# A Mathematical Model for the COVID-19 Pandemic in Tokyo through Changing Point Calculus
## 2 Materials and Methods
### Conceptual Model
#### 2.1.3 Model Equations  
The classical SIR model proposed by [PERSON] and [PERSON] is expressed by the following system of differential equations [22]:  
\[\left\{ \begin{aligned} \frac{dS}{dt}&=-\beta IS,\\ \frac{dI}{dt}&=\beta IS-\gamma I,\\ \frac{dR}{dt}&=\gamma I.\end{aligned} \right. \tag{4}\]  
When we consider, as in this case, that the population mixes at random, so that each individual has a small and equal chance of coming into contact with any other individual, the transmission of infection is represented by the term \(\beta IS/N\)[23], where \(N\) is the total population. Thus, we have the following system of equations:  
\[\left\{ \begin{aligned} \frac{dS}{dt}&=-\frac{\beta IS}{N},\\ \frac{dI}{dt}&=\frac{\beta IS}{N}-\gamma I,\\ \frac{dR}{dt}&=\gamma I.\end{aligned} \right. \tag{5}\]  
Taking into account that the data regarding the COVID-19 pandemic are discrete, since the authorities publish the measures every day, it is more convenient to work with a discrete form of the SIR model. A fast and direct way to discretize the SIR model is by the forward Euler method [24], using the formula:  
\[y_{n+1}=y_{n}+\Delta tf(t,y(t)),\quad\text{for }n=0,1,2... \tag{6}\]  
Thus, applying Equation (6) to Equation (5), with \(\Delta t=1\)_day_, we obtain the following system of equations for the discrete SIR model:  
\[\left\{ \begin{aligned}  S_{t+1}&=S_{t}-\beta\,I_{t}\,S_{t} /N_{t},\\ I_{t+1}&=I_{t}+\beta\,I_{t}\,S_{t}/N_{t}-\gamma\,I_{t},\\ R_{t+1}&=R_{t}+\gamma\,I_{t},\end{aligned} \right. \tag{7}\]  
where \(N_{t}\) is the total population in any given time \(t\), and \(N_{t}=S_{t}+I_{t}+R_{t}\). Following the same reasoning, we decide to formulate the SEIAMD model as a discrete-time model. Thus, applying the same procedure described above, the resulting system of difference equations that rule the SEIAMD model behavior is:\[\left\{ \begin{array}{l}S_{t+1}=S_{t}-\beta_{i}\,S_{t}\,I_{t}/N_{t}-\beta_{a}\,S_ {t}\,A_{t}/N_{t}-\varphi_{t}+\sigma\,M^{*},\\ E_{t+1}=E_{t}+\beta_{i}\,S_{t}\,I_{t}/N_{t}+\beta_{a}\,S_{t}\,A_{t}/N_{t}- \lambda\,E_{t},\\ I_{t+1}=I_{t}+\alpha\,\lambda\,E_{t}-\gamma\,I_{t}-\mu\,I_{t},\\ A_{t+1}=A_{t}+\left(1-\alpha\right)\lambda\,E_{t}-\gamma\,A_{t},\\ M_{t+1}=M_{t}+\gamma\,I_{t}+\gamma\,A_{t}+\varphi_{t}-\sigma\,M^{*},\\ D_{t+1}=D_{t}+\mu\,I_{t},\end{array} \right. \tag{8}\]  
where \(N_{t}=S_{t}+E_{t}+I_{t}+A_{t}+M_{t}+D_{t}\). Furthermore, we consider \(N_{t}\) to be constant, and, calculating the mean value from official data [25], \(N_{t}=14,\)039,312. Taking into account the inputs and outputs of each compartment that is part of the SEIAMD model (see Figure 2), we can follow step by step the construction of the equations that we have just presented. By doing so, we have that the Susceptible compartment (\(S\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(S_{t+1}=S_{t}\dots\)), minus the individuals that have moved to the Exposed compartment (\(E\)), whose number depends on the parameters \(\beta_{i}\) and \(\beta_{a}\) (\(-\beta_{i}\,S_{t}\,I_{t}/N_{t}-\beta_{a}\,S_{t}\,A_{t}/N_{t}\)), minus the \(\varphi\) individuals that have moved to the Immunized compartment (\(M\)) (\(-\varphi_{t}\)), plus the individuals that have entered from the Immunized (\(M\)) compartment, whose number depends on the parameter \(\sigma\) (\(+\sigma\,M^{*}\)).