# Limiting behaviour of an SIS epidemic model with environmental stochasticity

Greenhalgh, David and Gray, Alison and Mao, Xuerong and Pan, Jiafeng (2012) Limiting behaviour of an SIS epidemic model with environmental stochasticity. In: 13th WSEAS International Conference on Mathematics and Computers in Biology and Chemistry, 2012-06-13 - 2012-09-15. (Unpublished)

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## Abstract

In this talk we extend the classical SIS (susceptible-infected-susceptible) epidemic model from a deterministic one to a stochastic one and formulate it as a stochastic differential equation (SDE) for I(t), the number of infectious individuals at time t. An SIS model is an epidemic model in which a typical individual starts off as susceptible, at some stage catches the disease and after an infectious period becomes susceptible again. Such models are often used for sexually transmitted diseases such as gonorrhoea, or bacterial diseases such as pneumococcus. We survey some relevant deterministic and stochastic models in the literature. We then formulate our basic model. The stochasticity is introduced as a Brownian motion in the disease transmission coefficient (equivalently in the contact rate of infected individuals). This models the effect of random environmental variation. After deriving the SDE for the spread of the disease we then prove that this SDE has a unique positive solution. For the deterministic model classical results show that there is a unique threshold value R0D, the deterministic basic reproduction number, such that if R0D is less than or equal to one then the disease will die out and if R0D exceeds one then the disease tends to a unique endemic equilibrium. We show that for the stochastic model there is a smaller threshold value R0S and provided that a condition involving the variance of the stochastic noise is satisfied then the disease will die out almost surely (a.s.) for R0S<1. We conjecture that in fact the variance condition is not necessary. If R0S>1 then we show that the disease will fluctuate about a strictly positive level a.s. We discuss the connection between some limiting values of the stochastic threshold R0S and the deterministic threshold R0D. We then show that if R0S>1 the SDE SIS model has a unique non-zero stationary distribution and derive expressions for the mean and variance of this stationary distribution. All the theoretical results are illustrated and confirmed by numerical simulations. We finish by discussing two real-life examples: first gonorrhoea amongst homosexuals and second pneumococcus amongst Scottish children under two years old.