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A competing-risks model explains hierarchical spatial coupling of measles epidemics en route to national elimination

Nature Ecology & Evolution volume 4, pages 934–939 (2020)Cite this article

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Abstract

Apart from its global health importance, measles is a paradigm for the low-dimensional mechanistic understanding of local nonlinear population interactions. A central question for spatio-temporal dynamics is the relative roles of hierarchical spread from large cities to small towns and metapopulation transmission among local small population clusters in measles persistence. Quantifying this balance is critical to planning the regional elimination and global eradication of measles. Yet, current gravity models do not allow a formal comparison of hierarchical versus metapopulation spread. We address this gap with a competing-risks framework, capturing the relative importance of competing sources of reintroductions of infection. We apply the method to the uniquely spatio-temporally detailed urban incidence dataset for measles in England and Wales, from 1944 to the infection’s vaccine-induced nadir in the 1990s. We find that despite the regional influence of a few large cities (for example, London and Liverpool), metapopulation aggregation in neighbouring towns and cities played an important role in driving national dynamics in the prevaccination era. As vaccination levels increased in the 1970s and 1980s, the signature of spatially predictable spread diminished: increasingly, infection was introduced from unidentifiable random sources possibly outside regional metapopulations. The resulting erratic dynamics highlight the challenges of identifying shifting sources of infection and characterizing patterns of incidence in times of high vaccination coverage. More broadly, the underlying incidence and demographic data, accompanying this paper, will also provide an important resource for exploring nonlinear spatiotemporal population dynamics.

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Fig. 1: A schematic illustration of the spatial dynamics and persistence of measles.
Fig. 2: Fit of the absence–presence–absence model in the competing-risks framework from 1944 to 1964.
Fig. 3: Prevaccination regional dynamics of measles, inferred from the absence–presence–absence model in a competing-risks framework.
Fig. 4: Impact of vaccination on the spatial spread of measles.

Data availability

The measles data are available at https://github.com/msylau/measles_competing_risks.

Code availability

The code is available at https://github.com/msylau/measles_competing_risks.

References

  1. Verguet, S. et al. Measles control in sub-Saharan Africa: South Africa as a case study. Vaccine 30, 1594–1600 (2012).

    Article  Google Scholar 

  2. Anselin, L. Local indicators of spatial association—LISA. Geogr. Anal. 27, 93–115 (1995).

    Article  Google Scholar 

  3. Sabbe, M. Measles resurgence in Belgium from January to mid-April 2011: a preliminary report. Eurosurveillance 16, 19848 (2011).

    PubMed  Google Scholar 

  4. Châtelet, I. P., du, Floret, D., Antona, D. & Lévy-Bruhl, D. Measles resurgence in France in 2008, a preliminary report. Eurosurveillance 14, 19118 (2009).

    Google Scholar 

  5. Grenfell, B. T., Bjørnstad, O. N. & Kappey, J. Travelling waves and spatial hierarchies in measles epidemics. Nature 414, 716–723 (2001).

    Article  CAS  Google Scholar 

  6. Finkenstädt, B. F. & Grenfell, B. T. Time series modelling of childhood diseases: a dynamical systems approach. J. R. Stat. Soc. Ser. C 49, 187–205 (2000).

    Article  Google Scholar 

  7. King, A. A. & Schaffer, W. M. The geometry of a population cycle: a mechanistic model of snowshoe hare demography. Ecology 82, 814–830 (2001).

    Article  Google Scholar 

  8. Wilson, H. B. & Hassell, M. P. Host–parasitoid spatial models: the interplay of demographic stochasticity and dynamics. Proc. R. Soc. Lond. B 264, 1189–1195 (1997).

    Article  Google Scholar 

  9. Xia, Y., Bjørnstad, O. N. & Grenfell, B. T. Measles metapopulation dynamics: a gravity model for epidemiological coupling and dynamics. Am. Nat. 164, 267–281 (2004).

    Article  Google Scholar 

  10. Keeling, M. J. & Grenfell, B. T. Disease extinction and community size: modeling the persistence of measles. Science 275, 65–67 (1997).

    Article  CAS  Google Scholar 

  11. Bolker, B. M. & Grenfell, B. T. Impact of vaccination on the spatial correlation and persistence of measles dynamics. Proc. Natl Acad. Sci. USA 93, 12648–12653 (1996).

    Article  CAS  Google Scholar 

  12. Bak, P. How Nature Works: The Science of Self-Organized Criticality (Springer Science & Business Media, 2013).

  13. Grenfell, B. & Harwood, J. (Meta)population dynamics of infectious diseases. Trends Ecol. Evol. 12, 395–399 (1997).

    Article  CAS  Google Scholar 

  14. Erlander, S. & Stewart, N. F. The Gravity Model in Transportation Analysis: Theory and Extensions (VSP, 1990).

  15. Murray, G. D. & Cliff, A. D. A stochastic model for measles epidemics in a multi-region setting. Trans. Inst. Br. Geogr. 2, 158–174 (1977).

    Article  Google Scholar 

  16. Ferrari, M. J. et al. A gravity model for the spread of a pollinator‐borne plant pathogen. Am. Nat. 168, 294–303 (2006).

    Article  Google Scholar 

  17. Viboud, C. et al. Synchrony, waves, and spatial hierarchies in the spread of influenza. Science 312, 447–451 (2006).

    Article  CAS  Google Scholar 

  18. Jandarov, R., Haran, M., Bjørnstad, O. & Grenfell, B. Emulating a gravity model to infer the spatiotemporal dynamics of an infectious disease. J. R. Stat. Soc. Ser. C 63, 423–444 (2014).

    Article  Google Scholar 

  19. Bjørnstad, O. N. & Grenfell, B. T. Hazards, spatial transmission and timing of outbreaks in epidemic metapopulations. Environ. Ecol. Stat. 15, 265–277 (2007).

