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A stochastic differential equation SIS epidemic model. (English) Zbl 1263.34068
Authors’ 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.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34F05 Ordinary differential equations and systems with randomness 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 92D25 Population dynamics (general) 92D30 Epidemiology
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