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.
Highlights
In this paper we reconsider a SIS epidemic model with maturation delay developed in [1]: ds d B n n e 1 s i (1) di s i n dnIn this paper we consider a special class of Equation (1) by assuming that the death rate in each stage prior to the adult stage and the disease induced death are negligible, i.e. 1 0
First we show that the discrete system has equilibria which are exactly the same as those of continuous model
It was concluded that the stability conditions of equilibria of the discrete system obtained by the Euler method are consistent with system (2) only if the numerical time-step is relatively small
Summary
In this paper we reconsider a SIS epidemic model with maturation delay developed in [1]: ds d. Using different method of analysis, Kunnawuttipreechachan [7] and the author [8] derived the sufficient conditions of the numerical step-size for equilibria to be asymptotically stable. It was concluded that the stability conditions of equilibria of the discrete system obtained by the Euler method are consistent with system (2) only if the numerical time-step is relatively small. To overcome the dependence of stability condition on the time-step size, we will apply a nonstandard finite difference (NSFD) scheme This method, which is developed by Mickens [9,10], has been applied to various problems; see e.g. We will show that the discrete SIS epidemic model with a delay obtained by the NSFD method maintains the stability properties of equilibria irrespective of the size of numerical time-step. The existence of bifurcation of the discrete model will be investigated
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