dc.contributor.supervisor 
Gumel, Abba (Mathematics) 
en 
dc.contributor.author 
Melesse, Dessalegn Yizengaw


dc.date.accessioned 
20100830T15:14:06Z 

dc.date.available 
20100830T15:14:06Z 

dc.date.issued 
20100830T15:14:06Z 

dc.identifier.uri 
http://hdl.handle.net/1993/4086 

dc.description.abstract 
The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and nonperiodic 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 diseasefree equilibrium (DFE) of the resulting SEI<sup>n</sup>RS model is shown to be globallyasymptotically 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 nonlinear Lyapunov function of GohVolterra type) to be globallyasymptotically 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 globallyasymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globallystable 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 nonautonomous SE<sup>m</sup>I<sup>n</sup>RS model is shown to have a globallystable diseasefree solution when the associated reproduction ratio is less than unity. Furthermore, the nonautonomous model has at least one positive (nontrivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the nonautonomous 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. 
en 
dc.format.extent 
1381569 bytes 

dc.format.mimetype 
application/pdf 

dc.language.iso 
en_US 

dc.subject 
infectious diseases 
en 
dc.subject 
Lyapunov function 
en 
dc.subject 
LaSalle's Invariance Principle 
en 
dc.subject 
Reproduction number 
en 
dc.subject 
Comparison Theorem 
en 
dc.subject 
Persistence theory 
en 
dc.subject 
uniformly persistence 
en 
dc.subject 
strongly uniformly persistence 
en 
dc.subject 
endemic equilibrium 
en 
dc.subject 
diseasefree equilibrium 
en 
dc.subject 
global asymptotic stability 
en 
dc.subject 
local asymptotic stability 
en 
dc.title 
Mathematical Analysis of an SEIRS Model with Multiple Latent and Infectious Stages in Periodic and Nonperiodic Environments 
en 
dc.degree.discipline 
Mathematics 
en 
dc.contributor.examiningcommittee 
Lui, Shaun (Mathematics) Wang, Liqun (Statistics) 
en 
dc.degree.level 
Master of Science (M.Sc.) 
en 
dc.description.note 
October 2010 
en 