Global threshold dynamics of a stochastic differential equation SIS model

In this paper, we further investigate the global dynamics of a stochastic differential equation SIS (Susceptible-Infected-Susceptible) epidemic model recently proposed in Gray et al. (2011) [8]. We present a stochastic threshold theorem in term of a stochastic basic reproduction number R0S: the disease dies out with probability one if R0S<1, and the disease is recurrent if R0S⩾1. We prove the existence and global asymptotic stability of a unique invariant density for the Fokker-Planck equation associated with the SDE SIS model when R0S>1. In term of the profile of the invariant density, we define a persistence basic reproduction number R0P and give a persistence threshold theorem: the disease dies out with large probability if R0P⩽1, while persists with large probability if R0P>1. Comparing the stochastic disease prevalence with the deterministic disease prevalence, we discover that the stochastic prevalence is bigger than the deterministic prevalence if the deterministic basic reproduction number R0D>2. This shows that noise may increase severity of disease. Finally, we study the asymptotic dynamics of the stochastic SIS model as the noise vanishes and establish a sharp connection with the threshold dynamics of the deterministic SIS model in term of a Limit Stochastic Threshold Theorem.