Last modified: 2014-03-31

#### Abstract

The SIS-model, where the susceptible and infective individuals are disjointcompartments of a total constant population, is the simplest representationfor a disease in which recovery does not provide immunity. Here, we follow Pvan den Driessche and J Watmough (2000): \textit{J. Math. Biol.}, \textbf{40}%, 523-540. The contact rate is a function of the infectives population andthe period of infectivity is incorporated into the model by considering theSIS-Volterra integral equation. Unlike the classical SIS-model, where thevalue $\mathcal{R}_{0}=1$ of the basic reproduction number is a forwardbifurcation, there exist now two treshold parameters $\mathcal{R}%_{0}^{c}\leq \mathcal{R}_{0}^{m},$ $\mathcal{R}_{0}^{m}\geq 1$, and theconsidered model can undergo a backward bifurcation as follows. Thedisease-free equilibrium (DFE) is the only equilibrium and it is globallyasymptotically stable (GAS) when $\mathcal{R}_{0}<\mathcal{R}_{0}^{c}$;there exists only one endemic equilibrium (EE), which is locallyasymptotically stable (LAS) when $\mathcal{R}_{0}>\mathcal{R}_{0}^{m}$ withDFE being unstable when $\mathcal{R}_{0}>1$; for $\mathcal{R}_{0}^{c}<%\mathcal{R}_{0}<1,$ the DFE is LAS and co-exists with at least one LASendemic equilibrium. \ We design a NSFD scheme that preserves positivity andboundedness of the solution as well as the stability properties ofequilibria. \