Electronic Thesis and Dissertation Repository


Doctor of Philosophy


Applied Mathematics


Pei Yu and Lindi Wahl


Recurrent disease, characterized by repeated alternations between acute relapse and long re- mission, can be a feature of both common diseases, like ear infections, and serious chronic diseases, such as HIV infection or multiple sclerosis. Due to their poorly understood etiology and the resultant challenge for medical treatment and patient management, recurrent diseases attract much attention in clinical research and biomathematics. Previous studies of recurrence by biomathematicians mainly focus on in-host models and generate recurrent patterns by in- corporating forcing functions or stochastic elements. In this study, we investigate deterministic in-host models through the qualitative analysis of dynamical systems, to reveal the possible intrinsic mechanisms underlying disease recurrence. Recurrence in HIV infection is referred to as “viral blips”, that is, transient periods of high viral replication separated by long periods of quiescence. A 4-dimensional HIV antioxidant- therapy model exhibiting viral blips is studied using bifurcation theory. Four conditions for the existence of viral blips in a deterministic in-host model are proposed. Guided by the four con- ditions, the simplest 2-dimensional infection model which shows recurrence is obtained. One key point for recurrence is identified, that is an increasing and saturating infectivity function. Furthermore, Hopf and generalized Hopf bifurcations, Bogdanov-Takens bifurcation, and ho- moclinic bifurcation are proved to exist in this 2-dimensional model. Bogdanov-Takens bifur- cation and homoclinic bifurcation provide a new mechanism for generating recurrence. From the viewpoint of modelling, the increasing and saturating infectivity function gives rise to a convex incidence rate, which further induces backward bifurcation and Hopf bifurcation, and allows the infection model to exhibit rich dynamical behavior, such as bistability, recurrence, and regular oscillation. The relapse-remission cycle in autoimmune disease is investigated based on a regulatory T cell model. By introducing a newly discovered class of regulatory T cells, Hopf bifurcation oc- curs in the autoimmune model with negative backward bifurcation, and gives rise to a recurrent pattern. The main insight of this thesis is that recurrent disease can arise naturally from the de- terministic dynamics of populations. It will provide a starting point for further research in dynamical systems theory, and recurrence in other physical systems.