Dependent random graphs and spatial

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 Z d. In these models, infected individuals eventually die (are removed) and are not replaced. We combine these results with certain continuity and re-scaling arguments in order to improve our knowledge of the phase diagram of a modiied 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 a certain two critical values, no coexistence is possible for suuciently small positive values of the recovery rate. This result provides a converse to results of Durrett{Neuhauser and Andjel{Schinazi. We show also that such an intermediate phase indeed exists for every d 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.