Models of infectious diseases are characterized by a phase transition between extinction and persistence. A challenge in contemporary epidemiology is to understand how the geometry of a host’s interaction network influences disease dynamics close to the critical point of such a transition. Here we address this challenge with the help of moment closures. Traditional moment closures, however, do not provide satisfactory predictions close to such critical points. We therefore introduce a new method for incorporating longer-range correlations into existing closures. Our method is technically simple, remains computationally tractable and significantly improves the approximation’s performance. Our extended closures thus provide an innovative tool for quantifying the influence of interaction networks on spatially or socially structured disease dynamics. In particular, we examine the effects of a network’s clustering coefficient, as well as of new geometrical measures, such as a network’s square clustering coefficients. We compare the relative performance of different closures from the literature, with or without our long-range extension. In this way, we demonstrate that the normalized version of the Bethe approximation–extended to incorporate long-range correlations according to our method–is an especially good candidate for studying influences of network structure. Our numerical results highlight the importance of the clustering coefficient and the square clustering coefficient for predicting disease dynamics at low and intermediate values of transmission rate, and demonstrate the significance of path redundancy for disease persistence.Read less <