Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model

dc.contributor.advisorLubuma, Jean M.-S.en
dc.contributor.emailu29542538@tuks.co.zaen
dc.contributor.postgraduateTerefe, Yibeltal Adane
dc.date.accessioned2015-11-25T09:53:38Z
dc.date.available2015-11-25T09:53:38Z
dc.date.created2015/09/01en
dc.date.issued2015en
dc.descriptionThesis (PhD)--University of Pretoria, 2015.en
dc.description.abstractWe design and investigate the reliability of various nonstandard nite di erence (NSFD) schemes for the SIS epidemiological model in three di erent settings. For the classical SIS model, we construct two new NSFD schemes which faithfully replicate the property of the continuous model of having the parameter R0, the basic reproduction number, as a threshold to determine the stability properties of equilibrium points: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R0 1; it is unstable when R0 > 1 and there appears a unique GAS endemic equilibrium (EE) in this case. These schemes also preserve the positivity and boundedness properties of solutions of the classical SIS model. The schemes are further used to derive NSFD schemes for the SIS-di usion model which constitutes the second setting of the study. The designed NSFD schemes are dynamically consistent with the global asymptotic stability of the disease-free equilibrium for R0 1 and the instability of the disease-free equilibrium for R0 > 1. In the latter case, the schemes replicate the global asymptotic stability of the endemic equilibrium. Positivity and boundedness properties of solutions of the SIS-di usion model are also preserved by the NSFD schemes. In a third step, the classical SIS model is extended into a SIS-Volterra integral equation model in which the contact rate is a function of fraction of infective individuals and allows a distributed period of infectivity. The qualitative analysis is now based on two threshold parameters Rc 0 1 Rm0 . The system can undergo the backward bifurcation phenomenon as follows. The DFE is the only equilibrium and it is GAS when R0 < Rc 0; there exists only one EE, which is GAS when R0 > Rm0 with the DFE being unstable when R0 > 1; for Rc 0 < R0 < 1, the DFE is locally asymptotically stable (LAS) and coexists with at least one LAS endemic equilibrium. We design a NSFD scheme and prove theoretically and computationally that it preserves the above-stated stability properties of equilibria as well as positivity and boundedness of the solutions of the continuous model.en
dc.description.availabilityUnrestricteden
dc.description.degreePhDen
dc.description.departmentMathematics and Applied Mathematicsen
dc.description.librariantm2015en
dc.identifier.citationTerefe, YA 2015, Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/50800>en
dc.identifier.otherS2015en
dc.identifier.urihttp://hdl.handle.net/2263/50800
dc.language.isoenen
dc.publisherUniversity of Pretoriaen_ZA
dc.rights© 2015 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria.en
dc.subjectUCTDen
dc.titleConstructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological modelen
dc.typeThesisen

Files

Original bundle

Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
Terefe_Constructive_2015.pdf
Size:
1.37 MB
Format:
Adobe Portable Document Format
Description:
Thesis