Spatio-temporal spread of epidemics
The earliest reference I know to the spatio-temporal spread of infectious diseases is in the History of the Peloponnesian War, by Thucydides (460-395 BC). In this passage (link), Thucydides describes the arrival into Athens of a "plague" (perhaps typhus).
Most of my work in this area uses metapopulations. An example: major cities in the world are connected by the global air transportation network. If an infection arises in a given city, then this infection can either continue locally, or it can be transported to another city by an infected traveler. I am a member of the BioDiaspora project, whose objective is to evaluate the vulnerability of Canada to the importation of infectious diseases in such a manner. In this project, we have data originating from OAG, IATA, ICAO, ACI, as well as other sources, that gives us a good description of the network.
For more details, see this page.
As part of my work on the spatio-temporal spread of epidemics, I have put together a file that puts together the WHO data on the spread of the Severe Acute Respiratory Syndrome (SARS) in 2003. The file can be downloaded here.
Understanding the way an influenza pandemic would spread in a population, and in particular, what would be the effect of various control measures, is very important.
Other problems in epidemiology
I have considered problems concerning the effect of vaccination in a human disease, in particular when the time during which the vaccine remains efficacious has an arbitrary distribution; the effect of pathogen resistance in the vectors of a vector-borne disease;
Other problems in population dynamics
I am interested in the description of the time individuals spend in the different classes in compartmental models. I have also considered the dynamics of an experimental device called a chemostat.
Intermediate filaments dynamics
Intermediate filaments make up one of the three networks that constitute the cytoskeleton, a framework present in all animal cells and giving them shape and structure.
Large systems of ordinary differential equations
Most of my work uses ordinary differential equations. Because of the number of populations to describe (for example, every city in the global air transportation network), I generally consider large systems. These are systems with an arbitrary (but finite) number of equations. Because of their size, they pose very specific challenges, and some techniques have to be developed.
Continuous-time Markov chains
Delay differential equations
Delay differential equations are used to model situations where the evolution of a system depends not only on its current state but also, on the state the system was in some time in the past. In population dynamics, this type of system arises when one models such things as the time individuals spend in various states.