key: cord-0747264-8ohzq3vq authors: Choi, Wongyeong; Shim, Eunha title: Optimal Strategies for Vaccination and Social Distancing in a Game-theoretic Epidemiological Model date: 2020-07-25 journal: J Theor Biol DOI: 10.1016/j.jtbi.2020.110422 sha: cdd6d346e93c499f540ab620445fab348c7f4b17 doc_id: 747264 cord_uid: 8ohzq3vq For various infectious diseases, vaccination has become a major intervention strategy. However, the importance of social distancing has recently been highlighted during the ongoing COVID-19 pandemic. In the absence of vaccination, or when vaccine efficacy is poor, social distancing may help to curb the spread of new virus strains. However, both vaccination and social distancing are associated with various costs. It is critical to consider these costs in addition to the benefits of these strategies when determining the optimal rates of application of control strategies. We developed a game-theoretic epidemiological model that considers vaccination and social distancing under the assumption that individuals pursue the maximization of payoffs. By using this model, we identified the individually optimal strategy based on the Nash strategy when both strategies are available and when only one strategy is available. Furthermore, we determined the relative costs of control strategies at which individuals preferentially adopt vaccination over social distancing (or vice versa). Infectious diseases represent an ongoing threat to global health, and their prevention can 37 significantly reduce economic and epidemiological burdens. Worldwide, an estimated 14.7 million 38 individuals are killed by infectious diseases each year, with the vast majority of death occurring in 39 developing regions (14.2 million out of 14.7 million) (Michaud, 2009 ). Specifically, the mortality 40 from infectious diseases is the highest in sub-Saharan Africa with 6.8 million annual deaths and South 41 Asia with 4.4 million annual deaths (Michaud, 2009) . It is estimated that over 400 million disability- Overall, human behavior plays an important role in the spread of infectious diseases. Therefore, Behavioral epidemiology is an inherently multidisciplinary field combining classical epidemiological 75 modeling with behavioral sciences, such as sociology, psychology, economics, and anthropology, to 76 understand the connection between human mechanics and infection mechanics (d'Onofrio and 77 Manfredi, 2020) . Frameworks for the computational modeling of behavioral epidemiology range from 78 classical models assuming homogeneously mixing (mean-field) populations to complex models that 79 account for behavioral feedback and population spatial/social structures. Many of these methods 80 originated from statistical physics models, such as lattice and network models (Wang et al., 2016) . The key issue in behavioral epidemiology is to understand the relationships between behavioral 82 changes and the dynamics of infectious diseases. For example, a past study quantified how increases 83 in population awareness can reduce disease transmission in well-mixed populations (Funk et al., 84 2010b). In another study on behavior epidemiology, a behavior-implicit susceptible-infectious-85 recovered (SIR) model with prevalence-dependent vaccination and prevalence-dependent contact 86 rates was developed (d' Onofrio and Manfredi, 2020) . This study presented the baseline perceived risk 87 conditions under minimal infection circulation at which the elimination of a disease can be achieved. Among the behavioral changes that can be used as strategies to limit the spread of infectious 89 diseases, we consider vaccination and social distancing, and incorporate these strategies into a 90 mathematical model of disease transmission. Specifically, a game-theoretic model was developed to 91 consider both the costs and benefits associated with disease intervention strategies to identify the 92 individually optimal strategy. By using game-theoretic models, one can not only examine the 93 transmission dynamics of infectious diseases, but also determine how individual decision-making is 94 affected by the perceived costs of actions and the resulting benefits. Given the definitions and assumptions presented above, the transmission dynamic model is 133 described by the following differential equations: We assume that the total population is asymptotically constant, meaning . Therefore, Here, we note that when and . Because , it follows that has a solution with 191 when . Figure distancing level that is required to achieve herd immunity is denoted as . where . To analyze the existence of the Nash vaccination strategy , we examine the potential real 277 roots of (12), which are denoted as and , where ( Figure S2) The Nash vaccination rate is presented in Figure 7 (a) as a function of the relative cost of 329 vaccination when the value of varies. One can seen that decreases with and has a 330 higher sensitivity to increasing costs when is greater. (16) To obtain the threshold value for the social distancing level that is required to achieve herd immunity 359 ( ), we set and solve for as follows: 372 In the presence of an endemic equilibrium, we define as the probability that a It should be noted that is a decreasing function of ( ) that 406 attains its maximum value when (Figure 8) . Therefore, never exceeds the 407 herd immunity threshold and equality is attained only when the relative cost of social distancing 408 is zero ( ). Furthermore, if , then the resulting optimal social distancing strategy is Similarly, the optimal social distancing strategy becomes a function of the vaccination rate . By substituting the expression for from (5) into (see (11) parameter values (Table 1) . -If and , then the black line (optimal level of social distancing) remains 444 above the gray line (optimal vaccination line). Therefore, social distancing is the dominant strategy 445 (Figure 9(a) ). -If and , then the gray line remains above the black line. Therefore, 447 vaccination is the dominant strategy (Figure 9(b) ). The cutoff values for the relative costs in the graph in Fig. 10 One limitation of our model is that it does not consider age heterogeneity in terms of disease 507 susceptibility or age-dependent mixing patterns. Additionally, our model assumes that the costs 508 associated with social distancing are limited to the personal costs, even though there are social costs 509 associated with a malfunctioning society (e.g., food supply and healthcare services breakdowns) when 510 social distancing is practiced by a relatively large fraction of a population over an extended period of 511 time. Furthermore, our analysis was conducted under the assumption that a system has already 512 reached an endemic state when individuals begin selecting protection strategies. In the real world, the 513 infection probability changes dynamically with disease prevalence and age-dependent susceptibility. For example, the optimal vaccination rates for all age groups are the highest at the beginning of a 515 seasonal influenza epidemic, and the optimal vaccination coverage differs between age groups (Shim, To study the existence of a Nash equilibrium , we solve (12), which has the following two How will country-548 based mitigation measures influence the course of the COVID-19 epidemic? 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Coronavirus disease (COVID-2019) situation reports  We propose a game-theoretic model for infectious disease transmission  Vaccination and social distancing are considered as control strategies  We identify individual optimal strategies by considering group behavior  We determine the threshold costs at which either vaccination or social distancing is preferred Wongyeong Choi: Methodology, Formal analysis, Visualization, Writing-Original draft preparation Writing-Reviewing and Editing Formal analysis, Visualization, Writing-Original draft Writing-Reviewing and Editing, Supervision, Funding acquisition ☒ The authors declare that they have no known competing financial interests or personal 669 relationships that could have appeared to influence the work ☐The authors declare the following financial interests/personal relationships which may 672 be considered as potential competing interests