Spatial heterogeneity, nonlinear dynamics and chaos in XXXX .

@entity1 @entity66 @entity4409 @entity281 @entity166
There is currently considerable interest in the role of nonlinear phenomena in the population dynamics of @entity166 . @entity66 such as measles are particularly well documented dynamically, and have recently been the subject of analyses (of both models and notification data) to establish whether the pattern of epidemics is chaotic. Though the spatial dynamics of measles have also been extensively studied, spatial and nonlinear dynamics have only recently been brought together. The present review concentrates mainly on describing this synthesis. We begin with a general review of the nonlinear dynamics of measles models, in a spatially homogeneous environment. Simple compartmental models (specifically the SEIR model) can behave chaotically, under the influence of strong seasonal 'forcing' of @entity281 rate associated with patterns of schooling. However, adding observed heterogeneities such as age structure can simplify the deterministic dynamics back to limit cycles. By contrast all current strongly seasonally forced stochastic models show large amplitude irregular fluctuations, with many more 'fadeouts' of @entity281 that is observed in real communities of similar size. This indicates that (social and/or geographical) spatial heterogeneity is needed in the models. We review the exploration of this problem with nonlinear spatiotemporal models. The few studies to date indicate that spatial heterogeneity can help to increase the realism of models. However, a review of nonlinear analyses of spatially subdivided measles data show that more refinements of the models (particularly in representing the impact of @entity1 demographic changes on @entity281 dynamics) are required. We conclude with a discussion of the implication of these results for the dynamics of @entity166 in general and, in particular, the possibilities of cross fertilization between @entity1 disease epidemiology and the study of plant and @entity4409 .
