Epidemic Models with Varying Infectivity on a Refining Spatial Grid—I—The SI Model

We consider a space–time SI epidemic model with infection age dependent infectivity and non-local infections constructed on a grid of the torus Td=[0,1)d, where the individuals may migrate from node to node. The migration processes in either of the two states are assumed to be Markovian. We establish a functional law of large numbers by letting the initial approximate number of individuals on each node, N, to go to infinity and the mesh size of the grid, ε, to go to zero jointly. The limit is a system of parabolic PDE/integral equations. The constraint on the speed of convergence of the parameters N and ε is that Nεd→∞ as (N,ε)→(+∞,0).