We present a detailed analysis of the pattern of measles outbreaks in the small isolated community of the Faroe Islands. Measles outbreaks in this population are characterized by frequent fade-out of infection resulting in long intervals when the disease is absent from the islands. Using an analysis of the distribution of epidemic sizes and epidemic durations we propose that the dynamical structure observed in the measles case returns reflects the existence of an underlying scaling mechanism. Consequently the dynamics are not as purely stochastic as is usually thought for epidemiological systems of this sort. We use a lattice-based epidemic model to provide a theoretical estimate of the scaling exponents and show that a conventional compartmental SEIR model is unable to reproduce this result. The methods discussed in this paper are general and represent a novel way to consider the dynamics of any other communicable disease where there is frequent fade-out in the case returns.