Stability and bifurcation of deterministic infectious disease models

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dc.contributor.advisor Supervisor Dr W J Walker en Korobeinikov, Andrei en 2006-11-30T01:19:53Z en 2006-11-30T01:19:53Z en 2001 en
dc.identifier.citation Thesis (PhD--Mathematics)--University of Auckland, 2001. en
dc.identifier.uri en
dc.description Subscription resource available via Digital Dissertations en
dc.description.abstract Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors. en
dc.language.iso en en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA972143 en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. en
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dc.source.uri en
dc.title Stability and bifurcation of deterministic infectious disease models en
dc.type Thesis en Mathematics en The University of Auckland en Doctoral en PhD en
dc.subject.marsden Fields of Research::230000 Mathematical Sciences en
dc.rights.holder Copyright: The author en
pubs.local.anzsrc 01 - Mathematical Sciences en Faculty of Science en
dc.identifier.wikidata Q111963940

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