Critical asymptotic behaviour in the SIR model

TL;DR

This paper analyzes a spatially-dependent SIR epidemic model, establishing convergence to PDE solutions, deriving the final survivor density, and revealing critical asymptotic behaviors in the mean field limit.

Contribution

It introduces a particle system model with spatial infection dependence and proves its convergence to PDEs, highlighting critical behaviors in the asymptotic regime.

Findings

Weak convergence of density fields to PDE solutions

Explicit expression for final survivor density

Identification of critical parameter values in asymptotics

Abstract

This article is devoted to the analysis of a particle system model for epidemics among a finite population with susceptible, infective and removed individuals (SIR). The infection mechanism depends on the relative distance between susceptibles and infected so that an infected individual is more likely to infect nearby sites than those further away. For fixed time, we prove that the density fields weakly converge to the solution of a PDE's system, as the number of particles increases. We find an implicit expression for the final survivor density of the limit equation and we analyze the asymptotics of the microscopic system, by taking first the time and after the number of particles to infinity, showing a critical behaviour for some values of the parameters when the system is set in the mean field regime.