Travelling waves and spatial hierarchies in measles epidemics


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.

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Figure 1: Wavelet time series analysis for the log-transformed weekly London measles time series (see Box 1 for details).
Figure 2: Wavelet phase analysis of weekly measles reports for Cambridge, Norwich and London in the pre-vaccine era.
Figure 3: Phase differences from London for the full urban data set (954 locations).
Figure 4: Pre-vaccination (black) and vaccine era (red).
Figure 5: Average biennial behaviour of a spatially coupled measles model (see Box 2 for details).


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We thank S. Cornell, T. Coulson, J. Gog, M. Keeling, A. Lloyd, G. Nason, P. Rohani, C. Torrence and C. Williams for helpful discussions. B.T.G. and J.K. were supported by the Wellcome Trust. O.N.B. was supported by the National Center for Ecological Analysis and Synthesis (a Centre funded by an NSF grant, the University of California Santa Barbara, and the State of California) and the Norwegian Science Foundation.

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Correspondence to B. T. Grenfell.

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