Research Interests

General Focus
My research interests lie in mathematical problems with applications in Neuroscience, Oncology, and Evolutionary Biology. More specifically, I use the techniques of parameter estimation, machine learning, dynamical systems and network theory to create models of biological phenomenon (e.g. cancer growth, sleep dynamics, cellular immune dynamics). My current research program facilitates truly interdisciplinary collaboration spanning a number of subject areas including mathematics, computer science, physics, engineering, evolutionary biology, and oncology.

Exploring the Impact of Ectoparasites on the Evolution of Social Systems
Ectoparasites
Here, we seek to understand how social systems evolve under a careful balancing act between living in collaborative groups that can accomplish complex tasks and avoiding patterns in social contact that would amplify the risks of infection transmission. We are developing and analyzing the mathematical systems needed to understand the interplay between sociality and ectoparasitic (parasites that live external to their host) infections. A number of models have investigated how organizational structure can emerge from individual behavioral dynamics. The effects of these dynamics are explored through models of social network theory where mathematical metrics are used to quantify the relative importance of nodes and the organization of the network. Network centrality theory is used to identify the most important individuals in a social network as well as to characterize the properties of the network as a whole. Within this context, our main focus for this project centers around characterizing the link between parasite load and sociality in dynamic networks. We aim to capture the key elements of these social systems in two ways : 1) using a system of differential equations; 2) via agent-based simulations.

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Temperature Effects on Rem/non-Rem Sleep Dynamics
Sleep Model
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. Mathematical models of sleep often neglect temperature properties and the effects of temperature on sleep, as they pose additional complexities to the already-intricate nature of sleep. However, if it is believed that temperature plays a key role in both the quality and quantity of human sleep, then including temperature as a model feature is imperative. Thus, we have developed a mathematical model that incorporates this key component and its effects on sleep, so that we may better describe and understand the mechanisms underlying sleep behavior when thermoregulation plays a role. We have developed a system of nonlinear, Morris-Lecar type, ordinary differential equations that model human sleepwake regulation with thermoregulation and temperature effects. Our results show that incorporating a biologically inspired representation of thermoregulation in a mathematical model for human sleep/wake cycling can account for several features observed in experimental data. Analysis of the model provides a more detailed understanding of the underlying mathematical mechanisms associated with sleep. We find that temperature effects could provide a crucial component of mathematical models of human sleep/wake cycling.

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Optimization of Combined Cancer Treatments : Anti-angiogenic Drugs and Chemotherapy
Mixed-Effect Cancer Model
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.

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