Spatial and Temporal Spread of Disease in Discrete Space

The SARS epidemic of 2003 and the 2009 H1N1 pandemic illustrated the capacity of infectious pathogens to spread quickly over vast distances. However, rapid dissemination is not the only spatial and temporal aspect in disease propagation. In today’s globalized world, public health systems of different qualities come in contact as individuals continuously move between them, inducing stresses on the different systems.
After a brief discussion of spatio-temporal aspects in the spread of infectious pathogens, I will introduce a modelling paradigm that allows to capture some of the mechanisms involved. These so-called metapopulation epidemic models consider space as consisting  of discrete geographical regions connected through mobility of individuals between the regions. Thus, metapopulation models consist of multi-digraphs in which vertices are endowed with dynamical systems describing the evolution of the population in the regions and multi-arcs represent the movement of the different types of individuals between the regions. I will present a few examples where the epidemic models used consist of ordinary differential equations. I will show that, despite the apparent complexity and high dimensionality, these models can be analysed in a manner very similar to their lower dimensional single population equivalents. Finally, I will discuss problems specific to the metapopulation context, namely the existence of “mixed equilibria” and the non-strongly connected movement case.