We consider an interacting particle process on a graph which, from a macroscopic point of view, looks like $Z^d$ and, at a microscopic level, is a complete graph of degree $N$ (called a patch). There are two birth rates: an inter-patch one $lambda$ a
nd an intra-patch one $phi$. Once a site is occupied, there is no breeding from outside the patch and the probability $c(i)$ of success of an intra-patch breeding decreases with the size $i$ of the population in the site. We prove the existence of a critical value $lambda_{cr}(phi, c, N)$ and a critical value $phi_{cr}(lambda, c, N)$. We consider a sequence of processes generated by the families of control functions ${c_i}_{i in N}$ and degrees ${N_i}_{i in N}$; we prove, under mild assumptions, the existence of a critical value $i_{cr}$. Roughly speaking we show that, in the limit, these processes behave as the branching random walk on $Z^d$ with external birth rate $lambda$ and internal birth rate $phi$. Some examples of models that can be seen as particular cases are given.
We consider two approaches to study the spread of infectious diseases within a spatially structured population distributed in social clusters. According whether we consider only the population of infected individuals or both populations of infected i
ndividuals and healthy ones, two models are given to study an epidemic phenomenon. Our first approach is at a microscopic level, its goal is to determine if an epidemic may occur for those models. The second one is the derivation of hydrodynamics limits. By using the relative entropy method we prove that the empirical measures of infected and healthy individuals converge to a deterministic measure absolutely continuous with respect to the Lebesgue measure, whose density is the solution of a system of reaction-diffusion equations.
