A SIR epidemic model on a refining spatial grid II-Central limit theorem

TL;DR

This paper develops a stochastic SIR epidemic model incorporating spatial heterogeneity, and establishes a central limit theorem showing the stochastic model's deviation from the deterministic PDE model converges to a Gaussian process.

Contribution

It introduces a functional central limit theorem for the spatial stochastic SIR model, linking it to a Gaussian process solution of a stochastic PDE.

Findings

The stochastic model converges to the deterministic PDE as population size grows.

The deviation is characterized by an Ornstein-Uhlenbeck Gaussian process.

The limit process solves a stochastic partial differential equation.

Abstract

A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein-Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.