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Mathematical Analysis of an SEIRS Model with Multiple Latent and Infectious Stages in Periodic and Non-periodic EnvironmentsMelesse, Dessalegn Yizengaw 30 August 2010 (has links)
The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and non-periodic environments. The classical SEIRS model, with standard incidence function, is, first of all, extended to incorporate multiple infectious stages. Using Lyapunov function theory and LaSalle's Invariance Principle, the disease-free equilibrium (DFE) of the resulting SEI<sup>n</sup>RS model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, this model has a unique endemic equilibrium point (EEP), which is shown (using a non-linear Lyapunov function of Goh-Volterra type) to be globally-asymptotically stable for a special case. The SEI<sup>n</sup>RS model is further extended to incorporate arbitrary number of latent stages. A notable feature of the resulting SE<sup>m</sup>I<sup>n</sup>RS model is that it uses gamma distribution assumptions for the average waiting times in the latent (m) and infectious (n) stages. Like in the case of the SEI<sup>n</sup>RS model, the SE<sup>m</sup>I<sup>n</sup>RS model also has a globally-asymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globally-stable for a special case) when the threshold exceeds unity. The SE<sup>m</sup>I<sup>n</sup>RS model is further extended to incorporate the effect of periodicity on the disease transmission dynamics. The resulting non-autonomous SE<sup>m</sup>I<sup>n</sup>RS model is shown to have a globally-stable disease-free solution when the associated reproduction ratio is less than unity. Furthermore, the non-autonomous model has at least one positive (non-trivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the non-autonomous model, the disease will always persist in the population whenever the reproduction ratio is greater than unity. One of the main mathematical contributions of this thesis is that it shows that adding multiple latent and infectious stages, gamma distribution assumptions (for the average waiting times in these stages) and periodicity to the classical SEIRS model (with standard incidence) does not alter the main qualitative dynamics (pertaining to the persistence or elimination of the disease from the population) of the SEIRS model.
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Mathematical modeling and analysis of HIV/AIDS control measuresGbenga, Abiodun J. January 2012 (has links)
>Magister Scientiae - MSc / In this thesis, we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyse a math-
ematical model that describes the dynamics of HIV infection among the im-
migrant youths and intervention that can minimize or prevent the spread of
the disease in the population. In particular, we are interested in the effects of
public-health education and of parental care.We consider existing models of public-health education in HIV/AIDS epidemi-ology, and provide some new insights on these. In this regard we focus atten-tion on the papers [b] and [c], expanding those researches by adding sensitivity analysis and optimal control problems with their solutions.Our main emphasis will be on the effect of parental care on HIV/AIDS epidemi-ology. In this regard we introduce a new model. Firstly, we analyse the model without parental care and investigate its stability and sensitivity behaviour.We conduct both qualitative and quantitative analyses. It is observed that
in the absence of infected youths, disease-free equilibrium is achievable and is
asymptotically stable. Further, we use optimal control methods to determine
the necessary conditions for the optimality of intervention, and for disease
eradication or control. Using Pontryagin’s Maximum Principle to check the
effects of screening control and parental care on the spread of HIV/AIDS, we
observe that parental care is more effective than screening control. However,
the most efficient control strategy is in fact a combination of parental care and screening control. The results form the central theme of this thesis, and are included in the manuscript [a] which is now being reviewed for publication.
Finally, numerical simulations are performed to illustrate the analytical results.
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