We introduce the ε-susceptible-infected-susceptible (SIS) spreading model, which is taken as a benchmark for the comparison between the N-intertwined approximation and the Pastor-Satorras and Vespignani heterogeneous mean-field (HMF) approximation of the SIS model. The N-intertwined approximation, the HMF approximation, and the ε-SIS spreading model are compared for different graph types. We focus on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions. Overall, the N-intertwined approximation is superior to the HMF approximation. The N-intertwined approximation is exactly the same as the HMF approximation in regular graphs. However, for some special graph types, such as the square lattice graph and the path graph, the two mean-field approximations are both very different from the ε-SIS spreading model.
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