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Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models
Dissertation (MSc)--University of Pretoria, 2012.
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| Format: | Thesis |
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University of Pretoria
2013
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| _version_ | 1867613507115024384 |
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| access_status_str | Open Access |
| author2 | Lubuma, Jean M.-S. |
| author_browse | Lubuma, Jean M.-S. |
| author_facet | Lubuma, Jean M.-S. |
| collection | Thesis |
| dc_rights_str_mv | © 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 Pretoria |
| description | Dissertation (MSc)--University of Pretoria, 2012. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/24917 |
| institution | University of Pretoria (South Africa) |
| last_indexed | 2026-06-10T12:37:14.671Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2013 |
| publishDateRange | 2013 |
| publishDateSort | 2013 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/24917 Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models Lubuma, Jean M.-S. Mureithi, Eunice W. yibadan@yahoo.com Terefe, Yibeltal Adane Sis and sir epidemiological models Nonstandard finite difference scheme Nsfd UCTD Dissertation (MSc)--University of Pretoria, 2012. 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 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. Mathematics and Applied Mathematics unrestricted 2013-09-06T18:49:17Z 2013-05-24 2013-09-06T18:49:17Z 2013-04-17 2012 2013-05-23 Dissertation Terefe, 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 > E13/4/513/gm http://hdl.handle.net/2263/24917 http://upetd.up.ac.za/thesis/available/etd-05232013-115911/ © 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 Pretoria application/pdf University of Pretoria |
| spellingShingle | Sis and sir epidemiological models Nonstandard finite difference scheme Nsfd UCTD Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title | Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title_full | Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title_fullStr | Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title_full_unstemmed | Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title_short | Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models |
| title_sort | bifurcation analysis and nonstandard finite difference schemes for kermack and mckendrick type epidemiological models |
| topic | Sis and sir epidemiological models Nonstandard finite difference scheme Nsfd UCTD |
| url | http://hdl.handle.net/2263/24917 http://upetd.up.ac.za/thesis/available/etd-05232013-115911/ |