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.