1. Zoology Department, University of Cambridge, Downing Street, Cambridge CB2 3EJ, UK
2. 501 ASI Building, Departments of Entomology and Biology, Penn State University, University Park, Pennsylvania 16802, USA

Correspondence to: B. T. Grenfell¹ Correspondence and requests for materials should be addressed to B.T.G. (e-mail: Email: b.t.grenfell@zoo.cam.ac.uk).

Abstract

Spatio-temporal travelling waves are striking manifestations of predator–prey and host–parasite dynamics. However, few systems are well enough documented both to detect repeated waves and to explain their interaction with spatio-temporal variations in population structure and demography. Here, we demonstrate recurrent epidemic travelling waves in an exhaustive spatio-temporal data set for measles in England and Wales. We use wavelet phase analysis, which allows for dynamical non-stationarity—a complication in interpreting spatio-temporal patterns in these and many other ecological time series. In the pre-vaccination era, conspicuous hierarchical waves of infection moved regionally from large cities to small towns; the introduction of measles vaccination restricted but did not eliminate this hierarchical contagion. A mechanistic stochastic model suggests a dynamical explanation for the waves—spread via infective 'sparks' from large 'core' cities to smaller 'satellite' towns. Thus, the spatial hierarchy of host population structure is a prerequisite for these infection waves.