My research

I obtained a Ph.D., under the direction of Professor James D. Murray, who is a faculty member of the Applied Math Department. The focus of my research is mathematical biology.

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 developement of models and their application to the medical field. I have included a few abstracts of my work.

Applied Mathematics

A model of intracellular delay used to study HIV pathogenesis.

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 antiretroviral 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. We consider a model which allows for less then perfect drug effects and which includes a delay process. We present detailed analysis of this delay differential equation model and compare results between a model with instantaneous behavior to a model with a constant delay between infection and viral production. Our analysis shows 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. Thus, previous estimates of infected cell loss rates can be improved upon by considering more realistic models of viral infection.

A second project in this area is examining a general form for a delay which may be continuous or discrete. We have also extended our previous work to look at the full non-linear model which includes a combination of drug treatments.

Mathematical Biology

Mathematical models of HIV dynamics in vivo.

Mathematical models have proven valuable in understanding the dynamics of HIV-1 infection in vivo. By comparing these models to data obtained from patients undergoing antiretroviral drug therapy, it has been possible to determine many quantitative features of the interaction between HIV-1, the virus that causes AIDS, and the cells that are infected by the virus. The most dramatic finding has been that even though AIDS is a disease that occurs on a time scale of about 10 years, there are very rapid dynamical processes that occur on time scales of hours to days, as well as slower processes that occur on time scales of weeks to months. We show how dynamical modeling and parameter estimation techniques have uncovered these important features of HIV pathogenesis and impacted the way in which AIDS patients are treated with potent antiretroviral drugs.


Models of macrophage activation in response to pathogens.

The immune response to infection can be classified into two compartments; innate and cell-mediated. Macrophages, part of the innate system, recognize and digest foreign particles. This leads to a cascade of events, one of which is the signalling of the cell-mediate system. In the past decade, mathematical models have become an integral part in the study of infection and the immune response. Models have been developed which examine the interactions of the innate and cell-mediated system to infection and have provided much insight into the disease dynamics. Unfortunetly, there have only been a few mathematical works which focus on the innate system's response. We study the changes in the dynamics of macrophages in response to a pathogen and extend the previous works by including two compartments for macrophages, resident and activated. The model is then applied to experimental data and estimates for certain, previously unknown, kinetic parameters are obtained.

Effect of the eclipse phase of the viral life cycle on estimation of HIV viral dynamic parameters.

Using potent antiretroviral therapy to perturb the steady state viral load in HIV-1 infected patients has yielded estimates of the lifespan of virally infected cells. Here we show that including a delay that accounts for the eclipse phase of the viral lifecycle in HIV dynamics models decreases the estimate of the productively infected cell lifespan. Thus, productively infected cells may have a half-life that is shorter than the estimate of 1.6 days published by Perelson et al.
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