Doctoral Dissertations

Gülşah Yeni

Abstract

"We study higher dimensional systems of first order dynamic equations on time scales together with their applications. In particular, we focus on epidemic models such as HIV (Human Immunodeficiency Virus), SIS (Susceptible-Infected-Susceptible) and SIR (Susceptible-Infected-Recovered).

First, we generalize the early studied continuous three dimensional linear model of drug therapy for HIV-1 decline on time scales in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. We compare these models to explore the impact of the theory of time scales. After fitting the models to the data collected at a clinical trial using nonlinear regression analysis, we show that the discrete systems result in the best fit. We extend our work, in which the efficacy of the drug therapy is assumed to be perfect, to the presence of combined imperfect drug therapy, and derive the unique solution for the model on time scales. We also discuss the stability of the trivial solution of this model on the set of integers.

Motivated by the fact that between discrete and continuous models of HIV-1 dynamics, the former is more appropriate, we formulate and solve two dimensional SIS and SIR epidemic models with nonlinear incidence and time dependent coefficients on time scales. Later on, we discuss the asymptotic behavior of susceptibles and infectives. In addition, we study three dimensional discrete SIR models with nonlinear incidence and time independent coefficients. Specifically, we show the local stability and global stability of equilibria by the linearization method and constructing a suitable Lyapunov function.

In all the work above, we show the applications of positive solutions of higher dimensional systems in epidemiology. Finally, we investigate four dimensional dynamic systems, in which solutions are classified based on the signs of their components, and find the criteria to ensure that these systems are oscillatory and nonoscillatory"--Abstract, page iv.

Advisor(s)

Akin, Elvan

Committee Member(s)

Bohner, Martin, 1966-
Dosla, Zuzana
Charatonik, W. J.
Gelles, Gregory M.

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

Missouri University of Science and Technology

Publication Date

Fall 2019

Journal article titles appearing in thesis/dissertation

  • Continuous and discrete modeling of HIV-1 decline on therapy
  • On exact solutions to epidemic dynamic models
  • Stability of discrete SIR models
  • Oscillation criteria for four-dimensional time-scale systems
  • Oscillation and nonoscillation criteria for four dimensional advanced and delay time-scale systems

Pagination

x, 123 pages

Note about bibliography

Includes bibliographic references.

Rights

© 2019 Gülşah Yeni, All rights reserved.

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 11656

Electronic OCLC #

1139525678

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Included in

Mathematics Commons

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