A Nonstandard Finite Difference Scheme for SIS Epidemic Model with Delay: Stability and Bifurcation Analysis

Agus Suryanto

Department of Mathematics, Faculty of Sciences, Brawijaya University, Malang, Indonesia.

DOI: 10.4236/am.2012.36080                

Abstract

A numerical scheme for a SIS epidemic model with a delay is constructed by applying a nonstandard finite difference (NSFD) method. The dynamics of the obtained discrete system is investigated. First we show that the discrete system has equilibria which are exactly the same as those of continuous model. By studying the distribution of the roots of the characteristics equations related to the linearized system, we can provide the stable regions in the appropriate parameter plane. It is shown that the conditions for those equilibria to be asymptotically stable are consistent with the continuous model for any size of numerical time-step. Furthermore, we also establish the existence of Neimark-Sacker bifurcation (also called Hopf bifurcation for map) which is controlled by the time delay. The analytical results are confirmed by some numerical simulations.

Keywords

SIS Epidemic Model with Delay; Stability; Nonstandard Finite Difference Method; Neimark-Sacker (Hopf) Bifurcation

Conflicts of Interest

The authors declare no conflicts of interest.

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