## Mathematics of HSV-2 Dynamics

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##### Date

2010-08-26T19:47:50Z

##### Authors

Podder, Chandra Nath

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##### Abstract

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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##### Keywords

Epidemiology, Mathematical Modeling, Disease-free Equilibrium, Local Stability, Global Stability, Basic Reproduction Number, Endemic Equilibrium, Lyapunov Function, Centre Manifold Theory, Backward Bifurcation