Research Interests

My research interests lie at the intersection of the area of Mathematics known as deterministic and stochastic dynamical systems and Biology, broadly construed. I view myself as both a mathematician and a scientist, developing and applying sophisticated mathematical techniques to answer critical questions in ecology, epidemiology, oncology and medicine. My work to date has touched on the how the structure of populations can affect the dynamics of disease spread, how the immune system can be excited to promote its ability to destroy the cancer cells associated to melanoma, and the paradox of biodiversity. Mathematically, I am interested in how our knowledge of deterministic dynamical systems, such as systems of ordinary differential equations, can be leveraged to understand related stochastic dynamical systems.

Population Movement and Disease Dynamics

As we have seen with the global SAR-CoV-2 pandemic, human defined boundaries of race, ethnicity, culture, nationality and geography do stop the spread of disease in our globally connected world. This connection is driven by the increased mobility of human and animal populations. When a person moves and takes on the attributes of the denizens of their new group, as is the case in migration, this movement can be captured using Eulerian movement modeling. However, it is common for residents of one community (as defined by the attributes of that community) may visit another, but retain the attributes of their home community. This is the case when we commute, or travel for work or pleasure. This latter form of movement can be captured by Lagrangian movement models. One current area of interest for me is investigating the effects of complex networks formed by combining Eulerian and Lagrangian movement in stochastic models of infectious disease dynamics.

Mathematical Medicine and Oncology

Advances in modern medicine have provided greater understanding to the complex biochemical reactions that relate to healthy function as well as disease. Our ability to collect and analyze clinical data has increased with equal speed. Mathematical models can leverage data from early experiments to help predict the dynamics of these biochemical reactions, design follow-up experiments and tailor dosing and other treatment strategies to individual patients. In work with L. Dickman and Y. Kuang, we studied the dynamics of dendritic cell therapy for treatment of melanoma. Dendritic cell therapy consists of exciting the body's natural immune response to this deadly form of cancer by loading the system with activated dendritic cells. These cells activate the CD4+ and CD8+ T cells responsible for destroying the cancer cells. Our work helps to understand potential outcomes for this method of treatment and the implications for tumor control or escape.

Competitive Exclusion and the Paradox of Biodiversity

Analysis of classic models of competition between two species supports the principle of competitive exclusion and suggest that coexistence, and therefore biodiversity, is either unlikely or impossible. However, biodiversity is commonly observed in nature. In response to this paradox, Stephen Hubbell developed what came to be the Unified Neutral Theory of Biodiversity and Biogeography. However, Hubbell's theory was not without its detractors. The word neutral in the name comes from the notion that competitors at each trophic level in the food web are neutral in the sense that they have demographic traits resulting in equal fitnesses. This notion of neutrality can be relaxed when competition is modeled using continuous time Markov chains (CTMCs). We can see this by analyzing models related to the classical Lotka-Volterra model of competition and exploitative competition for a single non-reproducing essential resource in a chemostat, respectively, using the recently developed approximation called local approximation in time and space (LATS).