  We study the threshold of epidemic models in quenched networks with degree
distribution given by a power-law. For the susceptible-infected-susceptible
(SIS) model the activity threshold lambda_c vanishes in the large size limit on
any network whose maximum degree k_max diverges with the system size, at odds
with heterogeneous mean-field (HMF) theory. The vanishing of the threshold has
not to do with the scale-free nature of the connectivity pattern and is instead
originated by the largest hub in the system being active for any spreading rate
lambda>1/sqrt{k_max} and playing the role of a self-sustained source that
spreads the infection to the rest of the system. The
susceptible-infected-removed (SIR) model displays instead agreement with HMF
theory and a finite threshold for scale-rich networks. We conjecture that on
quenched scale-rich networks the threshold of generic epidemic models is
vanishing or finite depending on the presence or absence of a steady state.
