## 3 Epidemiological Model and Evolutionary Game Theory
### Evolutionary Game Theory
#### 3.2.2 Fermi Rule  
Following the usual evolutionary game dynamics, whether agent \(i\) changes to agent \(j\)'s strategy depends on the payoffs of both [17]. To express the probability that an agent switches its strategy, we employ the Fermi rule in this study. The Fermi rule expresses the probability \(\Theta\big{(}\pi_{i},\pi_{j}\big{)}\) of agent \(i\) changing to agent \(j\)'s strategy as in Equation (4), where \(\pi_{i}\) and \(\pi_{j}\) are the payoffs of agents \(i\) and \(j\), and \(k\) can measure the degree of noise.  
\[\Theta\big{(}\pi_{i},\pi_{j}\big{)}=\frac{1}{1+e^{-(\pi_{j}-\pi_{i})/k}} \tag{4}\]  
Mathematical Model Expressing the Relationship between Human Behavior and Infectious Disease Outbreaks  
[PERSON] et al. [17] used evolutionary game theory to represent human behavior in infectious disease models. In their study, agents choose whether to voluntarily isolate themselves to reduce the risk of infection or to behave as in a normal situation. These actions are denoted as strategy \(Q\) (quarantine) and strategy \(N\) (normal), respectively. In addition, it is assumed that the risk of infection is lower for strategy \(Q\) than for strategy \(N\). In their extended SIR model, susceptible individuals and infected individuals can choose one of the strategy. The model diagram is shown in Figure 5. The model can be expressed by equations in system (5).  
Figure 4: Invasion of mutations into the population.  
Figure 5: SIR model expressing the relationship between human behavior and infectious disease outbreaks.  
\[\left\{\begin{array}{l}S_{N}^{\prime}=-S_{N}\big{(}\beta_{N}I_{N}+\beta_{a}I_{Q} \big{)}+\tau\varphi_{S}\\ S_{Q}^{\prime}=-S_{Q}\big{(}\beta_{a}I_{N}+\beta_{Q}I_{Q}\big{)}-\tau\varphi_{S }\\ I_{N}^{\prime}=S_{N}\big{(}\beta_{N}I_{N}+\beta_{a}I_{Q}\big{)}-\gamma I_{n}+ \tau\varphi_{I}\\ I_{Q}^{\prime}=S_{Q}\big{(}\beta_{a}I_{N}+\beta_{Q}I_{Q}\big{)}-\gamma I_{Q}- \tau\varphi_{I}\\ R^{\prime}=\gamma\big{(}I_{N}+I_{Q}\big{)}\end{array}\right. \tag{5}\]  
In system (5), we denote the infection rate of \(S_{Q}\) as \(\beta_{Q}\) and that of \(S_{N}\) as \(\beta_{N}\). In this model, interaction between the two strategies may occur, such as when \(S_{Q}\) and \(I_{N}\) are infected by contact. The infection rate in such a case is denoted by \(\beta_{\alpha}\). The magnitude of each infection rate is \(\beta_{Q}<\beta_{\alpha}<\beta_{N}\).  
\(\varphi_{S}\), \(\varphi_{I}\) denote the strategy transition rate in each state as expressed in Equations (6) and (7). The parameter \(\tau\) represents how quickly the new strategy is adopted with respect to the timescale of the infection epidemic.  
\[\varphi_{S}=S_{Q}(S_{N}+I_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-S_{N}\big{(} S_{Q}+I_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{6}\]  
\[\varphi_{I}=I_{Q}(S_{N}+I_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-I_{N}\big{(} S_{Q}+I_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{7}\]  
where \(\pi_{Q},\ \pi_{N}\) denote the payoffs of each strategy. \(\Theta\big{(}\pi_{Q},\pi_{N}\big{)}\) represents the probability of changing from strategy \(Q\) to strategy \(N\), and \(\Theta\big{(}\pi_{N},\pi_{Q}\big{)}\) represents the probability of changing from strategy \(N\) to strategy \(Q\).