# A Mathematical Model for the COVID-19 Pandemic in Tokyo through Changing Point Calculus
## 3 Results
\begin{table}
\begin{tabular}{c l} \hline \hline
**Date** & **Event** \\ \hline
2020-04-07 & Declaration of state of emergency in Tokyo [29]. \\
2020-04-16 & Declaration of state of emergency in Japan [30]. \\
2020-05-07 & Extension of the state of emergency (planned end: 6 May) [30]. \\
2020-05-26 & First day after the end of the state of emergency [30]. \\
2020-08-08 & Obon Festival. \\
2020-11-01 & Resumption of cross-border travel [31]. \\
2020-11-23 & Labor Thanksgiving Day. \\
2020-12-23 & First case of Alpha variant [32]. \\
2021-01-08 & Declaration of state of emergency in Tokyo [33]. \\
2021-01-14 & Extension of the state of emergency to other prefectures [34]. \\
2021-03-22 & First day after the end of the state of emergency [35]. \\
2021-04-25 & Declaration of state of emergency in Tokyo [36]. \\
2021-06-21 & First day after the end of the state of emergency [37]. \\
2021-07-12 & Declaration of state of emergency in Tokyo [38]. \\
2021-08-15 & Obon Festival. \\
2021-09-01 & Disaster Prevention Day. \\
2021-10-01 & First day after the end of the state of emergency [39]. \\
2021-12-24 & First community transmission of Omicron variant [40]. \\
2022-01-21 & Establishment of quasi-emergency measures [41]. \\
2022-01-30 & Weekend. \\
2022-02-23 & The Emperor’s Birthday. \\
2022-03-26 & Hanami (cherry blossom viewing). \\
2022-04-10 & Weekend. \\
2022-05-03 & Golden Week holidays. \\
2022-05-28 & Weekend. \\
2022-06-17 & Friday. \\
2022-07-26 & Summer vacation. \\
2022-09-01 & Disaster Prevention Day. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Change points in the behavior of the Tokyo population, shown in Figure 3.  
Figure 4: Official and calculated data of identified people obtained with the SIR model in the prefecture of Tokyo during the first seven waves of the COVID-19 pandemic. In Figure 3, we can notice that around the last months of 2020, there was a higher concentration of change points (i.e., the change points were closer to each other). This is explained by a fast change in the behavior of the population. Thus, people stressed by different and changing conditions ended up in disparate outcomes. However, just after that period, as is normal, the behavior of the population stabilized; therefore, there was a longer period with no change point. This also resulted in a quite long period with very sparse change points, even covering the totality of the fifth wave (around August 2021). At this moment, the population reacted more strictly to the NPIs, as demonstrated by the fact that the change points reflected the change in official policy. The stable period ended around December 2021 with the first occurrence of the Omicron variant. This change point was succeeded by a higher concentration of change points, although never as close as the previous case. Our model experienced higher deviations, as shown at the end of January 2022. This is because of the high number of people infected, which made it more difficult to adjust with high precision. The same occurred around July 2022. However, the concentration of change points around both spikes was very different. There was a noticeably lower concentration of change points around the last spike in comparison to the previous spike, which means that people did not react to the rise in cases by changing their behavior; so, it is normal that this last spike resulted in a higher number of persons infected. Regarding the periods and values of the effective contact rates shown in Table 2, it is worth noting the stability of the resulting values of the parameters. This is particularly important for the parameter \(\beta_{i}\), which responds to the effective contact rate of people identified as infected, which was always lower than 0.9, although most values (75-percentile) were lower than 0.3 (the mean is 0.26, and the standard deviation is 0.2).