Central limit theorem for a spatial stochastic epidemic model with mean field interaction

TL;DR

This paper establishes a central limit theorem for a spatial stochastic epidemic model with mean field interactions, showing that fluctuations around the law of large numbers limit converge to a linear stochastic PDE.

Contribution

It introduces a novel semigroup and Hilbert space approach to prove the CLT for a complex spatial epidemic model with mean field interactions.

Findings

Empirical measure converges to a nonlinear McKean-Vlasov equation.

Fluctuations converge to a solution of a linear stochastic PDE.

Method provides a new proof technique distinct from coupling approaches.

Abstract

In this article, we study an interacting particle system in the context of epidemiology where the individuals (particles) are characterized by their position and infection state. We begin with a description at the microscopic level where the displacement of individuals is driven by mean field interactions and state-dependent diffusion, whereas the epidemiological dynamic is described by the Poisson processes with an infection rate based on the distribution of other nearby individuals, also of the mean-field type. Then under suitable assumptions, a form of law of large numbers has been established to show that the associated empirical measure to the above system converges to the law of the unique solution of a nonlinear McKean-Vlasov equation. As a natural follow-up question, we study the fluctuation of this stochastic system around its limit. We prove that this fluctuation process converges to a limit process, which can be characterized as the unique solution of a linear stochastic PDE. Unlike the existing literature using a coupling approach to prove the central limit theorem for interacting particle systems, the main idea in our proof is to use a semigroup formalism and some appropriate estimates to directly study the linearized evolution equation of the fluctuation process in a suitable weighted Sobolev space and follows a Hilbertian approach.