## 4 Problem Description and Model Formulation
### Model Formulation
#### 4.2.1 SEIR Model with Incubation Period and Viral Mutation  
There is an incubation period before symptoms develop after infection with COVID-19. In addition, the number of cases in Japan shows that the scale of the infectious epidemic is gradually increasing. This is mainly due to the effect of virus mutation [30]. In this section, we propose the model to examine the effect of the incubation period of COVID-19 infection and the increase in the infection rate due to virus mutation on the number of infected persons. The model proposed in this section consists of four states: susceptible (\(S\)), infected (\(I\)), incubation period (\(E\)), and recovered (\(R\)). In states \(S\), \(E\), and \(I\), agents have two strategies: (1) voluntarily refraining from going out and staying at home (strategy \(Q\)), and (2) acting as they would under normal circumstances (strategy \(N\)). The former strategy have lower probability of infection than the latter. Let \(S_{Q}\) denote the agent that takes strategy \(Q\) in state \(S\) and \(S_{N}\) denote the agent that takes strategy \(N\). Similarly, in states \(E\) and \(I\), we denote by \(E_{Q}\), \(E_{N}\), \(I_{Q}\), and \(I_{N}\), respectively. A diagram of the model is shown in Figure 6.  
Here, the infection rate of \(S_{Q}\) is represented by \(\beta_{Q}\) and the infection rate of \(S_{N}\) is denoted by \(\beta_{N}\). In this model, it can be assumed that \(\beta_{Q}<\beta_{N}\) because the risk of infection is smaller when voluntary isolation is used [17]. In rare cases, the interaction between the two strategies may occur, such as when \(S_{Q}\) and \(I_{N}\) come into contact with each other. The infection rate in such a case is denoted by \(\beta_{a}\), where \(\beta_{Q}<\beta_{a}<\beta_{N}\). Since \(\beta_{Q}\) and \(\beta_{N}\) are considered to increase with the prevalence of infection, they are expressed as in Equations (8) and (9), respectively. \(\beta_{Q0}\) and \(\beta_{N0}\) represent the initial values of \(\beta_{Q}\) and \(\beta_{N}\). \[\beta_{Q}=\beta_{Q0}+a\big{(}I_{Q}+I_{N}+R\big{)} \tag{8}\]  
\[\beta_{N}=\beta_{N0}+b\big{(}I_{Q}+I_{N}+R\big{)} \tag{9}\]  
For simplicity, \(\beta_{a}\) is expressed as a weighted average of \(\beta_{Q}\) and \(\beta_{N}\) using the parameter \(c\) with \(0<c<1\) as in Equation (10). \[\beta_{a}=\frac{c\big{(}\beta_{Q}+\beta_{N}\big{)}}{2} \tag{10}\]  
Since the incubation period of the disease is the same regardless of the strategy, the incidence rate for both \(E_{Q}\) and \(E_{N}\) is \(\varepsilon\). The recovery rate is also \(\gamma\) for both \(I_{Q}\) and \(I_{N}\). \(\varphi_{S}\), \(\varphi_{E}\), and \(\varphi_{I}\) denote the rate of change in strategy in each state. The payoffs for each strategy are shown in Equations (11) and (12). \[\pi_{Q}=-\Omega \tag{11}\]  
\[\pi_{N}=-\delta\beta_{N}I \tag{12}\]  
Figure 6: SEIR model with incubation period and viral mutation. where \(\Omega\) represents the cost over the isolation period and \(\delta\) is the risk of the disease as perceived by the individual. The probability of whether agent \(i\) adopts agent \(j^{\prime}\)s strategy using the Fermi rule in Equation (4) [31]. Using Equation (4), the strategy transition rates for each of the states \(S\), \(E\), and \(I\) are expressed in Equations (13) to (15) as follows:  
\[\varphi_{S}=S_{Q}(S_{N}+E_{N}+I_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-S_{N} \big{(}S_{Q}+E_{Q}+I_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{13}\]  
\[\varphi_{E}=E_{Q}(S_{N}+E_{N}+I_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-E_{N} \big{(}S_{Q}+E_{Q}+I_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{14}\]  
\[\varphi_{I}=I_{Q}(S_{N}+E_{N}+I_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-I_{N} \big{(}S_{Q}+E_{Q}+I_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{15}\]  
The SEIR model with incubation period and viral mutation is shown in system (16) as follows:  
\[