A SIR Stochastic Epidemic Model in Continuous Space: Law of Large Numbers and Central Limit Theorem

TL;DR

This paper models the spatial spread of an epidemic using a stochastic SIR model in continuous space, deriving large population limits and analyzing fluctuations via the central limit theorem, with results expressed as stochastic PDEs.

Contribution

It introduces a stochastic SIR model with spatial movement, establishes large population approximations, and characterizes fluctuations through a distribution-valued Ornstein-Uhlenbeck process.

Findings

Large population limit described by reaction-diffusion equations

Fluctuations characterized by a stochastic PDE

Distribution-valued Ornstein-Uhlenbeck process derived

Abstract

The impact of spatial structure on the spread of an epidemic is an important issue in the propagation of infectious diseases. Recent studies, both deterministic and stochastic, have made it possible to understand the importance of the movement of individuals in a population on the persistence or extinction of an endemic disease. In this paper we study a compartmental SIR stochastic epidemic model for a population that moves on $R^{d}$ following SDEs driven by independent Brownian motions. We define the sequences of empirical measures, which describe the evolution of the positions of the susceptible, infected and removed individuals. Next, we obtain large population approximations of those sequence of measures, as weak solution of a system of reaction-diffusion equations. Finaly we study the fluctuations of the stochastic model around its large population limit via the central limit theorem. The limit is a distribution valued Ornstein-Uhlenbeck process and can be represented as the solution of system of stochastic partial differential equations.