Weak Convergence of a Seasonally Forced Stochastic Epidemic Model

TL;DR

This paper extends classical epidemic process convergence results to models with explicit seasonal time dependence, demonstrating weak convergence to mean-field ODEs and diffusion processes as population size grows.

Contribution

It generalizes weak convergence results to seasonally forced stochastic epidemic models with explicit time dependence.

Findings

Weak convergence to mean-field ODEs as population size increases.

Fluctuations converge to a diffusion process under proper scaling.

Extends Kurtz's classical results to time-inhomogeneous models.

Abstract

In this study we extend the results of Kurtz (1970,1971) to show the weak convergence of epidemic processes that include explicit time dependence, specifically where the transmission parameter,$\beta(t)$, carries a time dependency. We first show that when population size goes to infinity, the time inhomogeneous process converges weakly to the solution of the mean-field ODE. Our second result is that, under proper scaling, the central limit type fluctuations converge to a diffusion process.