One of my main research interests is the probabilistic modeling of infectious disease spread in a population. Knowing how preventive strategies can change the spread and the final impact of the disease is of central interest.
A common feature of epidemic models is to exhibit the so-called threshold phenomena.
In general, if the reproductive number is below one, the major epidemic is impossible or unlikely,
whereas when it is above one, a major epidemic is possible. In the sub-critical case the epidemics dies out fast and often closely follows mean field approximation to extinction. In the supercritical case the infection can become endemic and it takes a long time before it becomes extinct. In the critical and nearly critical case one can sometimes uncover intricate limiting behavior. The main tools that are employed in the study of spread and final outcome of an infectious disease are the weak convergence theory and the theory of large deviations.