## 3 Results  
To model the propagation of the COVID-19 pandemic in Japan as a diffusion process of the infected population from Tokyo and to estimate effective distances, we first recapitulated a time series using the SEIR model in which the number of infected people in Tokyo depended on the distinct phases of the pandemic, or periods P1, P2, and P3 (Figure 1). We calibrated the parameters for the number of potentially newly infected individuals in Tokyo during each period using the Markov chain Monte Carlo (MCMC) algorithm, and the summary of the estimated parameters is listed in Table S1, which were generally consistent with previous studies on COVID-19 [37; 42]. Models using the estimated parameters accurately recapitulated the temporal dynamics and amplitude of the outbreaks and  
Figure 1: Timelineof the pandemic of coronavirus disease 2019 (COVID-19) in Japan. The period from 18 March 2020 to 12 March 2021 was considered. The top row indicates the period during which a state of emergency (i.e., “soft lockdown”, or mild restrictions on movement, alcohol-serving businesses, and large events) was declared. The second row indicates the period of a travel campaign, “Go To Travel”, during which the Japanese government encouraged the citizens to travel domestically by subsidizing travel expenses. The travel campaign had two stages: (1) the first from 22 July 2020 to 30 September 2020, during which Tokyo was excluded from the campaign, and (2) the second from 1 October 2020 to 27 December 2020, during which Tokyo was included and the subsidy was increased from stage 1. The main panel exhibits the 7-day backward moving average of the number of newly confirmed cases in each prefecture, which are highlighted in blue for Tokyo. Depending on the phase of the pandemic, the period was divided into six different periods. surges in infections (Figure S1a-c). Leveraging these models as sources, we subsequently set up diffusion equations that were calculated as one-dimensional partial differential equations (PDEs) (see Section 5). Thus, human mobility outcomes differed and depended on whether the pandemic was on a spreading or shrinking trend; hence, we solved the diffusion equations dividing each phase into periods before and after a peak of infection in Tokyo, which generated six periods (i.e., P1a, P1b, P2a, P2b, P3a, and P3b) (Figure 1). After numerically solving the PDEs, we fit the number of newly confirmed cases in each of the 47 prefectures to the solutions and estimated the effective distance from Tokyo (Figures 2 and S2). The calculated results at the estimated effective distance, particularly during the first part of each period (i.e., P1a, P2a, and P3a), recapitulated the temporal evolution of the number of newly confirmed cases in most prefectures (Figure 2). Consequently, by considering the propagation of the pandemic as a diffusion process, we estimated the effective distance from Tokyo based on only the time series of newly confirmed cases of COVID-19 in each prefecture. The Effective Distance Estimated Based on the Diffusion Process from Tokyo Revealed the Mobility of Individuals during the Pandemic  
To characterize the estimated effective distance based on the diffusion process from Tokyo, we applied linear regression in which the geographical distances from Tokyo or effective distances based on inter-regional passenger traffic are explanatory variables [31]. As for the effective distance derived from passenger traffic data, we used inter-prefecture passenger volume data obtained in Japan in 2019, built a weighted directed network of mobility probability between prefectures, and estimated it as the shortest path from Tokyo in the graph. The effective distance based on the passenger traffic network generally predicted the effective distance based on the diffusion process, whereas the geographical distance did not, indicating that human mobility from Tokyo to other prefectures largely explained the time series of infections (Figures 3a and S3a-c).