  We develop an analytical approach to the susceptible-infected-susceptible
(SIS) epidemic model that allows us to unravel the true origin of the absence
of an epidemic threshold in heterogeneous networks. We find that a delicate
balance between the number of high degree nodes in the network and the
topological distance between them dictates the existence or absence of such a
threshold. In particular, small-world random networks with a degree
distribution decaying slower than an exponential have a vanishing epidemic
threshold in the thermodynamic limit.
