Qualitative Behaviour of a Model of an SIRS Epidemic: Stability and Permanence

We consider a classical model of a SIRS epidemic in an open population. The positivity and permanence are studied and explicit formulae are obtained by which the eventual lower bound of the density of infectives can be computed. The stability of the model is studied. We mainly use the Lyapunov functional to established the global stability of disease-free and endemic equilibrium points for both the deterministic and stochastic models. In addition we illustrate the dynamic behaviour of the deterministic and stochastic models via a numerical example.