Epidemic Game Theory

My dissertation focused primarily on characterizing uniqueness properties of two simple distancing-type epidemic games. Despite the relative maturity of epidemic games, conditions for uniqueness of equilibrium strategies in these games are not well-established, yet there are few distinct examples of non-uniqueness. One of my important results was the discovery of a simple example that clearly exhibits three equilibrium strategies.

I am continuing to work on plumbing the depths of uniqueness and multiplicity of equilibria in epidemic games. One topic of particular interest is the relationship between equilibria over the general piecewise-continuous strategy space as compared to the delay-strategy restriction, in particular when optimal strategies lie outside of the delay space.

Societal Dynamics

Human societies are complex systems of many interacting individuals each making decisions based up intuition, incentive, and whim. How people live, and what they choose to do day-to-day, are influenced by history, culture, and geography. As these societies develop through time, notions of authority change alongside organizational structures. All these factors make for a very rich space for inquiry and investigation that has been largely neglected from a mathematical modeling perspective, with the key exception of those involved in the founding of this domain such as Dr. Peter Turchin.

One topic I am particularly interested in is how individual behavior relates to institutions. Questions along this line include how do institutions affect individual behavior, and how does collective behavior both inhibit and prevent institutions from acting upon a society. On more historical timescales, how to fundamental social arrangements determine the types of institutions and their structures that arise through further social development.

Spatial Vaccination Heterogeneity

Practical vaccination mechanisms inherently create heterogeneity in vaccination spatially through differential accessibility. These processes can create pockets of susceptibility that go unnoticed when observed through population level statistics like total vaccination coverage. This has significant implications for the elimination of pathogens like measles, where pockets of susceptibility can both sustain the pathogen and provide the conditions for an outbreak.

The role of the vaccination mechanism on driving this heterogeneity, and how it affects epidemic outcomes, has largely been unexplored in the rich and broad epidemiology literature. While naïve vaccination can be seen as removing susceptibility, and thus the effect on an epidemic dynamic is comparable to an epidemic on a sparser network, the spatial aspect is more subtle.

Capturing Spatial Dynamics

One of the great complexities in mathematical modeling is profitably capturing spatial aspects of a system without introducing excessive complexity. While many physical systems must deal with space fully, sparse systems, exemplified by many biological and sociological systems, do not necessarily require a full representation in space to capture the same dynamics to a sufficient extent.

There have been results along this line encoding network structures into ODEs for SIR-type immunizing epidemics which demonstrate the flexibility of these equations for encoding space in a weak sense without resorting to more complex models like PDEs or network-based simulations. While there are many particularities of these epidemic models that make such an encoding possible, I am interested in exploring how these techniques can be generalized to even simple models like the Lotka-Volterra predator-prey equations.