in the lattice, but three scenarios are considered. In the first scenario, the vaccinated individuals are distributed randomly in the population \[[Fig. 12(a)](#fig12){ref-type="fig"}\], in the second one, the vaccinated individuals are evenly distributed in the lattice \[[Fig. 12(b)](#fig12){ref-type="fig"}\] and in the third scenario, the vaccinated individuals grouped into larger clusters (of nine individuals) that are evenly distributed in the lattice \[[Fig. 12(c)](#fig12){ref-type="fig"}\]. In all scenarios, the number of vaccinated and infected individuals is the same. These experiments are done in a $\documentclass[12pt]{minimal}
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[Table II](#table2){ref-type="table"} shows the result of each scenario. In all the three scenarios, the number of initially vaccinated is 144 which is approximately 5% of the population. We can see from [Table II](#table2){ref-type="table"} that if the individuals are evenly distributed, we will have few numbers of infection and vaccination and more free riders. This result is the effect of accessibility of vaccinated individuals in all cooperation groups; so, individuals can decide sooner to get vaccine confronting a disease epidemic and can also save others from being infected. Moreover, the result shows that group vaccination is less effective while there might be some vaccination not necessary for people who are not at risk of being infected, and also more initial vaccinations are needed to support this society. However, if we distribute groups such that they are more reachable for other individuals for imitating their behavior, the result will be improved.TABLE IIParameters of the ModelScenarioFree-RidersVaccinatedInfectedLength of Epidemic1 (a)5171347636982 (b)7591240501903 (c)23212051063100

E.. Effect of Sensitivity Factor {#sec3e}
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In order to examine the behavior of the model to changing sensitivity factor, the steady-state results of updating three different starting lattices under different sensitivity factors are presented in [Fig. 13](#fig13){ref-type="fig"}. [Fig. 13](#fig13){ref-type="fig"} shows that as the sensitivity factor increases the infection has a lower chance to affect other people while individuals respond sooner to epidemic. In these experiments, we have considered $\documentclass[12pt]{minimal}
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We have also examined the result of changing sensitivity factor when updating a lattice with distribution of players for three different vaccination costs ([Figs. 14](#fig14){ref-type="fig"}--[](#fig15){ref-type="fig"}[](#fig16){ref-type="fig"}[](#fig17){ref-type="fig"}[18](#fig18){ref-type="fig"}). In the diagrams, the blue lines show the experiments for $\documentclass[12pt]{minimal}
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[Fig. 14](#fig14){ref-type="fig"} illustrates the total cost of each lattice of players. This cost is calculated using the following formula which shows the total cost of being infected or vaccinated for the society:$$\documentclass[12pt]{minimal}
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We can see that as the number of free riders increases, and the ratio of infected-to-vaccinated individuals decreases, the total cost decreases. This means that the society could control the spread of disease with minimum number of vaccinated individuals. In [Fig. 14](#fig14){ref-type="fig"}, the total cost decreases for all three different vaccination costs as the sensitivity factor increases but it does not change much for $\documentclass[12pt]{minimal}
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[Fig. 15](#fig15){ref-type="fig"} shows the number of vaccinated players in the last lattice for each scenario. We can see that for sensitivity factors greater than 4, the number of vaccinated individuals does not change notably. However, the final number of vaccinated individuals is greater when the cost of vaccination is low.

[Fig. 16](#fig16){ref-type="fig"} shows the number of recovered individuals in the last lattice. In the last lattice, the epidemic season is over, and all the infected individuals have changed their state to "recovered." Thus, the number of recovered individuals is used to represent the number of individuals infected during each run. [Fig. 16](#fig16){ref-type="fig"} shows that the number of recovered individuals has the same behavior as the total cost; this is due to the high cost of infection relative to the cost of vaccination and the effect of the high number of infected individuals on the total cost. Also, [Fig. 16](#fig16){ref-type="fig"} demonstrates again that more people will be infected as the cost of vaccine increases.

