# A Mathematical Model for the COVID-19 Pandemic in Tokyo through Changing Point Calculus
## 3 Results  
To showcase the feasibility of the proposed SEIAMD model to the COVID-19 pandemic in Tokyo, we used the implementation discussed in Section 2.2. The closer the calculated data are to the official number of persons infected, the more suitable our SEIAMD model is for the purpose discussed in this paper. The nearness between the official and calculated data is shown in Figure 3. This figure shows the number of persons identified (\(y\)-axis) for each day (\(x\)-axis). The values of the plotted curve of the official data are obtained directly from the official data provided by health authorities [26]. The values of the plotted curve of the calculated data are obtained by executing the last function described in Section 2.2. As we can see, both curves are highly overlapped for most days. To assess numerically the accuracy of our model, we calculated the \(MdAPE\) (Median Absolute Percentage Error) value as follows [28]:  
\[MdAPE=\left(median\left(\frac{\left|\text{official data}-\text{calculated data}\right|}{\text{official data}}\right)\right)\times 100. \tag{9}\]  
By doing so, we obtained an \(MdAPE\) value of 9.3%, which indicates that the calculated data fit very well to the official data. In order to compare the fit of our model to the data with the fit of the classical SIR model, we implemented the SIR model in Python following the same procedure described in Section 2.2 for the SEIAMD model, using the equations described in Equation (7), and the parameter \(\gamma_{SIR}\) shown in Equation (3). The resulting projection is shown in Figure 4.  
The \(MdAPE\) value obtained for the SIR model was 60.6%, which indicates that the calculated data fit poorly to the official data. In view of these results, we can affirm that the SEIAMD model fits the official data better than the SIR model. Table 1 shows the change points obtained with the SEIAMD model. We determined that on each of the dates listed in this table, there was a change in the behavior of Tokyo's inhabitants, which implies a change in the effective contact rate of the population. Between each pair of change points, there is a period with consistent (similar) behavior, which is justified by showing the good fitting of the values obtained with our model. Table 2 shows these periods and the values obtained for the parameters \(\beta_{i}\) and \(\beta_{a}\) associated with each period. As mentioned above, our fitting procedure iterated among the periods to obtain the value of \(\beta_{i}\) and \(\beta_{a}\). Figure 3: Official and calculated data of identified people obtained with the SEIAMD model in the prefecture of Tokyo during the first seven waves of the COVID-19 pandemic. As we can see in Table 1, the change points obtained with the SEIAMD model can be linked to actual events related to the Tokyo population, most of which can be identified as Non-Pharmaceutical Interventions (NPIs) or events clearly related to the habits of the population such as holidays and weekends. This strengthens the validity of the assumptions we considered while defining our SEIAMD model, as well as the validity of the SEIAMD model as a tool for evaluating the effectiveness of NPIs.