The Accuracy of Mean-Field Approximation for Susceptible-Infected-Susceptible Epidemic Spreading with Heterogeneous Infection Rates

The epidemic spreading over a network has been studied for years by applying the mean-field approach in both homogeneous case, where each node may get infected by an infected neighbor with the same rate, and heterogeneous case, where the infection rates between different pairs of nodes are also different. Researchers have discussed whether the mean-field approaches could accurately describe the epidemic spreading for the homogeneous cases but not for the heterogeneous cases. In this paper, we explore if and under what conditions the mean-field approach could perform well when the infection rates are heterogeneous. In particular, we employ the Susceptible-Infected-Susceptible (SIS) model and compare the average fraction of infected nodes in the metastable state, where the fraction of infected nodes remains stable for a long time, obtained by the continuous-time simulation and the mean-field approximation. We concentrate on an individual-based mean-field approximation called the N-intertwined Mean Field Approximation (NIMFA), which is an advanced approach considered the underlying network topology. Moreover, for the heterogeneity of the infection rates, we consider not only the independent and identically distributed (i.i.d.) infection rate but also the infection rate correlated with the degree of the two end nodes. We conclude that NIMFA is generally more accurate when the prevalence of the epidemic is higher. Given the same effective infection rate, NIMFA is less accurate when the variance of the i.i.d. infection rate or the correlation between the infection rate and the nodal degree leads to a lower prevalence. Moreover, given the same actual prevalence, NIMFA performs better in the cases: 1) when the variance of the i.i.d. infection rates is smaller (while the average is unchanged); 2) when the correlation between the infection rate and the nodal degree is positive. Our work suggests the conditions when the mean-field approach, in particular NIMFA, is more accurate in the approximation of the SIS epidemic with heterogeneous infection rates.