We explore how heterogeneity in the intensity of interactions between people affects epidemic spreading. For that, we study the susceptible-infected-susceptible model on a complex network, where a link connecting individuals i and j is endowed with an infection rate β(ij)=λw(ij) proportional to the intensity of their contact w(ij), with a distribution P(w(ij)) taken from face-to-face experiments analyzed in Cattuto et al. [PLoS ONE 5, e11596 (2010)]. We find an extremely slow decay of the fraction of infected individuals, for a wide range of the control parameter λ. Using a distribution of width a we identify two large regions in the a-λ space with anomalous behaviors, which are reminiscent of rare region effects (Griffiths phases) found in models with quenched disorder. We show that the slow approach to extinction is caused by isolated small groups of highly interacting individuals, which keep epidemics alive for very long times. A mean-field approximation and a percolation approach capture with very good accuracy the absorbing-active transition line for weak (small a) and strong (large a) disorder, respectively.