Temporal changes in the incidence of measles virus infection within large urban communities in the developed world have been the focus of much discussion in the context of the identification and analysis of nonlinear and chaotic patterns in biological time series. In contrast, the measles records for small isolated island populations are highly irregular, because of frequent fade-outs of infection, and traditional analysis does not yield useful insight. Here we use measurements of the distribution of epidemic sizes and duration to show that regularities in the dynamics of such systems do become apparent. Specifically, these biological systems are characterized by well-defined power laws in a manner reminiscent of other nonlinear, spatially extended dynamical systems in the physical sciences. We further show that the observed power-law exponents are well described by a simple lattice-based model which reflects the social interaction between individual hosts.