  We use the pair heterogeneous mean-field (PHMF) approximation for an
asynchronous version of the susceptible-infected-removed (SIR) model to
estimate the epidemic thresholds on complex quenched networks. Our results
indicate an improvement compared to the heuristic heterogeneous mean-field
theory developed for one vertex (HMF) when the dynamic evolves on top random
regular and power-law networks. However, there is a slight overestimation of
the transition point for the later network type. We also analyze scaling for
random regular networks near the thresholds. For this region, collapses were
shown at the subcritical and supercritical phases.
