## 5 Methods
### Model of the Spread of the Infected Population from Tokyo and Estimation of Effective Distance  
The spread of the COVID-19 pandemic in prefectures other than Tokyo was modeled as the diffusion process of infected individuals from Tokyo. We denoted the number of newly confirmed cases as \(u(x,t)\), as follows:  
\[\frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}},\ x\in(0, \ L],\ t\in(0,\ T],\]  
where \(D\) was the diffusion coefficient. The parameters used in the simulation are summarized in Table S2. For simplicity, the initial conditions for the PDE were expressed as:  
\[u(0,\ 0) =\xi I_{0}\] \[u(x,\ 0) =0,\ x\in(0,\ L],\]  
which meant that the infectious population was located only in Tokyo in the initial condition. For the boundary conditions, we applied the Dirichlet boundary condition for \(x=0\) and the Neumann boundary condition for \(x=L\), or  
\[u(0,\ t)=\xi I\] \[\left.\frac{\partial u}{\partial x}\right|_{x=L}=0.\]  
The parameters for the PDE are summarized in Table S2. The PDE was replaced by a series of ordinary differential equations (ODEs) by finite difference approximations, which were solved using the SciPy software. The solutions of the PDE, or the diffusion process of the infectious population from Tokyo depending on \(x\) and \(t\) values, are shown in Figure 2a,b.  
To estimate the effective distance from Tokyo at different stages of the pandemic, we considered the diffusion process for six different periods depending on the period P1 to P3 and the peaks of the number of new cases in Tokyo during the periods: P1a, P1b, P2a, P2b, P3a, and P3b. The fitting of the observed data to the simulated data was also conducted for each period separately, and the \(x\), or the effective distance of each prefecture from Tokyo based on the diffusion process, was determined by choosing \(x\) with a minimum root mean square error (RMSE) value. The estimated effective distance for all periods are summarized in Table S3. If the effective distance of Tokyo, \(x_{T}\), estimated from the fitting was not equal to zero, the effective distance of all prefectures was adjusted by subtracting \(x_{T}\) for subsequent analyses.  
Effective Distance Based on the Inter-Prefecture Network of Traffic and Mobility Data of Individuals  
Since the contagion process is predominantly caused by the movement of people, we can also estimate the effective distance between prefectures based on passenger traffic data. The effective distance derived by this method was used to test the validity of the effective distance derived from the diffusion equation and for its characterization (Figures 3a,b and S3a). We defined the connectivity matrix \(\mathbf{P}\) as follows:  
\[P_{mm}=\frac{F_{nm}}{F_{m}}\ (0\leq P_{mm}\leq 1)\] \[F_{m}=\sum_{n}F_{nm},\]  
where \(F_{nm}\) represented the inter-prefecture passenger traffic on railways, cars, ships, and airlines. Thus, the matrix \(\mathbf{P}\) quantified the fraction of the passenger flux from pre\(n\) to \(m\). Given this flux-fraction matrix \(\mathbf{P}\), the effective distance \(d_{nm}\) from prefecture \(n\) to \(m\) is defined as  
\[d_{nm}=(1-\log P_{nm})\geq 1.\]  
We then constructed a weighted directed graph network consisting of 47 nodes, representing the prefectures in Japan, and edges from prefecture \(n\) to \(m\) with weights of \(d_{nm}\) (i.e., \(x\zeta_{2}=1081\) edges in total). Consequently, the effective distance from Tokyo to prefecture \(n\), or \(D_{n}\), was estimated as the length of the shortest path from Tokyo to the prefecture \(n\) in the graph network.  
Linear Regression of the Effective Distance Based on the Diffusion Process by Other Distance Metrics  
The effective distance derived from the diffusion equations was predicted by a linear model without intercept, or \(y=ax\), as Tokyo is at the origin. Models were fit to minimize the residual sum of squares. The coefficient of determination (\(R^{2}\)) was determined as \(R^{2}=1-\frac{u}{v}\), where \(u\) is the residual sum of squares and \(v\) is the total sum of squares.