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 (R0) 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 R0 = 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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We thank our colleagues V. Dubyanski and V. Ageyev for their assistance with interpreting publications and discussions about the PreBalkhash plague system, its surveillance and the host population of great gerbils. We thank J. Taxidis for help with coding the network model. We also thank J. Rees for her advice and encouragement. The work of S.D. and J.A.P.H. was supported by the Netherlands Organisation for Scientific Research (NWO/ZonMw grant 918.56.620). We also acknowledge the support of the UK Joint Environment and Human Health Programme (funders: Natural Environment Research Council, Defra, Environment Agency, Ministry of Defence, Medical Research Council).
Author Contributions S.D. proposed and pursued the question of what mechanism generated the plague threshold, constructed and analysed the network model and wrote the paper. P.T. introduced S.D. to percolation theory and contributed to the writing of the Supplementary Information. All authors discussed percolation and the plague system and made comments on the manuscript.
The file contains Supplementary Discussion, Supplementray Figures S1-S4 with Legends and additional references. This file contains a more comprehensive discussion of the network model, its assumptions, its limitations and the sensitivity of the results to changes in the parameters. The additional text is accompanied by four additional figures. (PDF 1881 kb)
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Davis, S., Trapman, P., Leirs, H. et al. The abundance threshold for plague as a critical percolation phenomenon. Nature 454, 634–637 (2008). https://doi.org/10.1038/nature07053
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