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Dependent random graphs and spatial epidemics. (English) Zbl 0946.92028

The authors study a model of spatial epidemics in \(\mathbb{Z}^d\), where each site is in one of 3 possible states: dead, susceptible or ill. Taking the death rate to be 1, it is characterized by the infection rate \(\alpha\) with \(0<\alpha<\infty\) and the rate \(\beta\) of the birth of new susceptibles with \(0\leq\beta\leq\infty\). R. Durrett and C. Neuhauser [Ann. Appl. Probab. 1, No. 2, 189-206 (1991; Zbl 0733.92022)] and E. Andjel and R. Schinazi [J. Appl. Probab. 33, No. 3, 741-748 (1996; Zbl 0859.92020)] showed coexistence for certain regions in the \((\alpha,\beta)\) phase space. In this paper the phase diagram is extended.
It is proved that for \(\alpha\) lying between two critical values, which are shown to be distinct, no coexistence is possible for sufficiently small \(\beta>0\). An important part of the proof are exponential decay estimates of subcritical percolations for a class of locally dependent random graphs, introduced by K. Kuulasmaa [ibid. 19, 745-758 (1982; Zbl 0509.60094)] for an epidemic model without recovery \((\beta=0)\). These results are of interest in their own.

MSC:

92D30 Epidemiology
60K35 Interacting random processes; statistical mechanics type models; percolation theory
05C80 Random graphs (graph-theoretic aspects)
82C43 Time-dependent percolation in statistical mechanics
82B43 Percolation
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