A nonstandard Volterra difference equation for the SIS epidemiological model

Abstract

By considering the contact rate as a function of infective individuals and by using a general distribution of the infective period, the SIS-model extends to a Volterra integral equation that exhibits complex behaviour such as the backward bifurcation phenomenon. We design a nonstandard finite difference (NSFD) scheme, which is reliable in replicating this complex dynamics. It is shown that the NSFD scheme has no spurious fixed-points compared to the equilibria of the continuous model. Furthermore, there exist two threshold parameters \(\mathcal {R}_0^c\) and \(\mathcal {R}_0^m,\; ~\mathcal {R}_0^c\le 1\le \mathcal {R}_0^m\), such that the disease-free fixed-point is globally asymptotically stable (GAS) for \(\mathcal {R}_0\), the basic reproduction number, less than \(\mathcal {R}_0^c\) and unstable for \(\mathcal {R}_0>1\), while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point for \(\mathcal {R}_0^c<\mathcal {R}_0<1\). A unique GAS endemic fixed-point exists when \(\mathcal {R}_0 >\mathcal {R}_0^m\) and \( \mathcal {R}_0^m <\infty \). Numerical experiments that support the theory are provided.