Stability of a Stochastic Model of an SIR Epidemic with Vaccination

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an analogue (mathcal{R}_sigma) of the basic reproduction number (mathcal{R}_0) of the underlying deterministic model, with (mathcal{R}_sigma le mathcal{R}_0). We prove that the disease-free equilibrium is almost sure exponentially stable if (mathcal{R}_sigma <1).