Functional law of large numbers and PDEs for epidemic models with infection-age dependent infectivity

TL;DR

This paper establishes a functional law of large numbers for epidemic models with infection-age dependent infectivity, describing the epidemic dynamics through deterministic PDEs and integral equations, and analyzes equilibrium points in the SIS model.

Contribution

It introduces a novel FLLN for epidemic models with infection-age dependent infectivity, linking stochastic processes to PDEs and integral equations, and characterizes equilibrium points.

Findings

FLLN limits are deterministic integral equations.

PDEs characterize the epidemic evolution under regularity conditions.

Equilibrium points are characterized for the SIS model with age-dependent infectivity.

Abstract

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time of infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.