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A swash–backwash model of the single epidemic wave

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Abstract

While there is a large literature on the form of epidemic waves in the time domain, models of their structure and shape in the spatial domain remain poorly developed. This paper concentrates on the changing spatial distribution of an epidemic wave over time and presents a simple method for identifying the leading and trailing edges of the spatial advance and retreat of such waves. Analysis of edge characteristics is used to (a) disaggregate waves into ‘swash’ and ‘backwash’ stages, (b) measure the phase transitions of areas from susceptible, S, through infective, I, to recovered, R, status (S → I → R) as dimensionless integrals and (c) estimate a spatial version of the basic reproduction number, R 0. The methods used are illustrated by application to measles waves in Iceland over a 60-year period from 1915 to 1974. Extensions of the methods for use with more complex waves are possible through modifying the threshold values used to define the start and end points of an event.

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Notes

  1. The 0.5 and the T + 0.5 in Stage IV, later, arise because time is measured as a series of integers (1, 2, 3,...), whereas the means are measured on a continuous scale.

  2. Sometimes called the basic reproduction rate or ratio.

References

  • Abramson G, Kenkre VM, Yates TL, Parmenter RR (2003) Travelling waves of infections in the Hantavirus epidemics. Bull Math Biol 65:519–534

    Article  Google Scholar 

  • Bailey NTJ (1975) The mathematical theory of infectious diseases. Griffin, London

    Google Scholar 

  • Chorley RJ, Schumm SA, Sugden DE (1984) Geomorphology. Methuen, London

    Google Scholar 

  • Cliff AD, Haggett P (1981) Methods for the measurement of epidemic velocity from time-series data. Int J Epidemiol 11:82–89

    Article  Google Scholar 

  • Cliff AD, Haggett P, Ord JK, Versey GR (1981) Spatial diffusion: an historical geography of epidemics in an island community. Cambridge University Press, Cambridge

    Google Scholar 

  • Cliff AD, Haggett P, Ord JK (1986) Spatial aspects of influenza epidemics. Pion, London

    Google Scholar 

  • Cliff AD, Haggett P, Smallman-Raynor MR (1993) Measles: an historical geography of a major human viral disease, from global expansion to local retreat, 1840–1990. Blackwell, Oxford

    Google Scholar 

  • Cliff AD, Haggett P, Smallman-Raynor MR (2000) Island epidemics. Oxford University Press, Oxford

    Google Scholar 

  • Cliff AD, Haggett P, Smallman-Raynor MR (2004) World atlas of epidemic diseases. Arnold, London

    Google Scholar 

  • Daley D, Gani J (1999) Epidemic modelling. Cambridge University Press, Cambridge

    Google Scholar 

  • Dietz K (1993) The estimation of the basic reproduction number for infectious diseases. Stat Meth Med Res 2:23–41

    Article  Google Scholar 

  • Duncan OD, Cuzzort RP, Duncan B (1961) Statistical geography: problems of analyzing areal data. Free Press, Glencoe

    Google Scholar 

  • Gilg AW (1973) A study in agricultural disease diffusion: the case of the 1970–71 fowl-pest epidemic, Institute of British Geographers, Publications 59:77–97

    Google Scholar 

  • Grenfell BT, Bjørnstad ON, Kappey J (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414:716–723

    Article  Google Scholar 

  • Haggett P (2000) The geographical structure of epidemics. Clarendon Press, Oxford

    Google Scholar 

  • Morrill R (1968) Waves of spatial diffusion. J Reg Sci 8:1–18

    Article  Google Scholar 

  • Morrill R (1970) The shape of diffusion in time and space. Econ Geogr 46:259–268

    Article  Google Scholar 

  • Strahler AN (1952) Hypsometric (area-altitude) of erosional topography. Bull Geol Soc Am 63:1117–1142

    Article  Google Scholar 

  • Trevelyan B, Smallman-Raynor M, Cliff AD (2005) The spatial dynamics of poliomyelitis in the United States: emergence to vaccine-induced retreat, 1910–1971. Ann Assoc Am Geogr 95:269–293

    Article  Google Scholar 

  • Watts DS, Muhamad R, Medina DC, Dodds PS (2005) Multiscale, resurgent epidemics in a hierarchical metapopulation model. Proc Natl Acad Sci USA 102:1157–1162

    Article  Google Scholar 

  • Zhao X-Q, Wang W (2004) Fisher waves in an epidemic model, Discrete Contin Dyn Syst B4:1117–1128

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Geography, University of Cambridge, Downing Place, Cambridge, CB2 3EN, UK

    Andrew D. Cliff

  2. Institute for Advanced Studies, University of Bristol, Royal Fort House, Bristol, BS8 1UJ, UK

    Peter Haggett

Authors
  1. Andrew D. Cliff
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  2. Peter Haggett
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Correspondence to Andrew D. Cliff.

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Cliff, A.D., Haggett, P. A swash–backwash model of the single epidemic wave. J Geograph Syst 8, 227–252 (2006). https://doi.org/10.1007/s10109-006-0027-8

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  • DOI: https://doi.org/10.1007/s10109-006-0027-8

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