  We describe the phenomenon of localization in the epidemic SIS model on
highly heterogeneous networks in which strongly connected nodes (hubs) play the
role of centers of localization. We find that in this model the localized
states below the epidemic threshold are metastable. The longevity and scale of
the metastable outbreaks do not show a sharp localization transition, instead
there is a smooth crossover from localized to delocalized states as we approach
the epidemic threshold from below. Analyzing these long-lasting local outbreaks
for a random regular graph with a hub, we show how this localization can be
detected from the shape of the distribution of the number of infective nodes.
