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A stochastic differential equation SIS epidemic model

Gray, Alison and Greenhalgh, David and Hu, L. and Mao, Xuerong and Pan, Jiafeng (2011) A stochastic differential equation SIS epidemic model. SIAM Journal on Applied Mathematics, 71 (3). pp. 876-902.

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Abstract

In this paper we extend the classical susceptible-infected-susceptible epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals $I(t)$. We then prove that this SDE has a unique global positive solution $I(t)$ and establish conditions for extinction and persistence of $I(t)$. We discuss perturbation by stochastic noise. In the case of persistence we show the existence of a stationary distribution and derive expressions for its mean and variance. The results are illustrated by computer simulations, including two examples based on real-life diseases.