PDE model for multi-patch epidemic models with migration and infection-age dependent infectivity

TL;DR

This paper develops a PDE-based framework for multi-patch epidemic models incorporating migration and infection-age dependent infectivity, providing a rigorous law of large numbers and a detailed PDE system for infection age densities.

Contribution

It introduces a novel PDE model for multi-patch epidemic dynamics with migration and age-dependent infectivity, derived from a stochastic process and large population limit.

Findings

Established a functional law of large numbers for the epidemic process.

Derived a PDE system describing the density of infected individuals by infection age.

Provided a mathematical foundation for analyzing complex epidemic interactions across patches.

Abstract

We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infectivity while the interactive infection process depends on the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infectivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals at each time that have been infected for a certain amount of time. From the limits, we derive a PDE model for the density of the number of infected individuals with respect to the infection age, which is a systems of linear PDE equations with a boundary condition that is determined by a set of integral equations.