  We study the SIR ("susceptible, infected, removed/recovered") model on
directed graphs with heterogeneous transmission probabilities within the
message-passing approximation. We characterize the percolation transition,
predict cluster size distributions and suggest vaccination strategies. All
predictions are compared to numerical simulations on real networks. The
percolation threshold which we predict is a rigorous lower bound to the
threshold on real networks. For large, locally tree-like networks, our
predictions agree very well with the numerical data.

Physics and Society;Statistical Mechanics;Social and Information Networks