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 Rc 0 andRm0 , Rc 0 ≤ 1 ≤ Rm0 , such that the disease-free fixed-point is globally asymptotically stable (GAS) for R0, the basic reproduction number, less than Rc 0 and unstable for R0 > 1, while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point forRc 0 < R0 < 1. A unique GAS endemic fixed-point exists whenR0 > Rm0 andRm0 < ∞. Numerical experiments that support the theory are provided.