  We use the connection between bond percolation and SIR epidemics to establish
lower bounds for the critical percolation probability in $2$ and $3$ dimensions
as the range becomes large. The bound agrees with the conjectured asymptotics
for the long range critical probability, refines results of M. Penrose, and
complements results of van der Hofstad and Sakai in dimensions greater than
$6$.
