A stochastic differential equation SIS epidemic model

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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.
Original languageEnglish
Pages (from-to)876-902
Number of pages27
JournalSIAM Journal on Applied Mathematics
Issue number3
Early online date2 Jun 2011
Publication statusPublished - 2011



  • susceptible-infected-susceptible model
  • pneumococcus
  • gonorrhea
  • stationary distribution
  • basic reproduction number
  • persistence
  • extinction
  • stochastic differential equations
  • Brownian motion

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