| The study of epidemic models for the dynamics of infectious diseases has been one of the important areas in the mathematical theory of epidemiology. An SEIR type model is a compartmental model that describes population transfers among compartments S (susceptible), E (exposed), I (infectious) and R (recovered). Compared to earlier SIR models in the literature, the SEIR models studied in this dissertation assume that the disease has a latent period. This assumption is more realistic for many infectious diseases such as Hepatitis B, Chagas' disease and AIDS, and changes the transfer dynamics of the disease described by an SIR model. Typically, the population transfer process for an SEIR model is: Once infected, each susceptible individual remains latent before becoming infectious, and then recovers with permanent immunity.; We first develop and describe necessary mathematical tools for our model analysis. A new criterion for local stability of steady states of a differential system is established using a simple spectral property of compound matrices. A global stability result for a three dimensional competitive system is stated and proved in the way that it can be easily used in our model analysis. A global stability result for an arbitrary finite dimensional differential equation without monotonicity is also described.; Diffusion-driven instability in reaction-diffusion systems has been one of the most interesting topics since Turing first pointed out that diffusion can give rise to instability in 1952. We derive a set of conditions, which we call the minors condition, that provide a systematic way for detecting the occurrence of diffusion-driven instability in a general diffusive system. By establishing some stability and instability results for matrices, their implications for the stability and instability of a constant steady state of a reaction-diffusion system are studied.; Using the mathematical theory and tools outlined above, several SEIR type models are systematically analyzed. Threshold results are established for each model by identifying a threshold number such that, if the disease-free equilibrium is globally stable and the disease always dies out, whereas if the endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic level if it is initially present.; Vaccination has proven to be a very effective and successful means to prevent a disease from becoming endemic. An optimal vaccination strategy is investigated as an optimal control problem, that takes into account both the number of infectious individuals and the cost. The existence and uniqueness of the optimal control are established. Necessary conditions on an optimal control are obtained using the Pontryagin's Maximum Principle. Numerical simulations using Mathematica are provided to support theoretical results. |
