Classic epidemiology models make several simplifying assumptions. These simplifications make the computations tractable, and in many cases provide results that are accurate enough. But sometimes more complex models are needed. One of the primary assumptions is that individuals meet each other randomly. This ignores both social and spacial dynamics: individuals are more likely to give a disease to a friend they see often than a stranger, and more likely to give it to a nearby stranger than one halfway around the world. Additionally, I'm more likely to be friends with an individual if he's a friend of a friend than if he is not: societies form clusters. Modelers havetried various techniques to generate predictions for populations with spatial and social complexity (although usually not simultaneously), with some success. However, the models only generate analytical solutions for symmetric systems, or on edge cases.

My research has focused on generating solutions for populations too complex for an analytical solution. To accomplish, I wrote a program (WorldSim) to explicitly model populations with spatial and social complexity.

More information:
Mathematical models in Population Biology and Epidemiology: introduces many basic population and epidemic models.
Small World: modeling populations with graph theory.

Acknowledgements:
Funding: Cornell Presidential Research Scholars
Charting Utility: JFreeChart
Research Mentor: Evan Cooch