    Article  Google Scholar 

  20. Birnbaum, Z. W. On the Mathematics of Competing Risks (US Dept. of Health, Education, and Welfare, Public Health Service, Office of the Assistant Secretary for Health, National Center for Health Statistics, 1979).

  21. Bjørnstad, O. N., Finkenstädt, B. F. & Grenfell, B. T. Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR model. Ecol. Monogr. 72, 169–184 (2002).

    Article  Google Scholar 

  22. Bharti, N., Xia, Y., Bjornstad, O. N. & Grenfell, B. T. Measles on the edge: coastal heterogeneities and infection dynamics. PLoS ONE 3, e1941 (2008).

    Article  Google Scholar 

  23. Bjørnstad, O. N. & Falck, W. Nonparametric spatial covariance functions: estimation and testing. Environ. Ecol. Stat. 8, 53–70 (2001).

    Article  Google Scholar 

  24. Graham, M. et al. Measles and the canonical path to elimination. Science 364, 584–587 (2019).

    Article  CAS  Google Scholar 

  25. Lloyd, A. L. & May, R. M. Spatial heterogeneity in epidemic models. J. Theor. Biol. 179, 1–11 (1996).

    Article  CAS  Google Scholar 

  26. Ramsay, M. E. et al. The elimination of indigenous measles transmission in England and Wales. J. Infect. Dis. 187, S198–S207 (2003).

    Article  Google Scholar 

  27. Jin, L., Brown, D. W., Ramsay, M. E., Rota, P. A. & Bellini, W. J. The diversity of measles virus in the United Kingdom, 1992–1995. J. Gen. Virol. 78, 1287–1294 (1997).

    Article  CAS  Google Scholar 

  28. Jansen, Va. A. et al. Measles outbreaks in a population with declining vaccine uptake. Science 301, 804–804 (2003).

    Article  CAS  Google Scholar 

  29. Gastañaduy, P. A. et al. Impact of public health responses during a measles outbreak in an Amish community in Ohio: modeling the dynamics of transmission. Am. J. Epidemiol. 187, 2002–2010 (2018).

    Article  Google Scholar 

  30. Metcalf, C. J. E. et al. Use of serological surveys to generate key insights into the changing global landscape of infectious disease. Lancet 388, 728–730 (2016).

    Article  Google Scholar 

  31. van Panhuis, W. G. et al. Contagious diseases in the United States from 1888 to the present. N. Engl. J. Med. 369, 2152–2158 (2013).

    Article  Google Scholar 

  32. Kermack, W. O. & McKendrick, A. G. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Math. Phys. Eng. Sci. 115, 700–721 (1927).

    Google Scholar 

  33. Keeling, M. J. & Rohani, P. Modeling Infectious Diseases in Humans and Animals (Princeton Univ. Press, 2008).

  34. Bjornstad, O. N. Package ‘ncf’: Spatial nonparametric covariance functions v.1.1–7 (R Foundation for Statistical Computing, 2016); http://CRAN.R-project.org/package=ncf

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Acknowledgements

We thank the RAPIDD Program of the US Department of Homeland Security and the Fogarty International Centre, National Institutes of Health (NIH). H.M.K. was also supported by the Eunice Kennedy Shriver National Institute of Child Health & Human Development of the NIH under award number P2CHD047879. A.D.B. was supported by a National Science Foundation Graduate Research Fellowship.

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Author notes

  1. These authors contributed equally: Max S. Y. Lau, Alexander D. Becker.

Authors and Affiliations

  1. Department of Biostatistics and Bioinformatics, Rollins School of Public Health, Emory University, Atlanta, GA, USA

    Max S. Y. Lau

  2. Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ, USA

    Alexander D. Becker, Hannah M. Korevaar, Quentin Caudron, C. Jessica E. Metcalf & Bryan T. Grenfell

  3. Royal (Dick) School of Veterinary Studies & The Roslin Institute, University of Edinburgh, Midlothian, UK

    Darren J. Shaw

  4. Center for Infectious Disease Dynamics, Department of Biology, Pennsylvania State University, University Park, PA, USA

    Ottar N. Bjørnstad

  5. Fogarty International Center, National Institutes of Health, Bethesda, MD, USA

    Bryan T. Grenfell

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Contributions

M.S.Y.L. designed the research. M.S.Y.L., A.D.B. and B.T.G. performed the research. M.S.Y.L. analysed the data. M.S.Y.L., A.D.B., H.M.K., Q.C., D.J.S., C.J.E.M., O.N.B. and B.T.G. wrote the paper.

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Correspondence to Max S. Y. Lau.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary Figs. 1 and 2, Tables 1 and 2, and Notes 1–3.

Reporting Summary

Spatiotemporal dynamics of measles in England and Wales 1944-1966

Pre-vaccination spatiotemporal dynamics of measles panning 1,451 urban and rural areas from 1944-1966. Dynamics remain strongly coupled throughout this era. Data are aggregated at the monthly scale, however weekly data are available within this manuscript.

Spatiotemporal dynamics of measles in England and Wales 1944-1994

Long-term spatiotemporal dynamics of measles panning 354 urban and rural areas from 1944 through to 1994. Dynamics are strongly coupled until the start of a mass measles-containing vaccination program in 1968, after which dynamics became de-coupled and erratic. Data are aggregated at the monthly scale, however weekly data are available within this manuscript.

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Lau, M.S.Y., Becker, A.D., Korevaar, H.M. et al. A competing-risks model explains hierarchical spatial coupling of measles epidemics en route to national elimination. Nat Ecol Evol 4, 934–939 (2020). https://doi.org/10.1038/s41559-020-1186-6

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