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Centre Manifold Theory for some continuous and Discrete Epidemiological models
Thesis (PhD)--University of Pretoria, 2019.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2019
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| _version_ | 1867613448396865536 |
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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 | © 2019 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, 2019. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/72721 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:36:18.633Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| 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:2263/72721 Centre Manifold Theory for some continuous and Discrete Epidemiological models Lubuma, Jean M.-S. u14459142@tuks.co.za Anguelov, Roumen Dukuza, Njengele Kenneth Kennedy UCTD Thesis (PhD)--University of Pretoria, 2019. In mathematical epidemiology, the threshold theory introduced by W.O. Kermack and A.G. McKendrick (1927) can be expressed in terms of the basic reproduction number R0. This is defined as the average number of secondary infections that occur when one infective is introduced into a susceptible host population. In this setting and for many diseases, the prediction of the likelihood of persistence or dying out of the disease within the population reads as follows: the disease-free equilibrium is locally asymptotically stable (LAS) when R0 < 1, it is unstable when R0 > 1 and at least one endemic equilibrium (EE) which is LAS is born in this case. In other words, at R0 = 1, a forward bifurcation occurs. However, some diseases undergo the backward bifurcation phenomenon whereby, for R0 < 1, the LAS disease-free equilibrium coexists with a small positive unstable EE and a large positive LAS EE. In this thesis, we study theoretically, numerically, and computationally the existence of the backward bifurcation phenomenon for dynamical systems, with emphasis on a “simple” SIS model with vaccination and a “complex” malaria model. We re-centre the reduction theorem in C. Castillo-Chavez and B. Song (2004) and highlight its advantage over the legendary power series approximations in the use of the Centre Manifold Theory (CMT). We propose and prove a Centre Manifold-based theorem for the existence of a backward bifurcation for discrete dynamical systems. We construct nonstandard finite difference (NSFD) schemes and prove that they preserve the backward bifurcation property of the continuous models. We make the results more specific for the SIS and malaria models for which we also provide numerical simulations that support the theory. In particular we prove for the malaria model a conjecture by Chitnis et al. (2006) for the existence of the backward bifurcation. Mathematics and Applied Mathematics PhD Unrestricted 2019-12-13T08:07:48Z 2019-12-13T08:07:48Z 2019/09/05 2019 Thesis Dukuza, KKK 2019, Centre Manifold Theory for some continuous and Discrete Epidemiological models, PhD Thesis, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/72721> S2019 http://hdl.handle.net/2263/72721 en © 2019 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 Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title | Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title_full | Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title_fullStr | Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title_full_unstemmed | Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title_short | Centre Manifold Theory for some continuous and Discrete Epidemiological models |
| title_sort | centre manifold theory for some continuous and discrete epidemiological models |
| topic | UCTD |
| url | http://hdl.handle.net/2263/72721 |