Applications of delay differential equations in physiology and epidemiology.

The primary aim of this thesis has been to study examples of the application of delay differential equations to both physiology and epidemiology. As such, the thesis has two main strands. The physiological application is represented by mathematical models of the glucose-insulin interaction in humans. We provide a detailed introduction to recent and current literature associated with this area, together with an overview of the physiological processes involved. Two systems explicitly incorporating a discrete delay are proposed and positivity and boundedness of solutions to these models are established. Sufficient conditions for global stability of the steady state of both systems are derived using both Lyapunov methods and comparison principles. Physiological interpretations of the analysis are provided. The simpler of the two models is then extended to represent a person being given periodic infusions of both insulin and glucose. Positivity of solutions of this system is established and the existence of a positive periodic solution is proved using the coincidence degree theory method. The epidemiological application of delay differential equations is represented by mathematically modelling the transmission dynamics of tuberculosis. A brief overview of the current impact of the disease is given and some of the problems public health officials face in combatting it are discussed. A summary of work in the literature on this subject is provided. The effect of migration on the spread of tuberculosis is considered. Patch type models consisting of just two patches are proposed, each patch considered to be a country. We allow for the possibility that migration between two countries is often more one way than the other so diffusion is not the discrete analogue of Fickian diffusion. Conditions for both local and global stability of the disease-free steady state are determined using a variety of methods. A model with a continuous representation of space incorporating Fickian diffusion is then proposed. This is assumed to be more appropriate for various animal species than for humans. The possibility of a travelling wave-front solution is investigated and the minimum speed of such a solution is determined. Numerical simulations support these results. Finally the model is adapted to incorporate the tendency to move towards a focal point. Using numerical simulations, the effect of random dispersal and purposeful movement towards a focal point are investigated