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Consider a graph where the sites are distributed in space according to aPoisson point process on $\mathbb R^n$. We study a population evolving on thisnetwork, with individuals jumping between sites with a rate which decreasesexponentially in the distance. Individuals give also birth (infection) and die(recovery) at constant rate on each site. First, we construct the process,showing that it is well-posed even when starting from non-bounded initialconditions. Secondly, we prove hydrodynamic limits in a diffusive scaling. Thelimiting process follows a deterministic reaction diffusion equation. We usestochastic homogenization to characterize its diffusion coefficient as thesolution of a variational principle. The proof involves in particular theextension of a classic Kipnis-Varadhan estimate to cope with thenon-reversiblity of the process, due to births and deaths. This work ismotivated by the approximation of epidemics on large networks and the resultsare extended to more complex graphs including percolation of edges.

Branching process and homogeneization for epidemics on spatial random graphs