A nonstandard Volterra difference equation for the SIS epidemiological model

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Authors

Lubuma, Jean M.-S.
Terefe, Yibeltal Adane

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Springer

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.

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Keywords

SIS model, Volterra integral equation, Dynamics preserving scheme, Nonstandard finite difference (NSFD) scheme

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Citation

Lubuma, JM-S & Terefe, YA 2015, 'A nonstandard Volterra difference equation for the SIS epidemiological model', Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A : Matematicas, vol. 109, no.2, pp. 597-602.