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We present a comparison between stochastic simulations and mean-fieldtheories for the epidemic threshold of the susceptible-infected-susceptible(SIS) model on correlated networks (both assortative and disassortative) withpower-law degree distribution $P(k)\sim k^{-\gamma}$. We confirm the vanishingof the threshold regardless of the correlation pattern and the degree exponent$\gamma$. Thresholds determined numerically are compared with quenchedmean-field (QMF) and pair quenched mean-field (PQMF) theories. Correlations donot change the overall picture: QMF and PQMF provide estimates that areasymptotically correct for large size for $\gamma<5/2$, while they only capturethe vanishing of the threshold for $\gamma>5/2$, failing to reproducequantitatively how this occurs. For a given size, PQMF is more accurate. Werelate the variations in the accuracy of QMF and PQMF predictions with changesin the spectral properties (spectral gap and localization) of standard andmodified adjacency matrices, which rule the epidemic prevalence near thetransition point, depending on the theoretical framework. We also show that,for $\gamma<5/2$, while QMF provides an estimate of the epidemic threshold thatis asymptotically exact, it fails to reproduce the singularity of theprevalence around the transition.

Spectral properties and the accuracy of mean-field approaches for epidemics on correlated networks