  Many real networks present a bounded scale-free behavior with a connectivity
cut-off due to physical constraints or a finite network size. We study epidemic
dynamics in bounded scale-free networks with soft and hard connectivity
cut-offs. The finite size effects introduced by the cut-off induce an epidemic
threshold that approaches zero at increasing sizes. The induced epidemic
threshold is very small even at a relatively small cut-off, showing that the
neglection of connectivity fluctuations in bounded scale-free networks leads to
a strong over-estimation of the epidemic threshold. We provide the expression
for the infection prevalence and discuss its finite size corrections. The
present work shows that the highly heterogeneous nature of scale-free networks
does not allow the use of homogeneous approximations even for systems of a
relatively small number of nodes.
