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 ${\mathbb{T}}^1=(0,1{]}^d$, where the individuals may migrate from node to another. The migration processes in either of the two states are assumed to be Markovian. We establish a functional law of large numbers by letting jointly $N$ the initial approximate number of individuals on each node go to infinity and $\epsilon$ the mesh size of the grid go to zero. The limit is a system of parabolic PDE/integral equations. The constraint on the speed of convergence of the parameters $N$ and $\epsilon$ is that $N{\epsilon }^d\to \mathrm{\infty }$ as $(N,\epsilon )\to (+\mathrm{\infty },0)$.