We are interested in modeling and understanding the dynamics of biological populations. Our main applications are to the spread of epidemics and to ecosystems. We ask how the geometry of the population, such as spatial extent or network structure, affects the system dynamics and how stochastic fluctuations at the level of individuals affect macroscropic behavior. Goals are better prediction and control of population processes. For infectious diseases, effective control would reduce disease cases or even drive a disease to extinction, while for ecosystems, it is desirable to preserve or restore native species. Our research topics are highly interdisciplinary, lying at the interface of mathematics, physics, and biology. Our tool set includes methods from statistical mechanics, nonlinear dynamics, and numerical simulation.

Marine population dynamics

Modeling of ecosystems is important in guiding future restoration efforts. We are modeling a key marine species, the native Chesapeake Bay oyster, which has dropped to about 1% of its previous levels due to overfishing and other factors. The oyster has a feedback interaction with sediment in the water column: large, healthy populations of oysters filter the water, reducing sediment and improving their habitat, while small populations of oysters can be choked by sediment. This leads to multiple stable states, which can present challenges for restoration efforts. We are also considering the effects of connectivity in extended marine populations.

Adaptive networks

One area of interest is populations with an adaptive network of contacts. The real world has abundant examples of complex networks, from the internet to human social networks to neurons in the brain. Many of these networks evolve over time, forming new connections and breaking old ones. At the same time, the nodes of the network may themselves display complicated dynamics. We are interested in adaptive networks, where the network geometry responds to the state of the nodes, while also the nodes are affected by the network geometry. That is, dynamics of the network and dynamics on the network are interrelated. We are modeling epidemic spread on social networks where the adaptive behavior consists of avoidance of infected individuals. On the other hand, we also model recruitment to a cause, where adaptation occurs to increase the number of recruiting contacts.

Epidemic metapopulation models

A coarse-grained approach to modeling is to treat the population as well-mixed rather than modeling individual contacts in a social network. At an intermediate level of coarse-graining, one can model several well-mixed subpopulations with a network of interactions between them. A major application is to dengue fever, a multistrain disease that already displays complex dynamics in a well-mixed subpopulation. By coupling together multiple subpopulations, we can study the resulting dynamics and compare with epidemiological observations such as traveling waves of infection across a region.

Stochastic extinction

In any finite population, there is a nonzero probability of extinction due to stochastic effects. However, the extinction probability may be small and observed only on long time scales. By assuming that extinction occurs due to a large, rare fluctuation, one can analytically compute the most probable path that the system will take to extinction. Knowledge of this path also leads to estimates for how parameters affect the extinction rate. We are interested in extending these methods to higher dimensional systems of biological importance to predict extinction risk, as well as developing new optimal control strategies for epidemics based on driving the system toward the most probable path to extinction.