Ergodicity and hydrodynamic limits for an epidemic model

TL;DR

This paper investigates epidemic spread in spatially structured populations using microscopic and hydrodynamic approaches, proving convergence of empirical measures to reaction-diffusion systems.

Contribution

It introduces two models for epidemic spread considering different population compositions and derives hydrodynamic limits using the relative entropy method.

Findings

Epidemic potential determined at microscopic level.

Empirical measures converge to reaction-diffusion equations.

Hydrodynamic limits established for both models.

Abstract

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 individuals 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.