Letter

Nature 454, 634-637 (31 July 2008) | doi:10.1038/nature07053; Received 31 January 2008; Accepted 30 April 2008

The abundance threshold for plague as a critical percolation phenomenon

S. Davis1, P. Trapman2, H. Leirs3,4, M. Begon5 & J. A. P. Heesterbeek1

  1. Theoretical Epidemiology, Faculty of Veterinary Medicine, University of Utrecht, Yalelaan 7, 3584 CL Utrecht, The Netherlands
  2. Julius Center for Health Sciences and Primary Care, University Medical Center Utrecht, PO Box 85500, 3508 GA Utrecht, The Netherlands
  3. Department of Biology, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium
  4. Danish Pest Infestation Laboratory, University of Aarhus, Faculty of Agricultural Sciences, Department of Integrated Pest Management, Skovbrynet 14, DK-2800 Kongens Lyngby, Denmark
  5. Host-Parasite Biology Research Group, School of Biological Sciences, University of Liverpool, Crown Street, Liverpool L69 7ZB, UK

Correspondence to: S. Davis1 Correspondence and requests for materials should be addressed to S.D. (Email: S.A.Davis@uu.nl).

Percolation theory is most commonly associated with the slow flow of liquid through a porous medium, with applications to the physical sciences1. Epidemiological applications have been anticipated for disease systems where the host is a plant or volume of soil2, 3, and hence is fixed in space. However, no natural examples have been reported. The central question of interest in percolation theory4, the possibility of an infinite connected cluster, corresponds in infectious disease to a positive probability of an epidemic. Archived records of plague (infection with Yersinia pestis) in populations of great gerbils (Rhombomys opimus) in Kazakhstan have been used to show that epizootics only occur when more than about 0.33 of the burrow systems built by the host are occupied by family groups5. The underlying mechanism for this abundance threshold is unknown. Here we present evidence that it is a percolation threshold, which arises from the difference in scale between the movements that transport infectious fleas between family groups and the vast size of contiguous landscapes colonized by gerbils. Conventional theory predicts that abundance thresholds for the spread of infectious disease arise when transmission between hosts is density dependent such that the basic reproduction number (R 0) increases with abundance, attaining 1 at the threshold. Percolation thresholds, however, are separate, spatially explicit thresholds that indicate long-range connectivity in a system and do not coincide with R 0 = 1. Abundance thresholds are the theoretical basis for attempts to manage infectious disease by reducing the abundance of susceptibles, including vaccination and the culling of wildlife6, 7, 8. This first natural example of a percolation threshold in a disease system invites a re-appraisal of other invasion thresholds, such as those for epidemic viral infections in African lions (Panthera leo), and of other disease systems such as bovine tuberculosis (caused by Mycobacterium bovis) in badgers (Meles meles).

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