Qualitative behaviour of a model of an SIRS epidemic: stability and permanence. (English) Zbl 1231.34083

Summary: 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.


34C60 Qualitative investigation and simulation of ordinary differential equation models
92D30 Epidemiology
37N25 Dynamical systems in biology
34D20 Stability of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness