  Consider a graph where the sites are distributed in space according to a
Poisson point process on $\mathbb R^n$. We study a population evolving on this
network, with individuals jumping between sites with a rate which decreases
exponentially 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 initial
conditions. Secondly, we prove hydrodynamic limits in a diffusive scaling. The
limiting process follows a deterministic reaction diffusion equation. We use
stochastic homogenization to characterize its diffusion coefficient as the
solution of a variational principle. The proof involves in particular the
extension of a classic Kipnis-Varadhan estimate to cope with the
non-reversiblity of the process, due to births and deaths. This work is
motivated by the approximation of epidemics on large networks and the results
are extended to more complex graphs including percolation of edges.
