Multi-patch epidemic models with general exposed and infectious periods

TL;DR

This paper develops a comprehensive mathematical framework for multi-patch epidemic models with general exposed and infectious periods, accounting for migration and spatial transmission, and establishes limit theorems describing their large population behavior.

Contribution

It introduces a novel multi-patch epidemic model with general period distributions and migration, providing rigorous law of large numbers and central limit theorems for the processes involved.

Findings

FLLN limit characterized by Volterra integral equations

Special case reduces to delay differential equations

FCLT describes fluctuations via stochastic Volterra equations

Abstract

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as "from distance" from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.