Spatiotemporal dynamics of epidemics: synchrony in metapopulation models.

Multi-patch models - also known as metapopulation models - provide a simple framework within which the role of spatial processes in disease transmission can be examined. An n-patch model which distinguishes between k different classes of individuals is considered. The linear stability of spatially homogeneous solutions of such models is studied using an extension of an analysis technique previously described for a population setting in which individuals migrate between patches according to a simple linear term. The technique considerably simplifies the analysis as it decouples the nk dimensional linearized system into n distinct k-dimensional systems. An important feature of the spatial epidemiological model is that the spatial coupling may involve non-linear terms. As an example of the use of this technique, the dynamical behavior in the vicinity of the endemic equilibrium of a symmetric SIR model is decomposed into spatial modes. For parameter values appropriate for childhood diseases, expressions for the eigenvalues corresponding to in-phase and out-of-phase modes are obtained, and it is shown that the dominant mode of the system is an in-phase mode. Furthermore, the out-of-phase modes are shown to decay much more rapidly than the in-phase mode for a broad range of coupling strengths.