Mathematical Modeling, Simulation, and Time Series Analysis of Seasonal Epidemics.

Seasonal and non-seasonal Susceptible-Exposed-Infective-Recovered-Susceptible (SEIRS) models are formulated and analyzed. It is proved that the disease-free steady state of the non-seasonal model is locally asymptotically stable if Rv \u3c 1, and disease invades if Rv \u3e 1. For the seasonal SEIRS model, it is shown that the disease-free periodic solution is locally asymptotically stable when R̅v \u3c 1, and I(t) is persistent with sustained oscillations when R̅v \u3e 1. Numerical simulations indicate that the orbit representing I(t) decays when R̅v \u3c 1 \u3c Rv. The seasonal SEIRS model with routine and pulse vaccination is simulated, and results depict an unsustained decrease in the maximum of prevalence of infectives upon the introduction of routine vaccination and a sustained decrease as pulse vaccination is introduced in the population. Mortality data of pneumonia and influenza is collected and analyzed. A decomposition of the data is analyzed, trend and seasonality effects ascertained, and a forecasting strategy proposed

Models Linking Epidemiology with Immunology and Ecology

Optimal control can be used to design intervention strategies for the control of infectious diseases and predator-prey systems. In this dissertation, we studied models encapsulating two relatively new areas of mathematical biology, which combine epidemiology with immunology and ecology. We formulated immuno-epidemiological models of coupled within-host model of ordinary differential equations and between-host model of ordinary differential equations and partial differential equations, using the Human Immunodeficiency Virus (HIV) for illustration, and set a framework for optimal control of immuno-epidemiological models. By constructing an iterative sequence from a representation formula for a solution to the linked model and using the fixed-point argument, existence and uniqueness of solution to the immuno-epidemiological model are obtained. An explicit expression for the basic reproduction number, R0 (R zero), of the linked model is derived, and local asymptotic and global stability results are obtained when R01, it is shown that the endemic equilibrium point is locally asymptotically stable. An optimal control problem with drug-treatment control on the within-host system is formulated and analyzed; these results are novel for optimal control of ODEs linked with such first order PDEs. Numerical simulations based on a forward-backward sweep method are obtained. Our analysis and control techniques give a new tool for investigating immuno-epidemiological models for other diseases. An eco-epidemiological model of predator and prey, motivated by cats and birds on the Marion Island, is formulated and analyzed. Basic and demographic reproduction numbers are obtained, and stability analysis of equilibria is investigated. An optimal control problem involving scalar and time-dependent controls is formulated and analyzed. Existence, characterization and uniqueness results are obtained. Numerical simulations based on a forward-backward sweep method illustrate the possibility of eradicating predators and conserving prey when a combination of control strategies are applied