Currently, my research can be divided into two areas. The first is Applied Mathematics, specifically the analysis of mathematical models which are applied to biology. The second is Mathematical Biology, specifically the development of models and their application to the medical field.
My work in applied mathematics deals with the analysis of delay differential equations. Delay differential equations arise very naturally in biological phenomenon but not many models have incorporated delays. I study analytical techniques for extracting information from models with delays, using numerical analysis, complex analysis and stability analysis. Currently, I am studying the effects that delays have on oscillator equations such as Van der Pol's and Duffing's equations using multiple scale techniques.
My second area of focus is mathematical biology. In this area my curiosity involves obtaining information about the fundamental workings of the immune response to infection. For example, mathematical modeling combined with experimental measurements have provided profound results in the study of HIV-1 pathogenesis. Experiments in which HIV-infected patients are given potent anti- retroviral drugs that perturb the infection process have provided data necessary for mathematical models to predict kinetic parameters such as the productively infected T cell loss and viral decay rates. Many of the models used to analyze data have assumed drug treatments to be completely efficacious and that upon infection a cell instantly begins producing virus. My work, which allows for less then perfect drug effects and which includes delay dynamics has improved upon the previous estimates of certain kinetic parameters crucial for understanding the progression of HIV. With proper mathematical analysis and comparison with previous works, we have been able to show that when drug efficacy is not 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy, and the length of the delay. A second example, includes the analysis of a non-linear population model to study the effects of reducing antibiotic use on the spread of drug resistant streptococcus pneumonia among children in daycare and the auto-immunity of Chagas Infection .
