# A Mathematical Model for the COVID-19 Pandemic in Tokyo through Changing Point Calculus
## 2 Materials and Methods
### Conceptual Model
#### 2.1.2 Model Parameters  
The effective contact rate (\(\beta\)) is defined as the product of the total contact rate and the transmission risk, i.e., the rate at which the contact between a susceptible individual and others infected will cause the susceptible individual to become infected. The parameters \(\beta_{i}\) and \(\beta_{a}\) represent the effective contact rate for the identified and for the asymptomatic, respectively. Considering the results obtained in [14], we assume that \(\beta_{a}=0.58\beta_{i}\). On the other hand, \(\beta_{i}\) is calculated by adjustment when running the model.  
The parameter \(\varphi\) is the number of individuals immunized by vaccine. We obtained the official data of vaccinations from [15], and we calculated \(\varphi\) taking into account the results obtained in [16]. Thus, we suppose that 14 days after the first dose, only 52 percent of vaccinated individuals will be immunized, and 7 days after the second, third, fourth, and fifth doses, 95 percent of vaccinated individuals will be immunized.  
The parameter \(\sigma\) represents the proportion of immunized individuals that are susceptible again after a period of time. From [17], we assume that \(\sigma=0.81\) for the B.1.1.529 (Omicron) variant and \(\sigma=0.15\) for the others variants. \(M_{t}\) is the number of immunized individuals at time \(t\), and \(M^{*}\) is the number of individuals immunized at \(t-p\), with \(p\) as the period of time needed for the loss of immunity; from [17], we assume that \(p=6\) months.  
The parameter \(\lambda\) is the latency rate, and we suppose that the incubation period (\(1/\lambda\)) is 5 days for the original variant of SARS-CoV-2 (\(\lambda=0.2\)) [18], 4 days for the B1.1.7 (Alpha) and B.1.617.2 (Delta) variants (\(\lambda=0.25\)) [19; 20], and 3 days for the B.1.1.529 (Omicron) variant (\(\lambda=0.33\)) [21].  
The parameter \(\alpha\) represents the proportion of infected individuals that are identified, and we assume that nonidentified individuals are asymptomatic; therefore, taking into account the results obtained in [14], we suppose that \(\alpha=0.83\).  
The parameter \(\gamma\) is the recovery rate, and it is calculated from the official data as follows:  
\[\gamma=\frac{1}{f}\sum_{t=1}^{f}\frac{D_{t}-D_{t-1}}{G_{t}}=0.1201\sim 0.12, \tag{1}\]  
where \(D_{t}\) is the number of discharged individuals at time \(t\), and \(G_{t}\) is the number of individuals at time \(t\) that have tested positive, minus the number of individuals that have been discharged, and minus the number of deaths.  
Finally, the parameter \(\mu\) is the mortality rate. It is calculated, also from the official data, as follows:  
\[\mu=\frac{1}{f}\sum_{t=1}^{f}\frac{H_{t}-H_{t-1}}{G_{t}}=0.00095\sim 0.001, \tag{2}\]  
where \(H_{t}\) is the number of deaths at time \(t\).  
For the SIR model, implemented for comparative purposes, as the R compartment contains individuals that have recovered as well as individuals that have died, we recalculate the \(\gamma\) parameter so that it includes the deceased. Thus, the parameter \(\gamma\) for the SIR model is calculated from the official data as follows:  
\[\gamma_{SIR}=\frac{1}{f}\sum_{t=1}^{f}\frac{(D_{t}+H_{t})-(D_{t-1}+H_{t-1})}{ G_{t}}=0.12105\sim 0.121. \tag{3}\]