Theoretical Approaches to the Susceptible-Infected-Susceptible Dynamics on Complex Networks: Mean-Field Theories and Beyond

Models for epidemic spreading are a paradigmatic example of the nontrivial effect of a topologically complex substrate on simple dynamics. The Susceptible-Infected-Susceptible (SIS) model has been recently the subject of considerable activity, which has uncovered that subtle effects of the quenched network structure and of dynamical correlations play a relevant role in the model behavior. Starting from the simplest mean-field theory we review the theoretical approaches which have been applied to SIS dynamics on networks, including the very recent results taking into account long-range dynamical correlations. The findings illustrate the highly nontrivial interplay between complex topology and dynamics and the need to include additional ingredients beyond the usual mean-field assumptions.