A Natural Stochastic SIS Model, Analysis of Moments and Comparison of Different Perturbation Techniques

Abstract

In this study, a new and natural way of constructing a stochastic Susceptible-Infected-Susceptible (SIS) model is proposed. This approach is natural in the sense that the disease transmission rate, $β$, is substituted with a generic, almost surely non-negative one-dimensional diffusion. The condition $β\geq 0$ is essential in the deterministic model but generally overlooked in stochastic counterparts (see [12, 16]). Under different conditions on the parameters, the dynamics of the infected population such as boundedness, extinction, and persistence are identified. The new stochastic model agrees with its deterministic version, where the basic reproduction number $R^D_0$ determines the limiting dynamics: extinction when $R_0^D < 1$ and persistence when $R_0^D > 1$. A novel analytic technique is also provided to approximate the expectation of any well-behaved function of the infected population, including its moments, using an increasing power of correction terms. This is useful since the average dynamics of stochastic SIS models are not tractable due to their nonlinearity. Finally, using the first-order correction terms, two different perturbations with the same expectations: (1.4) performed in [12] and the Cox-Ingersoll-Ross (CIR) perturbation proposed here are compared in terms of their expected effect on the infected population dynamics. This comparison provides insight into how different small perturbations affect the overall dynamics of the model.