Epidemic threshold and localization of the SIS model on directed complex networks

We study the susceptible-infected-susceptible (SIS) model on directed complex networks within the quenched mean-field approximation. Combining results from random matrix theory with an analytic approach to the distribution of fixed-point infection probabilities, we derive the phase diagram and show that the model exhibits a nonequilibrium phase transition between the absorbing and endemic phases for c≥λ^{-1}, where c is the mean degree and λ is the average infection rate. Interestingly, the critical line is independent of the degree distribution but is highly sensitive to the form of the infection-rate distribution. We further show that the inverse participation ratio of infection probabilities diverges near the epidemic threshold, indicating that the disease may become localized on a small fraction of nodes. These results provide a systematic characterization of how network heterogeneities shape epidemic spreading on directed contact networks within the quenched mean-field approximation.