Global stability of a discrete SIR epidemic model with saturated incidence rate and death induced by the disease

Abstract

We discuss a numerical discretization for an SIR epidemic model, where the incidence rate is assumed to be saturated. The numerical discretization is performed by employing a nonstandard finite difference (NSFD) method to discretize the continuous SIR epidemic model where the denominator function is chosen such that the scheme maintains the population conservation law. This discretization leads to a numerical scheme which can be considered as a discrete system. The dynamics of the obtained discrete system is then analyzed. It is found that the disease-free equilibrium of the discrete system is globally asymptotically stable if the basic reproduction number is less than or equals to one. On the other hand, if the basic reproduction number is greater than one, then the endemic equilibrium is globally asymptotically stable. Such global stability properties have been confirmed by our numerical simulations. Furthermore, our numerical simulations show that the proposed conservative NSFD is more accurate than the NSFD without considering the population conservation law.