A nonstandard Volterra difference equation for the SIS epidemiological model
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Date
Authors
Lubuma, Jean M.-S.
Terefe, Yibeltal Adane
Journal Title
Journal ISSN
Volume Title
Publisher
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.
Description
Keywords
SIS model, Volterra integral equation, Dynamics preserving scheme, Nonstandard finite difference (NSFD) scheme
Sustainable Development Goals
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.