Global Threshold Dynamics of a Stochastic Differential Equation SIS Model

TL;DR

This paper analyzes the global behavior of a stochastic SIS epidemic model, establishing threshold theorems for disease extinction and persistence, and revealing how noise can amplify disease severity compared to deterministic models.

Contribution

It introduces stochastic threshold theorems using a stochastic basic reproduction number and explores the impact of noise on disease prevalence and stability in the SIS model.

Findings

Disease dies out if R_0^S<1 with probability one.

Existence of a unique invariant density when R_0^S>1.

Stochastic prevalence exceeds deterministic prevalence when R_0^D>2.

Abstract

In this paper, we further investigate the global dynamics of a stochastic differential equation SIS (Susceptible-Infected-Susceptible) epidemic model recently proposed in [A. Gray et al., SIAM. J. Appl. Math., 71 (2011), 876-902]. We present a stochastic threshold theorem in term of a \textit{stochastic basic reproduction number} $R_0^S:$ the disease dies out with probability one if $R_0^S<1,$ and the disease is recurrent if $R_0^S\geqslant1.$ We prove the existence and global asymptotic stability of a unique invariant density for the Fokker-Planck equation associated with the SDE SIS model when $R_0^S>1.$ In term of the profile of the invariant density, we define a \textit{persistence basic reproduction number} $R_0^P$ and give a persistence threshold theorem: the disease dies out with large probability if $R_0^P\leqslant1,$ while persists with large probability if $R_0^P>1.$ Comparing the \textit{stochastic disease prevalence} with the \textit{deterministic disease prevalence}, we discover that the stochastic prevalence is bigger than the deterministic prevalence if the deterministic basic reproduction number $R_0^D>2.$ This shows that noise may increase severity of disease. Finally, we study the asymptotic dynamics of the stochastic SIS model as the noise vanishes and establish a sharp connection with the threshold dynamics of the deterministic SIS model in term of a \textit{Limit Stochastic Threshold Theorem}.