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 $\varepsilon$-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 ${\tau }_c$ in the star graph, which is found as ${\tau }_c=\alpha {\tau }_c^{\left(1\right)}$, where $\alpha =\sqrt{\frac{1}{2}\log N+\frac{3}{2}\log \log N}$ and where ${\tau }_c^{\left(1\right)}=\frac{1}{\sqrt{N}}<{\tau }_c$ 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.

• Received 2 August 2012

DOI:https://doi.org/10.1103/PhysRevE.87.012811