  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 $\varepsilon$ 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 $\varepsilon$ is that $N\varepsilon^d \to \infty$ as
$(N, \varepsilon)\to (+\infty, 0)$.
