  The structure of a network dramatically affects the spreading phenomena
unfolding upon it. The contact distribution of the nodes has long been
recognized as the key ingredient in influencing the outbreak events. However,
limited knowledge is currently available on the role of the weight of the edges
on the persistence of a pathogen. At the same time, recent works showed a
strong influence of temporal network dynamics on disease spreading. In this
work we provide an analytical understanding, corroborated by numerical
simulations, about the conditions for infected stable state in weighted
networks. In particular, we reveal the role of heterogeneity of edge weights
and of the dynamic assignment of weights on the ties in the network in driving
the spread of the epidemic. In this context we show that when weights are
dynamically assigned to ties in the network an heterogeneous distribution is
able to hamper the diffusion of the disease, contrary to what happens when
weights are fixed in time.
