Abstract
Consider a graph where the sites are distributed in space according to a Poisson point process on . We study a population evolving on this network, with individuals jumping between sites with a rate which decreases exponentially in the distance. Individuals also give 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-reversibility 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 those obtained by percolation of edges.
Funding Statement
This work was partially funded by the ANR Cadence ANR-16-CE32-0007 and Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole polytechnique-MNHN-F.X and by the MUR Excellence Department Project MatMod@TOV, CUP E83C23000330006, awarded to the Department of Mathematics, University of Rome Tor Vergata.
Acknowledgments
The authors are very grateful to Jerome Coville, Alessandra Faggionato and Chi Tran Viet for stimulating discussions and relevant suggestions.
The authors would also like to thank two anonymous referees for an extremely careful check of the first version of the manuscript and for helping to improve it.
M.S. also thanks the INdAM unit GNAMPA.
Citation
Vincent Bansaye. Michele Salvi. "Branching processes and homogenization for epidemics on spatial random graphs." Electron. J. Probab. 29 1 - 37, 2024. https://doi.org/10.1214/24-EJP1175
Information