## 3 Results
### Effective Distance Changes Dynamically Depended on Stages of the Pandemic  
Another advantage of the effective distance derived from time series data of infections over that derived from passenger volume data is that human mobility can be estimated dynamically without the need for extensive surveys. Such dynamic estimations based on different stages of the pandemic are particularly important to evaluate the effects of specific policies (e.g., \"soft lockdown\" and travel campaigns) and to define a better strategy. Thus, to investigate changes in human mobility over different stages of the pandemic, we first examined the distribution of the effective distance over the respective periods. Histograms of effective distances revealed that the shapes of the distributions shifted de  
Figure 3: Effective distance derived from diffusion equations contains both passenger traffic and other local information that contribute to the spread of the pandemic. (**a,b**) Relationship between effective distance based on diffusion process from Tokyo and passenger traffic data in 2019 for all prefectures (**a**) and the prefectures in the Kanto region (**b**). A point represents each prefecture, and the prefectures whose effective distance is larger than 100 were excluded. Linear regression by least mean square and correlation coefficients are shown for each panel. pending on different periods, particularly during spreading periods P1a, P2a, and P3a (Figures 4a and 56a). Regarding the distribution of the effective distance, P2, during which the number of infected people declined without the need for any strict movement restrictions, our model showed that the largest mean and effective distance of each prefecture distributed widely around the mean (Figure 4a). Conversely, P1 exhibited the smallest mean and deviation from the mean, whereas P3 exhibited an intermediate mean, but the distribution was not unimodal, indicating the existence of distinct trends in human mobility between prefectures (Figure 4a). Considering that the Japanese government declared a state of emergency during P1 and P3 to contain the pandemic, the distribution of the effective distance of P2 would be considered desirable to limit the diffusion of the coronavirus pandemic while maintaining passenger mobility. Furthermore, during P3, the effective distance in some prefectures became smaller while others did not, presumably reflecting the increase in travelers from Tokyo to other prefectures due to the second stage travel campaign in which travels to and from Tokyo became subject to subsidization. These results demonstrated that infection-related human mobility could be captured during a pandemic. Subsequently, we investigated the changes in the effective distance for each prefecture by region. The general trend of the effective distance across the entire country was conserved over time (Figures 4b and 56b), which indicated that, other than diffusion from Tokyo, there was local connectivity between nearby prefectures that contributed to the spreading of the virus and altering their distance from Tokyo. Furthermore, considering the combination with other plots (Figures 4a and 55a,b), it can be implied that closer effective distances, in particular those found in major cities in each region, chiefly contribute to the rapid increase in coronavirus cases against which the government needed to impose restrictions. Consequently, the effective distance estimated from the time series of the newly confirmed cases evolved temporally, which might provide useful information for determining the appropriate mobility level during the pandemic. Figure 4: Effective distance changes dynamically depending on the stages of the pandemic. (**a**) Distribution of effective distances in periods P1a, P2a, and P3a. The vertical line in magenta indicates the mean value of the distribution. Prefectures whose effective distance is larger than 100 were excluded.