Determining travel fluxes in epidemic areas

Infectious diseases attack humans from time to time and threaten the lives and survival of people all around the world. An important strategy to prevent the spatial spread of infectious diseases is to restrict population travel. With the reduction of the epidemic situation, when and where travel restrictions can be lifted, and how to organize orderly movement patterns become critical and fall within the scope of this study. We define a novel diffusion distance derived from the estimated mobility network, based on which we provide a general model to describe the spatiotemporal spread of infectious diseases with a random diffusion process and a deterministic drift process of the population. We consequently develop a multi-source data fusion method to determine the population flow in epidemic areas. In this method, we first select available subregions in epidemic areas, and then provide solutions to initiate new travel flux among these subregions. To verify our model and method, we analyze the multi-source data from mainland China and obtain a new travel flux triggering scheme in the selected 29 cities with the most active population movements in mainland China. The testable predictions in these selected cities show that reopening the borders in accordance with our proposed travel flux will not cause a second outbreak of COVID-19 in these cities. The finding provides a methodology of re-triggering travel flux during the weakening spread stage of the epidemic.

Before applying the gravity model in main text, we first adjust the data to the same order of magnitude. Here we reduce the value of GDP by 100 times, and then use the mean of it and population size as the "mass" of each city. That is where GDP i and P opulation i are the GDP per capita and resident population in city Ω i , respectively.

The ODE models for parameter estimation
Assume that the infected individuals and net growth rate are homogeneous in each city Ω i , we integrate the original model (3) on the initial outbreak city Ω 0 , and get The divergence theorem derives Note that the right term in this equation is not well defined before we introduce a Dirac vector function δ(x) [2] on the boundary ∂Ω 0 . With this Dirac vector function, the gradient of p(t, x) on the discontinuous boundary ∂Ω 0 is In the early stages of the COVID-19 outbreak in China, the epidemic in Wuhan is far worse than other cities. Thus, we approximate the gradient as Consequently, Therefore, we get a ODE model for the epidemic in the city Ω 0 where γ 0 = −|∂Ω 0 |/|Ω 0 |γ is the total outflow rate from the initial outbreak city Ω 0 .
Before the lockdown in Wuhan, the infected individuals mainly spread from Wuhan to other cities. We generalize the ODE model for city Ω 0 to a general form for each city where γ i is the movement rate of population form the initial outbreak city Ω 0 to another city Ω i .

Source-sink method
Based on the behavior-related growth rate R(t 0 , Ω i , ), we distinguish the subregions into source subregion with R(t 0 , Ω i , ) > 0 and sink subregion with , we first give a basic matrix J with the element where D(Ω i , Ω j ) is the diffusion distance between Ω i and Ω j . And then, we test the patch model (15) using this matrix J as a mobility matrix. For every i = 1, ..., m, we calculate the right term of patch model (15). For where eps > 0 is small enough to ensure that the denominator is not zero. And then, we In this process, we promote the movement of population to subregions with low potential, while ensuring that the elements of the mobility matrix do not exceed the basic matrix to avoid population gathering in specific areas.

Theory Supplement
Here, we list the theoretical results used to determine the available subregions and travel fluxes in epidemic areas. Proof. We decompose the function g(x) by If B = 0 (i.e., g − (x) = 0), then g(x) = 0. We take h(x) = 0.
Proof. When the sample size m are large enough, the law of large numbers yeilds Using the Taylor expansion of f (y), we have where ∇f = ( ∂f ∂x 1 , ∂f ∂x 2 , ..., ∂f ∂x k ) T is the gradient of f and H is the corresponding Hessian.
First, for i = j, Second, for i = j, These equations (A.14-A.16) immediately imply that Similarly, we can calculate d α,δ (x) by Consequently, the shift operator P α,δ , which is the limitation of the transition matrix P m for infinite samples, has the asymptotic expansion as P α,δ φ(x) = 1 d α,δ (x) Ω k α,δ (x, y)φ(y)f (y)dy = φ(x) + 1 4δ Therefore, we have the infinitesimal generator L α of the diffusion as