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Travelling waves and spatial hierarchies in measles epidemics

Nature volume 414, pages 716–723 (2001)Cite this article

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.

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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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Acknowledgements

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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Authors and Affiliations

  1. Zoology Department, University of Cambridge, Downing Street, Cambridge, CB2 3EJ, UK

    B. T. Grenfell, O. N. Bjørnstad & J. Kappey

  2. Departments of Entomology and Biology, 501 ASI Building, Penn State University, University Park, 16802, Pennsylvania, USA

    O. N. Bjørnstad

Authors
  1. B. T. Grenfell
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Corresponding author

Correspondence to B. T. Grenfell.

Supplementary information

Figure I

(GIF 9.32 KB)

The figure (which accompanies Box 2, figure a) shows the mean (+/s.e.) biennial wavelet phase of 60 simulations of the spatial TSIR model with small centres, defined in Box 2 of the paper. To explore the phase relationships arising from approximately phase-locked biennial cycles, the simulations were run for 10 years, starting from the phase-locked deterministic attractor of the system.

Notes

  1. 1.

    The phase lag is not an ‘edge’ effect, since wrapped boundary condition also generate a phase lag near the city.

  2. 2.

    As we move away from the city, note that the phase error bars widen. This is partly due to greater irregularity nearer the edge, but also partly because occasional simulations drop onto the opposite attractor (with 180o phase lag). As noted in Box 2, this partly reflects distance from the city, but also a genuine edge effect (note that Norwich and its environs are near the coast, hence an ‘edge’ in terms of measles transmission).

  3. 3.

    Previous work (Bjørnstad et al. 2001, op. cit.) shows that the relative measles transmission rate (Ro) in the England and Wales data set is constant across 3 orders of magnitude of population size. In fact, the crude models used here show slight increases in transmission rate with coupling (because tight coupling in the crude way we have defined it can increase Ro).

    This is illustrated in Figure (II), which presents a phase analysis of the case when large centres are coupled (cf Box 2, figure (b) in the paper). Increased coupling between boroughs increases their effective Ro and causes them to lead the ‘periphery’ slightly, by 1 time step (2 weeks) – note also that the phase analysis picks up this small phase lag more sensitively than the crude plot of average biennia in Box 2.

    This effect – which is a byproduct of our coupling model, cannot cause the progressive phase lag of the periphery seen in Figure (I); this latter is driven by ‘sparks’ of infection in the hierarchical system, as described in Box 2. However, the effect of coupling on effective transmission does complicate the analysis of the relationship between waves and coupling. We are currently exploring this issue: preliminary analyses indicate that hierarchical waves will occur at a range of ‘intermediate coupling rates, as long as the overall dynamics are biennial and major epidemics occur in the same year. Recent work (1) indicates that a more mechanistic representation of coupling can remove the spurious slight increase in R0 in tightly-coupled centres.

    (1) Keeling, M.J. & Rohani, P. (2001) Spatial Coupling in Epidemiology: A Mechanistic Approach. Ecol. Letts. (in press).

Movie (FLI, 5.5 MB)

Structure of the movie frame

The movie shows the dynamics of measles in England and Wales during the period 1955-1965. The data are from the Registrar General's weekly notifications of measles in 954 urban locations; the movie plots a frame every 2 weeks.

The upper left quadrant shows observed measles dynamics, using colour coding for the logarithm of the number of reported infected cases (all data are scaled on (0-1) before plotting). Zero case reports are shown as white circles and circle size is proportional to {population size}0.2. The movie captures the overall synchonised biennial 'pulse' of measles epidemics, superimposed on the seasonal swing of infection. There are also more irregular epidemics in smaller places and many other interesting patterns, for example distinctive behaviour in many coastal towns.

The observed data also show travelling waves of measles, moving away from large centres such as London. These are revealed more clearly in the upper right quadrant, which shows the smoothed major epidemic cycles, filtered using wavelets in a window of 1.75 to 3 years. As discussed in the paper, the filtered series reveal dramatic waves of measles, moving away from large centres, especially London and the North West conurbation. The lower left quadrant shows the phase difference from London of these filtered series (see the paper for more details). The arrows and the colour of the arrows are indicators of the phase difference, as shown on the circular key (green denotes a zero phase difference from London, blue denotes a lag and yellow a lead). These give a more stable picture of the phase relationships -- notably, the wave moving away from London and a relatively early wave from the North West. Finally, the lower right quadrant plots a moving time bar against the London and Manchester series.

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Grenfell, B., Bjørnstad, O. & Kappey, J. Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716–723 (2001). https://doi.org/10.1038/414716a

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