## Mathematical Analysis of an SEIRS Model with Multiple Latent and Infectious Stages in Periodic and Non-periodic Environments

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##### Date

2010-08-30T15:14:06Z

##### Authors

Melesse, Dessalegn Yizengaw

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##### Abstract

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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##### Keywords

infectious diseases, Lyapunov function, LaSalle's Invariance Principle, Reproduction number, Comparison Theorem, Persistence theory, uniformly persistence, strongly uniformly persistence, endemic equilibrium, disease-free equilibrium, global asymptotic stability, local asymptotic stability