Contagion processes have been proven to fundamentally depend on the structural properties of the interaction networks conveying them. Many real networked systems -- especially social ones -- are characterized by clustered substructures representing either collections of all-to-all pair-wise interactions (cliques) and/or group interactions, involving many of their members at once. In this work we focus on interaction structures represented as simplicial complexes, in which a group interaction is identified with a face. We present a microscopic discrete-time model of complex contagion for which a susceptible-infected-susceptible dynamics is considered. Introducing a particular edge clique cover and a heuristic to find it, the model accounts for the high-order state correlations among the members of the substructures (cliques/simplices). The mathematical tractability of the model allows for the analytical computation of the epidemic threshold, thus extending to structured populations some primary features of the critical properties of mean-field models. Overall, the model is found in remarkable agreement with numerical simulations.