Susceptible-infected-susceptible epidemics on the complete graph and the star graph: exact analysis

Since mean-field approximations for susceptible-infected-susceptible (SIS) epidemics do not always predict the correct scaling of the epidemic threshold of the SIS metastable regime, we propose two novel approaches: (a) an ε-SIS generalized model and (b) a modified SIS model that prevents the epidemic from dying out (i.e., without the complicating absorbing SIS state). Both adaptations of the SIS model feature a precisely defined steady state (that corresponds to the SIS metastable state) and allow an exact analysis in the complete and star graph consisting of a central node and N leaves. The N-intertwined mean-field approximation (NIMFA) is shown to be nearly exact for the complete graph but less accurate to predict the correct scaling of the epidemic threshold τ(c) in the star graph, which is found as τ(c)=ατ(c)((1)), where α=√[1/2 logN + 3/2 log logN] and where τ(c)((1))=1/√[N]<τ(c) is the first-order epidemic threshold for the star in NIMFA and equal to the inverse of the spectral radius of the star's adjacency matrix.