Modeling of HIV, SIR and SIS epidemics on time scales and oscillation theory
"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.
Missouri University of Science and Technology
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