Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Pei Yu

Abstract

This work is concerned with bifurcation and stability in models related to various aspects of infections diseases.

First, we study the dynamics of a mathematical model on primary and secondary cytotoxic T-lymphocyte responses to viral infections by Wodarz et al. This model has three equilibria and the stability criteria of them are discussed. We analytically show that periodic solutions may arise from the third equilibrium via Hopf bifurcation. Numerical simulations of the model agree with the theoretical results. These dynamical behaviours occur within biologically realistic parameter range.

After studying the single-strain model, we analyze the bifurcation dynamics of an in vivo multi-strain model of Plasmodium falciparum. Main attention of this model is focused on the dynamics of cross-reactivity from antigenic variation. We apply the techniques of coupled cell systems to the model and find that synchrony-breaking Hopf bifurcation occurs from a nontrivial synchronous equilibrium. We prove the existence and calculate the condition for the bifurcation.

Aside from studying the previous models, we also study the clustering behaviour found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-state, periodic, or even chaotic motions can be self-organized into clusters. We show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcations from the quotient network of the models. We also calculate the stability criteria for different clustering patterns. Finally, the biological implications of these results are discussed.

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