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May 1998 Dependent random graphs and spatial epidemics
Geoffrey R. Grimmett, Rinaldo B. Schinazi, J. van den Berg
Ann. Appl. Probab. 8(2): 317-336 (May 1998). DOI: 10.1214/aoap/1028903529


We extend certain exponential decay results of subcritical percolation to a class of locally dependent random graphs, introduced by Kuulasmaa as models for spatial epidemics on $\mathbb{Z}^d$. In these models, infected individuals eventually die (are removed) and are not replaced. We combine these results with certain continuity and rescaling arguments in order to improve our knowledge of the phase diagram of a modified epidemic model in which new susceptibles are born at some positive rate. In particular, we show that, throughout an intermediate phase where the infection rate lies between two certain critical values, no coexistence is possible for sufficiently small positive values of the recovery rate. This result provides a converse to results of Durrett and Neuhauser and Andjel and Schinazi. We show also that such an intermediate phase indeed exists for every $d \geq 1$ (i.e., that the two critical values mentioned above are distinct). An important technique is the general version of the BK inequality for disjoint occurrence, proved in 1994 by Reimer.


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Geoffrey R. Grimmett. Rinaldo B. Schinazi. J. van den Berg. "Dependent random graphs and spatial epidemics." Ann. Appl. Probab. 8 (2) 317 - 336, May 1998.


Published: May 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0946.92028
MathSciNet: MR1624925
Digital Object Identifier: 10.1214/aoap/1028903529

Primary: 60K35

Keywords: critical value , Dependent percolation , Interacting particle system , percolation , spatial epidemic

Rights: Copyright © 1998 Institute of Mathematical Statistics


Vol.8 • No. 2 • May 1998
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