Development and applications of mathematical tools in models of infectious diseases and biological phenomenon

The interactions between infectious agents and immune cells have been extensively studied. Despite remarkable advances in the field, there are gaps in understanding the dynamics of host-pathogen interaction which prevent us from creating efficient drug therapies and preventive vaccines. The focus of this dissertation is to develop mathematical models that aim to explain the cell population dynamics in two viral infections. I start by developing and analyzing a model for the HIV primary infection. I use statistical techniques to show the need of experimental data that are consistent with the framework of our theoretical model. The next model incorporates important biological features of the Hepatitis B acute infection. Estimates of parameters will help establish the importance of immune response in the outcome of the disease. Lastly, I will discuss the van der Pol nonlinear delayed equation. I am interested in the effect of the delay in the promotion or suppression of limit cycle oscillations.