The primary focus of the project will be the study of stochastic models for epidemics and related random processes in which infection, information, or some other transmissible quantity is passed randomly among the nodes of a network. The dominant theme of the research will be the effect of the network geometry on the behavior of the epidemic processes, especially near critical values of the transmission rate parameter(s). Two large classes of network geometries, Euclidean and hyperbolic, will be studied. Euclidean geometries are appropriate for models of geographically structured populations, such as plants along a river bed; here, for instance, the network might consist of individual nodes arranged in communities placed at the vertices of a regular lattice, with interactions restricted to nodes in the same or neighboring communities. Hyperbolic geometries (such as those arising in expander graphs or ?small worlds? models) are in many instances more appropriate for human populations and computer networks, where interactions do not follow a regular geographic pattern. Both finite and infinite networks will be studied. The primary objective will be the description of large-population and/or large network limiting behavior.
Stochastic epidemic models are closely related to a number of other large classes of stochastic processes, and these will play a large role in the research project. Epidemics of SIR type (susceptible, infected, removed) are at least in simple cases equivalent to bond percolation on the network. Epidemics of SIS type are contact processes. In Euclidean geometries, many of these processes have measure-valued scaling limits that obey stochastic partial differential equations of reaction-diffusion type. A secondary objective of the research will be to understand the threshold phenomena and phase transitions that occur in some of these related processes, and to explore how these are mirrored in epidemic models.
Epidemics -- in human and animal populations, but also in computer and communications networks -- arise from chance events. These are, in many cases, simple but uncontrollable events: A passenger on a bus might or might not touch a guard rail that has been contaminated by MIRSA bacteria; a computer user might or might not click on an attachment to an email message informing him that a wealthy businessman in Nigeria is asking for his help in transferring 30 million dollars out of the country. These chance encounters are, however, repeated in large numbers, by thousands of people, hundreds of times, and so they occur with statistically predictable frequencies. This makes it possible -- at least in principle -- to predict the course of an epidemic in a large population.
This research project is concerned with the study of simple mathematical models of epidemic processes. The particular emphasis of the study will be on understanding how the underlying geometry of the network of individuals or communications nodes through which the epidemic propagates affects its course.
Research was conducted in several areas of probability theory and its connections with other areas of mathematics and the sciences, notably (1) random walk on non-Euclidean geometric structures; (2) statistical regularities of geodesics on negatively curved surfaces; (3) models of epidemic propagation; and (4) multi-player games in economic models of voting. The work on epidemic models and related random processes was mostly done in collaboration with Ph. D. students, two of whom received their degrees under my supervision, and two others who will likely finish their disssertations in 2015. The work on multi-player voting games was done in collaboration with an economist (Glen Weyl of Microsoft Research) who has advocated the use of "pay-for-votes" systems in corporate governance. Results of major importance were obtained in problem areas (1) and (2). First, the long-time behavior of return probabilities of a random walk on a "hyperbolic" group (for instance, an "Escher tessellation") was shown to be essentially determined by the ambient geometry, regardless of the particular rules of evolution of the random walk. Second, the self-intersection counts of long geodesics on negatively curved surfaces were shown to obey statistical laws analogous to the law of large numbers and central limit theorem of classical probability.