  We study the spreading of a disease on top of structured scale-free networks
recently introduced. By means of numerical simulations we analyze the SIS and
the SIR models. Our results show that when the connectivity fluctuations of the
network are unbounded whether the epidemic threshold exists strongly depends on
the initial density of infected individuals and the type of epidemiological
model considered. Analytical arguments are provided in order to account for the
observed behavior. We conclude that the peculiar topological features of this
network and the absence of small-world properties determine the dynamics of
epidemic spreading.
