Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models

dc.contributor.advisorLubuma, Jean M.-S.en
dc.contributor.advisorMureithi, Eunice W.en
dc.contributor.emailyibadan@yahoo.comen
dc.contributor.postgraduateTerefe, Yibeltal Adane
dc.date.accessioned2013-09-06T18:49:17Z
dc.date.available2013-05-24en
dc.date.available2013-09-06T18:49:17Z
dc.date.created2013-04-17en
dc.date.issued2012en
dc.date.submitted2013-05-23en
dc.descriptionDissertation (MSc)--University of Pretoria, 2012.en
dc.description.abstractThe 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 reaction–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 disease–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.en
dc.description.availabilityunrestricteden
dc.description.departmentMathematics and Applied Mathematicsen
dc.identifier.citationTerefe, YA 2012, Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://hdl.handle.net/2263/24917 >en
dc.identifier.otherE13/4/513/gmen
dc.identifier.upetdurlhttp://upetd.up.ac.za/thesis/available/etd-05232013-115911/en
dc.identifier.urihttp://hdl.handle.net/2263/24917
dc.language.isoen
dc.publisherUniversity of Pretoriaen_ZA
dc.rights© 2012 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 Pretoriaen
dc.subjectSis and sir epidemiological modelsen
dc.subjectNonstandard finite difference schemeen
dc.subjectNsfden
dc.subjectUCTDen_US
dc.titleBifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological modelsen
dc.typeDissertationen

Files

Original bundle

Now showing 1 - 1 of 1
Loading...
Thumbnail Image
Name:
Terefe_Bifurcation_2012.pdf
Size:
3.82 MB
Format:
Adobe Portable Document Format
Description: