Functional Limit Theorems for Non-Markovian Epidemic Models

TL;DR

This paper establishes functional limit theorems for non-Markovian epidemic models with general infectious period distributions, providing deterministic and stochastic process limits as population size grows.

Contribution

It introduces a novel framework for analyzing non-Markovian epidemic models using Volterra integral equations and Gaussian process limits.

Findings

Proves a functional law of large numbers for non-Markovian epidemic models.

Derives a functional central limit theorem with Gaussian process limits.

Provides a Poisson random measures representation for the stochastic fluctuations.

Abstract

We study non-Markovian stochastic epidemic models (SIS, SIR, SIRS, and SEIR), in which the infectious (and latent/exposing, immune) periods have a general distribution. We provide a representation of the evolution dynamics using the time epochs of infection (and latency/exposure, immunity). Taking the limit as the size of the population tends to infinity, we prove both a functional law of large number (FLLN) and a functional central limit theorem (FCLT) for the processes of interest in these models. In the FLLN, the limits are a unique solution to a system of deterministic Volterra integral equations, while in the FCLT, the limit processes are multidimensional Gaussian solutions of linear Volterra stochastic integral equations. In the proof of the FCLT, we provide an important Poisson random measures representation of the diffusion-scaled processes converging to Gaussian components driving the limit process.