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University of Pretoria

1. Terefe, Yibeltal Adane. Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models.

Degree: Mathematics and Applied Mathematics, 2013, University of Pretoria

URL: http://hdl.handle.net/2263/24917

The classical SIR and SIS epidemiological models are
extended by considering the number of adequate contacts per
infective in unit time as a function of the total population in
such a way that this number grows less rapidly as the total
population increases. A diffusion term is added to the SIS model
and this leads to a reactionâ€“diffusion equation, which governs the
spatial spread of the disease. With the parameter R0 representing
the basic reproduction number, it is shown that R0 = 1 is a forward
bifurcation for the SIR and SIS models, with the diseaseâ€“free
equilibrium being globally asymptotic stable when R0 is less than
1. In the case when R0 is greater than 1, for both models, the
endemic equilibrium is locally asymptotically stable and traveling
wave solutions are found for the SIS diffusion model. Nonstandard
finite difference (NSFD) schemes that replicate the dynamics of the
continuous SIR and SIS models are presented. In particular, for the
SIS model, a nonstandard version of the Runge-Kutta method having
high order of convergence is investigated. Numerical experiments
that support the theory are provided. On the other hand the SIS
model is extended to a Volterra integral equation, for which the
existence of multiple endemic equilibria is proved. This fact is
confirmed by numerical simulations.
*Advisors/Committee Members: Lubuma, Jean M.-S. (advisor), Mureithi, Eunice W. (advisor).*

Subjects/Keywords: Sis and sir epidemiological models; Nonstandard finite difference scheme; Nsfd; UCTD

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Terefe, Y. A. (2013). Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models. (Masters Thesis). University of Pretoria. Retrieved from http://hdl.handle.net/2263/24917

Chicago Manual of Style (16^{th} Edition):

Terefe, Yibeltal Adane. “Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models.” 2013. Masters Thesis, University of Pretoria. Accessed November 19, 2019. http://hdl.handle.net/2263/24917.

MLA Handbook (7^{th} Edition):

Terefe, Yibeltal Adane. “Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models.” 2013. Web. 19 Nov 2019.

Vancouver:

Terefe YA. Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models. [Internet] [Masters thesis]. University of Pretoria; 2013. [cited 2019 Nov 19]. Available from: http://hdl.handle.net/2263/24917.

Council of Science Editors:

Terefe YA. Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models. [Masters Thesis]. University of Pretoria; 2013. Available from: http://hdl.handle.net/2263/24917