The thesis is based on using dynamical systems theories and techniques to study the qualitative dynamics of herpes simplex virus type 2 (HSV-2), a sexually-transmitted disease of major public health significance. A deterministic model for the interaction of the virus with the immune system in the body of an infected individual (in vivo) is designed first of all. It is shown, using Lyapunov function and LaSalle's Invariance Principle, that the virus-free equilibrium of the model is globally-asymptotically stable whenever a certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present equilibrium when the threshold quantity exceeds unity. Using persistence theory, it is shown that the virus will always be present in vivo whenever the reproduction threshold exceeds unity. The analyses (theoretical and numerical) of this model show that a future HSV-2 vaccine that enhances cell-mediated immune response will be effective in curtailling HSV-2 burden in vivo. A new single-group model for the spread of HSV-2 in a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less than unity. The model has a unique endemic equilibrium, which is shown to be globally-stable for a special case, when the reproduction number exceeds unity. The model is extended to incorporate an imperfect vaccine with some therapeutic benefits. Using centre manifold theory, it is shown that the resulting vaccination model undergoes a vaccine-induced backward bifurcation (the epidemiological importance of the phenomenon of backward bifurcation is that the classical requirement of having the reproduction threshold less than unity is, although necessary, no longer sufficient for disease elimination. In such a case, disease elimination depends upon the initial sizes of the sub-populations of the model). Furthermore, it is shown that the use of such an imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity). The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies. Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it is shown that the risk-structured model undergoes backward bifurcation under certain conditions (the backward bifurcation property can be removed if the susceptible population is not stratified according to the risk of acquiring infection). Thus, one of the main findings of this thesis is that risk structure can induce the phenomenon of backward bifurcation in the transmission dynamics of HSV-2 in a population.

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Human T-lymphotropic virus type I (HTLV-I) is a persistent human retrovirus characterized by life-long infection and risk of developing HAM/TSP, a progressive neurological and inflammatory disease. Despite extensive studies of HTLV-I, a complete understanding of the viral dynamics has been elusive. Previous mathematical models are unable to fully explain experimental observations. Motivated by a new hypothesis for the mechanism of HTLV-I infection, a three dimensional compartmental model of ordinary differential equations is constructed that focusses on the highly dynamic interactions among populations of healthy, latently infected, and actively infected target cells. Results from mathematical and numerical investigations give rise to relevant biological interpretations. Comparisons of these results with experimental observations allow us to assess the validity of the original hypothesis. Our findings provide valuable insights to the infection and persistence of HTLV-I in vivo and motivate future mathematical and experimental work. / Applied Mathematics