## 3 Results
### Effective Distance Changes Dynamically Depended on Stages of the Pandemic
(**b**) Effective distance of each prefecture in periods P1a, P2a, and P3a, ordered by “Prefecture Code” defined International Organization for Standardization. Regions to which prefectures belong are colored differently. A Distorted Map of Japan, Based on Effective Distances and Local Interactions, Revealed the Non-Uniform Spread of the Pandemic across the Country  
To visualize how the pandemic spread across Japan across different periods, we combined information on effective distances and local interactions between prefectures, which we defined as the geographical distance between adjacent prefectures that had large passenger traffic in the 2019 survey, to create a distorted map of the host country. The position (i.e., longitude and latitude) of a capital city in each prefecture was determined as the solution of a mathematical optimization equation using the non-linear least squares calculation [43]. In general, regions to the west of Tokyo, largely the Kinki and Kyusyu regions, reduced in size, whereas prefectures around Tokyo were expanded (for example, the Chubu regions were stretched in the northwest direction) in the distorted maps compared to the original one (Figure 5a-d). These deformations indicated that the coronavirus tended to rapidly spread to the western regions of the country, while it required additional time to spread to the north and north-western directions, particularly towards the Tohoku and Chubu regions. Thus, distorting maps based on effective distance and local interactions would be useful to characterize the spread of the coronavirus geographically and to develop specific policies aimed at preventing a large spike in the infectious population. Figure 5: Distorted maps of Japan based on effective distance and local interactions reveal non-uniform spreading of the virus across Japan. (**a**) Original map of Japan. Each point indicates the capital of each prefecture, colored by region. (**b–d**) Distorted maps of Japan based on effective distance and local connectivity in periods P1a, P2a, and P3a (Figure S7). Quantitative Estimation of the Scale of the Pandemic in Each Prefecture in Different Sets of Effective Distances  
Although effective distance allows us to estimate how the virus spreads across the country, it is still unclear how much the effective distance contributes to determining the scale of the pandemic quantitatively, which would be a useful parameter when developing policies that balance both economic activity and medical resources in each prefecture. To evaluate the effects of the effective distance, we conducted simulations in which only the effective distance was changed from P3. If the effective distance was the same as P1, the total number of infected people would increase to more than twice in peripheral or rural areas, but would not increase to a similar degree in populated prefectures (Figure 6a). Conversely, if the effective distance was the same as P2, there would be no clear differences between the total number of infected individuals between the central or peripheral prefectures (Figure 6a). These results indicated that the effective distance could largely affect the scale of the pandemic, and these effects were heterogeneous between prefectures. Finally, we considered the increase in human mobility that could result from a very large public event such as the 2020 Olympic and the Paralympic Games, which were to be held in Tokyo and several other prefectures, and estimated the impact on the pandemic (Figure S9). Our findings showed that prefectures that held several matches, particularly prefectures that usually did not share strong connectivity with Tokyo, would experience a large increase in the number of patients (Figure 6b). For example, the number of infected cases in Fukushima prefecture would increase by 80%, while other host prefectures would increase by 40 to 60%. This indicated that the impact of an increase in human mobility due to huge public events was more severe in prefectures that usually had a larger effective distance from Tokyo.