## 4 Problem Description and Model Formulation
### Model Formulation
#### 4.2.2 SEIR Model for Consideration of Hospital Care  
When infected with COVID-19, patients with severe symptoms should be treated in a hospital. In Japan, the basic policy is that only those with severe symptoms should receive treatment at a hospital, while those with mild symptoms or asymptomatic patients should be cared for at home. However, there are not enough hospital beds in an epidemic of COVID-19 infection, and there are cases in which patients with severe illness cannot be treated in a hospital. We propose a model in this section to examine the level of symptoms at which infected patients should be admitted to hospitals so that those with severe illnesses can be treated in hospitals on a priority basis. The same parameters and conditions as in the model presented in Section 4.2.1 are omitted in this section. Agents choose their own strategies only for the set of states \(S\) and \(E\). We assume that there are two types of infected patients: those who are cared for in the hospital and those who are cared for at home. The person who recovers at home is denoted as state \(I_{m}\). The person who recovers in the hospital is denoted by state \(H\). The proposed model is shown in Figure 7.  
The infection rates \(\beta_{Q}\) and \(\beta_{N}\) are expressed by Equations (8) and (9) in Section 4.2.1. Parameter \(\alpha\) is a parameter that expresses how many people in the incubation period will be treated in the hospital when they develop the disease and become infected. The recovery rate for state \(H\) is denoted by \(\gamma_{A}\), and that for state \(I_{m}\) by \(\gamma_{B}\). The payoffs for each strategy are expressed in Equations (11) and (12).  
Using Equation (4) in Section 3.2.1, the strategy transition rates for each state of \(S\) and \(E\) are expressed in Equations (17) and (18) as  
\[\varphi_{S}=S_{Q}(S_{N}+E_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-S_{N}\big{(}S_{ Q}+E_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{17}\]  
\[\varphi_{E}=E_{Q}(S_{N}+E_{N})\Theta\big{(}\pi_{Q},\pi_{N}\big{)}-E_{N}\big{(}S _{Q}+E_{Q}\big{)}\Theta\big{(}\pi_{N},\pi_{Q}\big{)} \tag{18}\]  
The equations of the model are shown in system (19).  
\[\left\{\begin{aligned} S_{N}^{\prime}&=-\beta_{N}S_{N }(H+I_{m})+\tau\varphi_{S}\\ S_{Q}^{\prime}&=-\beta_{Q}S_{Q}(H+I_{m})-\tau\varphi_{S }\\ E_{N}^{\prime}&=\beta_{N}S_{N}(H+I_{m})-\varepsilon E _{N}+\tau\varphi_{E}\\ E_{Q}^{\prime}&=\beta_{Q}S_{Q}(H+I_{m})-\varepsilon E _{Q}-\tau\varphi_{E}\\ H^{\prime}&=\alpha\varepsilon(E_{N}+E_{Q})-\gamma H \\ I_{m}^{\prime}&=\varepsilon(1-\alpha)(E_{N}+E_{Q})- \gamma I_{m}\\ R^{\prime}&=\gamma_{A}H+\gamma_{B}I_{m}\end{aligned}\right. \tag{19}\]  
The model (19) represents the relationship between susceptible, incubators, infected, and recovering agents. In addition, \(\varphi_{S}\) and \(\varphi_{E}\) represent the change in the agent's strategy due to the evolutionary game. It can be said that the agent's behavior of self-isolation is incorporated into the SEIR model, which represents the infection epidemic. In addition, by adopting state \(H\) and parameter \(\alpha\), the number of patients in the hospital, which cannot be represented by the ordinary SEIR model, can be expressed.