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Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model
Thesis (PhD)--University of Pretoria, 2015.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2015
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| _version_ | 1867613553991614464 |
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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 | © 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. |
| description | Thesis (PhD)--University of Pretoria, 2015. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/50800 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:37:58.997Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2015 |
| publishDateRange | 2015 |
| publishDateSort | 2015 |
| 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:2263/50800 Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model Lubuma, Jean M.-S. u29542538@tuks.co.za Terefe, Yibeltal Adane UCTD Thesis (PhD)--University of Pretoria, 2015. We 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. tm2015 Mathematics and Applied Mathematics PhD Unrestricted 2015-11-25T09:53:38Z 2015-11-25T09:53:38Z 2015/09/01 2015 Thesis Terefe, 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> S2015 http://hdl.handle.net/2263/50800 en © 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. application/pdf University of Pretoria |
| spellingShingle | UCTD Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title | Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title_full | Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title_fullStr | Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title_full_unstemmed | Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title_short | Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model |
| title_sort | constructive treatment of reaction diffusion and volterra integral equations for the sis epidemiological model |
| topic | UCTD |
| url | http://hdl.handle.net/2263/50800 |