## 4 Discussion  
In this study, we calculated the effective distance of the spread of COVID-19 in Japan by applying the SEIR model to Tokyo, the capital city of Japan. Despite the presence of social factors that altered human mobility, such as two declarations of a state of emergency and travel campaigns, the spread of COVID-19 was demonstrated mainly by assuming  
Figure 6: Simulations quantitatively estimating the effects of effective distance on a scale of the pandemic in each prefecture. **(a)** Estimated increase or decrease of total infection during an equivalent period as P3 when the effective distance from Tokyo is set to the same as P1 (left) or P2 (right) and the source (Tokyo) is the same as P3. **(b)** Estimated increase in infections of infection during an equivalent period as P3 when every prefecture approaches Tokyo because of a large-scale event such as the Olympics. Some prefectures where some games are held are closer to Tokyo than others. The results of the simulated shifts are shown in Figure S9. the spread from Tokyo during all phases of the pandemic (Figure S1). More specifically, our SEIR model-based predictions revealed that the effective distance from Tokyo to other prefectures changed over each phase, in which the effective distance between Tokyo and Hokkaido was close for P1 and P3, but far for P2, for example (Figure S1). Comparing the fitting parameter and effective distance in P1a, P2a, and P3a, we can see the effect of government-sponsored travel promotion. In particular, the effective distance between Tokyo and prefectures with many tourist attractions has increased. These findings provided an explanation for the spread of the two outbreaks in Hokkaido and many other prefectures, which could not be explained merely by passenger traffic data alone. In Figure 6, we estimated the impact of a huge public event such as the Tokyo Olympics by creating an infection model that took into account the increase in the mobility of people between prefectures having event venues. Since the mobility of people between Tokyo and each prefecture was predicted to be a critical factor for pandemic control [24; 25; 26], the first priority for policymakers should be to prevent the infectious situation in Tokyo from being transferred to local areas, and in this sense, the decision of the Japanese government to hold the Olympic games without spectators is commendable. Conversely, although spectators have been excluded from participating in the events, athletes and media personnel remained a group of individuals moving around the event venues, so it is important to adequately consider the control strategy. Estimating the effective distance from actual infection data is also a useful method for constructing an infection network for the spread of infection, constructing all infection networks based on the SEIR model, and simulating the spread of infection when an outbreak occurs in a city is computationally infeasible. However, once the network structure is drawn, a feasible simulation becomes available. For example, in the case of this study, by assuming the spread of infection from Tokyo, we were able to explain the infection situation in other prefectures with high accuracy (Figure S3), suggesting that the infection network in the country was a simple radial structure centered on Tokyo. Conversely, if the same method is applied to the United States or European countries, where multiple cities or countries, respectively, experienced surges of infection simultaneously [44; 45], it may be possible to obtain a more complex structural organization of the infection network. By combining the effective distance estimated from infection data with the effective distance based on a transportation network and movement data, we believe that it is possible to establish more effective preventive policies to avoid the spread of infection. It is difficult to quantitatively evaluate the contribution of various modes of transportation, such as airplanes, cars, and trains, to infectious diseases.