[Fig. 17](#fig17){ref-type="fig"} shows the number of free riders in each experiment. As can be seen, this number increases as sensitivity increases but levels off after $\documentclass[12pt]{minimal}
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[Fig. 18](#fig18){ref-type="fig"} shows the duration of the epidemic season in each experiment. Based on this result, we can conclude that the length of an episode is mainly affected by the cost of vaccination; since cheaper vaccination results in an increase in the number of vaccinated individuals and cause the epidemic to end sooner. Similarly, if the sensitivity factor is increased, the duration of an epidemic is reduced. This is explained by the society that exhibits a higher sensitivity to the risk of being infected and is more proactive in vaccinating.

IV.. Conclusion {#sec4}
===============

In this paper, a public goods game-based model for modeling the behavior of population of players in response to an epidemic is illustrated. In this model, the payoff of each player is calculated using a function in which every cost of individuals in a $\documentclass[12pt]{minimal}
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However, increasing the sensitivity of individuals can result in more vaccination in the same situation. This behavior is very similar to the effect of fear of being infected in the real-world epidemics. Moreover, increasing the sensitivity can be beneficial for the society as individuals react to the epidemic sooner and decide faster to get vaccine in order to save themselves and their community, but increasing the sensitivity factor too much does not lead to an optimal cost for the society. The results show that increasing the sensitivity factor to more than 5 does reduce cost while the number of free riders dose not increase and the number of vaccinated and infected individuals does not change. This behavior is the result of dissuasive effect of selecting the strategy of the neighbors who have a higher payoff than the payoff of the player itself on the candidate list.

In addition, the model tests the effect of the infection transmission rate, and surprisingly, the epidemic length is lower when facing a disease with high transmission rate. This is explained by the fact that individuals respond sooner to the disease spread when there is a higher probability of being infected (represented as a strategy with a higher potential cost).

In this model, we also examined the effect of the number of initially vaccinated individuals on the epidemic which shows that mandatory vaccination can be beneficial when it does not force too many individuals to get the vaccine. Also, the distribution of vaccinated players in the lattice can affect the final result, when the players are distributed evenly in the lattice, more people are in contact with vaccinated individuals and this can cause them to get vaccine sooner when facing an epidemic and can result a better control of epidemic compared to the same number of vaccinated individuals who are randomly distributed.

![](solta-2896227.gif)

[Marzieh Soltanolkottabi](#contrib1){ref-type="contrib"} received the B.S. degree in industrial engineering and the M.S. degree in socio-economic systems engineering from the Isfahan University of Technology, Isfahan, Iran, in 2008 and 2011, respectively. She is currently pursuing the Ph.D. degree in industrial engineering with Kansas State University, Manhattan, KS, USA.

Her current research interests include the mathematical modeling and game theoretical approaches to healthcare and social complex systems.

![](benar-2896227.gif)

[David Ben-Arieh](#contrib2){ref-type="contrib"} received the B.S. degree in industrial engineering and the M.S. degree from Ben-Gurion University, Beersheba, Israel, in 1980 and 1982, respectively, and the Ph.D. degree from Purdue University, West Lafayette, IN, USA, in 1985.

In 1990, he joined the Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS, USA, as a Faculty Member, where he has been a Full Professor since 2003. Since 1985, he has been with the AT&T Bell Laboratories, Columbus, OH, USA, and later became a faculty member with Ben-Gurion University. His current research interests include healthcare operations and dynamics of human-based systems including propagation of diseases and behavioral changes due to health risks and predator prey dynamics.

Dr. Ben-Arieh is a member of the Institute of Industrial and Systems Engineers and the Society for Health Systems. In 2016, he became a fellow of the Institute of Industrial and Systems Engineers.

![](wu-2896227.gif)

[Chih-Hang John Wu](#contrib3){ref-type="contrib"} received the B.S. degree in industrial management from National Chung Cheng Kung University, Tainan, Taiwan, in 1984, and the M.S. and Ph.D. degrees in operations research from The Pennsylvania State University, State College, PA, USA, in 1993.

Since 1993, he has been a Faculty Member with the Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS, USA. His current research interests include human behaviors and their impacts the disease epidemics, mathematical modeling and simulation of human immune responses and sepsis, and the development of high-precision surface strains using noncontact measurements.

Dr. Wu is a member of the Institute for Operations Research and the Management Sciences, Institute of Industrial and Systems Engineers, the Society for Health Systems, and Society for Industrial and Applied Mathematics.
