## 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

### Citations:

Zbl 0733.92022; Zbl 0859.92020; Zbl 0509.60094
Full Text:

### References:

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