Deterministic and stochastic models of virus dynamics

A variety of mathematical models ranging from very simple ones to complicated ones have been developed and analyzed in order to capture different phenomena associated with the spread of diseases. Even though none of these models behave exactly according to the observed clinical data, major features of disease dynamics can be captured merely by means of a simple model. The model introduced by Nowak and May [12] is such simple deterministic model of which a stability analysis has not been done.

Our objectives in this endeavor are two-fold. The first objective of this thesis is to carry out a thorough analysis of the aforementioned deterministic model of virus dynamics while obtaining the related system of Ito stochastic differential equations which has not been obtained to date.

The motivation for obtaining the related stochastic model is also two-fold. The first reason is the capability of stochastic models to capture the randomness associated with the disease dynamics. The second reason is while a deterministic model predicts a single outcome for a given set of parameter values, a stochastic model predicts an infinite set of possible outcomes weighed by their likelihoods and probabilities.

Any mathematical model which describes virus dynamics, is not complete until it describes the immune response. With analogy to a predator-prey model, immune cells play the role of the predator while the virus plays the role of the prey. The immune response is triggered by encountering a foreign antigen. The role of the immune system is to fight off invasion by foreign pathogens. In this endeavor, our interest is a special kind of T cell, namely cytotoxic T lymphocyte (CTL) which can also identify and eliminate infected cells.

Then the immune response is incorporated with the aforementioned simple model of virus dynamics. This is done under three different assumptions on the CTL proliferation rate. This evidently results in three different models. The second objective of this thesis is to carry out a thorough stability analysis of the three deterministic models of virus dynamics with the CTL response while obtaining respective systems of Ito stochastic differential equations.