  An upper bound for the critical probability of long range bond percolation in
$d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR
epidemic model, thus complementing the lower bound result in Frei and Perkins
arXiv:arch-ive/1603.04130. A key ingredient is that we establish a uniform
bound for the local times of branching random walk by calculating their
exponential moments and by using the discrete versions of Tanaka's formula and
Garsia's Lemma.
