Appears in collection : 2022 - T1 - WS3 - Mathematical models in ecology and evolution

Joint work with Michele Salvi.

We are interested in population dynamics and epidemics for large random metapopulations. The sites of the metapopulation are described by a Poisson point process on the plane and transition rates between the sites depend on their distances. In such a non-homogeneous context, when the number of sites in a given box becomes large, homogenization occurs, leading to a non trivial diffusion of coeficient and spread of epidemics. Our motivations come from the spread of epidemics on networks (farms, cities, patches...) We will introduce an individual-based model including births, deaths and contaminations. We will first justify the existence of such a stochastic process starting from a (spatially) unbounded population distribution, which requires to control what can come from large distances. We will then prove the convergence of the renormalized stochastic process to a reaction diffusion models, with homogenized diffusion coefficients. We may discuss further new and multi-scaling or extensions to more complex large random graphs.