Functional Central Limit Theorem and SPDE for epidemic model with memory of the last infection and waning immunity

TL;DR

This paper establishes a functional central limit theorem and derives an SPDE for a stochastic epidemic model with memory, waning immunity, and varying infectivity, providing a detailed understanding of its fluctuations around the deterministic limit.

Contribution

It introduces the first FCLT for a measure-valued epidemic model with memory and waning immunity, and derives an associated SPDE for the fluctuations.

Findings

FCLT captures fluctuations around the deterministic limit.

Limit is characterized by a nonlinear stochastic integral equation.

Derived an SPDE describing the fluctuations in the model.

Abstract

We study the fluctuations of a stochastic epidemic model with memory of the last infections, varying infectivity, and waning immunity, as introduced in Guerin and Zotsa-Ngoufack:arXiv preprint arXiv:2505.00601. The dynamics of the epidemic model are described by a measure-valued process with respect to infection age and individual traits. The Functional Law of Large Numbers (FLLN) is formulated as an integral equation, which is solved by a deterministic measure. In this article, we establish the Functional Central Limit Theorem (FCLT), capturing the fluctuations of the stochastic model around its deterministic limit. The limit of the FCLT is given by a nonlinear stochastic integral equation which is solved by a random signed-measure. We further derive the weak solution in the form of a stochastic partial differential equation (SPDE) and propose an alternative representation of the FCLT, as fluctuations in the average total force of infection and average susceptibility.