  We develop a theory for the susceptible-infected-susceptible (SIS) epidemic
model on networks that incorporate both network structure and dynamic
correlations. This theory can account for the multistage onset of the epidemic
phase in scale-free networks. This phenomenon is characterized by multiple
peaks in the susceptibility as a function of the infection rate. It can be
explained by that, even under the global epidemic threshold, a hub can sustain
the epidemics for an extended period. Moreover, our approach improves
theoretical calculations of prevalence close to the threshold in heterogeneous
networks and also can predict the average risk of infection for neighbors of
nodes with different degree and state on uncorrelated static networks.
