*(English)*Zbl 1036.92029

Summary: 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 nonlinear 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.

##### MSC:

92D30 | Epidemiology |