Research Interests

My research interests lie in mathematical problems with applications in the biological sciences. I use and develop mathematical tools in the areas of Dynamical Systems, Numerical Analysis, Statistical Parameter Estimation. My research projects include :
  • Exploring the Impact of Ectoparasites on the Evolution of Social Systems
    We aim to capture the behavior of evolved social systems and the dynamics of ectoparasite transmission that result from social contact-based exposures. We model these social systems in two ways : 1) using a system of differential equations; 2) via agent-based simulations. We focus on how ectoparasites, those parasites that live on the outside of the host (e.g. fleas and ticks), may spread via social group interactions and how that impacts the group's social structure and evolutionary success.

  • Temperature Effects on Rem/non-Rem Sleep Dynamics
    Sleep is a behavioral state in which we spend nearly one third of our lives. This biological phenomenon clearly serves an important role in the lives of most species. While much effort has been put forth in understanding the nature of sleep, many aspects of sleep are still not well understood. We have developed a nonlinear, Morris-Lecar type, ODE model of of human sleep–wake regulation with thermoregulation and temperature effects. Simulations of this model show features previously presented in experimental data. The model highlights how temperature effects interact with sleep history to effect sleep regulation.

  • Optimization of Combined Cancer Treatments : Anti-angiogenic Drugs and Chemotherapy
    My postdoctoral appointment was with the NUMED Team of INRIA Grenoble - Rhône-Alpes. Here, we use longitudinal murine tumor growth data to develop nonlinear, mixed-effect, ODE models to study the effects of combined anti-angiogenic and chemotherapeutic treatments. Upon model validation, we run numerical experiments to determine optimal treatment protocols for the administration of anti-angiogenics and chemotherapy. The results of these theoretical experiments are then followed up with biological studies to determine the effectiveness of the theoretically obtained protocols.

  • Mathematical Models of Immune Regulation and Cancer Vaccines
    My graduate research was conducted within the Applied Mathematics & Statistics, and Scientific Computation program at the University of Maryland, College Park. Under the direction of Prof. Doron Levy, I made use of a number of mathematical tools in the areas of Dynamical Systems, Perturbation Methods, Numerical Analysis, Simulations and Statistics to investigate two main projects:
    • The enhancement of tumor vaccine efficacy by immunotherapy.
    • Modeling the functional stability of the regulatory T cell population.

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Undergraduate Research

Many important principals in mathematical modeling can be understood with undergraduate-level knowledge of ordinary differential equations and numerical simulations, as such, portions of my research and can be grasped by advanced undergraduate students. My current research program facilitates truly interdisciplinary collaboration spanning a number of subject areas including mathematics, computer science, physics, engineering, pharmacy, and oncology. Below, I offer brief descriptions of a selected undergraduate projects that I’ve directed.
  • Modeling Within-Host Dynamics of Schistosomiasis I was the co-lead on this project which was done in the context of the University of Maryland's MAPS-REU. Schistosomiasis, also known as snail fever, is a disease transmitted by several species of flatworms that are carried by freshwater snails. The primary goal of the present work was to model disease progression using systems of nonlinear ODEs. Parasite components considered are larvae, immature worms, mature worms, and eggs. Immune components modeled are resting macrophages, active macrophages and T lymphocytes (T-Cells). We investigated the sensitivity of the model with respect to relevant parameters. We then use the Quasi-Steady State Assumption (QSSA) to derive two reduced models of schisto-immune dynamics. We compared the behavior of the reduced models to the behavior of the original system.

  • Analytically Understanding Population Dynamics of the Interaction between T-cells and HIV
  • Here, a nonlinear mathematical model of differential equations with piecewise constants highlights the pessimistic relationship HIV population growth has with the T-cell population. We were able to find analytical solutions to a simple model of HIV-Tcell dynamics as well as run numerical simulations that highlight important parameter spaces for a more complex model of these dynamics. This work has been presented at a number of national conferences/meetings, even winning a student poster presentation award at the 2016 Joint Mathematics Meetings. The positive outcomes of this project were also used to support this student’s acceptance into a graduate program.

  • ODE Model Reduction Using Quasi-Steady State Approximation (QSSA)
    In an ODE model of a chemical reaction network, the QSSA is used to remove the highly-reactive, low concentration chemicals from the model. In this study, we explored when the QSSA is appropriate and when there is a guaranteed solution to our reduced system. This project involved topics of solvability of polynomial systems as well as numerical approximations and simulation. The project was particularly successful and the student was subsequently chosen as “Student of the Year” at Morehouse College for the 2013-2014 academic year. The positive outcomes of this project were also used to support this student’s acceptance into a graduate program.

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