## The Annals of Applied Probability

### Dependent random graphs and spatial epidemics

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 2 (1998), 317-336.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903529

Digital Object Identifier
doi:10.1214/aoap/1028903529

Mathematical Reviews number (MathSciNet)
MR1624925

Zentralblatt MATH identifier
0946.92028

#### Citation

van den Berg, J.; Grimmett, Geoffrey R.; Schinazi, Rinaldo B. Dependent random graphs and spatial epidemics. Ann. Appl. Probab. 8 (1998), no. 2, 317--336. doi:10.1214/aoap/1028903529. https://projecteuclid.org/euclid.aoap/1028903529

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• CWI, KRUISLAAN 413 16 MILL LANE 1098 SJ AMSTERDAM CAMBRIDGE CB2 1SB THE NETHERLANDS UNITED KINGDOM E-MAIL: jvdberg@cwi.nl E-MAIL: g.r.grimmett@statslab.cam.ac.uk R. B. SCHINAZI MATHEMATICS DEPARTMENT UNIVERSITY OF COLORADO
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