  Effects of heterogeneity in the suspected-infected-susceptible model on
networks are investigated using quenched mean-field theory. The emergence of
localization is described by the distributions of the inverse participation
ratio and compared with the rare-region effects appearing in simulations and in
the Lifschitz tails. The latter, in the linear approximation, is related to the
spectral density of the Laplacian matrix and to the time dependent order
parameter. I show that these approximations indicate correctly Griffiths Phases
both on regular one-dimensional lattices and on small world networks exhibiting
purely topological disorder. I discuss the localization transition that occurs
on scale-free networks at $\gamma=3$ degree exponent.
