# 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
Finally, combining all these factors, we obtain the Susceptible equation:  
\[S_{t+1}=S_{t}-\beta_{i}\,S_{t}\,I_{t}/N_{t}-\beta_{a}\,S_{t}\,A_{t}/N_{t}- \varphi_{t}+\sigma\,M^{*}.\]  
In the same way, we have that the Exposed compartment (\(E\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(E_{t+1}=E_{t}\dots\)), plus the individuals that have moved from the Susceptible compartment (\(S\)), 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 individuals that have moved to the Identified compartment (\(I\)) (\(-\alpha\,\lambda\,E_{t}\)), minus the individuals that have moved to the Asymptomatic compartment (\(A\)) (\(-(1-\alpha)\,\lambda\,E_{t}\)). These last two factors can be simplified into \(\lambda\,E_{t}\), which are the individuals who have left the Exposed compartment (\(E\)). Combining all these factors, we obtain the Exposed equation:  
\[E_{t+1}=E_{t}+\beta_{i}\,S_{t}\,I_{t}/N_{t}+\beta_{a}\,S_{t}\,A_{t}/N_{t}- \lambda\,E_{t}.\]  
On the other hand, the Identified compartment (\(I\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(I_{t+1}=I_{t}\dots\)), plus the individuals that have moved from the Exposed compartment (\(E\)) (\(+\alpha\,\lambda\,E_{t}\)), minus the individuals that have moved to the Immunized compartment (\(M\)) (\(-\gamma\,I_{t}\)), and minus the individuals that have moved to the Deceased compartment (\(D\)) (\(-\mu\,I_{t}\)). Combining all these factors, we obtain the Identified equation:  
\[I_{t+1}=I_{t}+\alpha\,\lambda\,E_{t}-\gamma\,I_{t}-\mu\,I_{t}.\]  
The Asymptomatic compartment (\(A\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(A_{t+1}=A_{t}\dots\)), plus the individuals that have moved from the Exposed compartment (\(E\)) (\(+(1-\alpha)\,\lambda\,E_{t}\)), minus the individuals that have moved to the Immunized compartment (\(M\)) (\(-\gamma\,A_{t}\)). Combining all these factors, we obtain the Asymptomatic equation:  
\[A_{t+1}=A_{t}+\left(1-\alpha\right)\lambda\,E_{t}-\gamma\,A_{t}.\]  
The Immunized compartment (\(M\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(M_{t+1}=M_{t}\dots\)), plus the individuals that have moved from the Identified compartment (\(I\)) (\(+(1-\alpha)\,\lambda\,E_{t}\)), plus the individuals that have moved from the Asymptomatic compartment (\(A\)) (\(+\gamma\,A_{t}\)), plus the \(\varphi\) individuals that have moved from the Susceptible compartment (\(S\)) (\(+\varphi_{t}\)), minus the individuals that have moved to the Susceptible compartment (\(S\)) (\(-\sigma\,M^{*}\)). Combining all these factors, we obtain the Immunized equation:  
\[M_{t+1}=M_{t}+\gamma\,I_{t}+\gamma\,A_{t}+\varphi_{t}-\sigma\,M^{*}.\]  
Finally, the Deceased compartment (\(D\)) in time \(t+1\) contains the individuals that were in time \(t\) (\(D_{t+1}=D_{t}\dots\)) plus the individuals that have moved from the Identified compartment (\(I\)) (\(+\mu\,I_{t}\)). Combining these factors, we obtain the Deceased equation:  
\[D_{t+1}=D_{t}+\mu\,I_{t}.\]