Physica A 354 (2005) 111–126 Stability of a stochastic SIR system Elisabetta Tornatore, Stefania Maria Buccellato, Pasquale Vetro à Dipartimento di Matematica ed Applicazioni, Universita´ di Palermo, via Archirafi, 34, 90123 Palermo, Italia Received 7 July 2004; received in revised form 1 December 2004 Available online 21 April 2005 Abstract We propose a stochastic SIR model with or without distributed time delay and we study the stability of disease-free equilibrium. The numerical simulation of the stochastic SIR model shows that the introduction of noise modifies the threshold of system for an epidemic to occur and the threshold stochastic value is found. r 2005 Elsevier B.V. All rights reserved. PACS: 92D30 Keywords: SIR model; Delay SIR model; Lyapunov function; Stochastic process; Numerical simulation; Stochastic stability 1. Introduction The SIR models of the spread of infectious diseases are studied by several authors. For the age-structured SIR epidemic model we have results of existence, uniqueness and stability of epidemic equilibrium [1], for models with diffusion on R we have results of geo-temporal spread of the diseases [2–8]. We consider the following ARTICLE IN PRESS www.elsevier.com/locate/physa 0378-4371/$-see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.02.057 à Corresponding author. E-mail address: vetro@math.unipa.it (P. Vetro).