Predicting the diffusion of real-world contagion processesrequires a simplified description of human-to-human interactions. Temporal networks offer a powerful means to developsuch a mathematically-transparent description. Through temporal networks, one may analytically study the co-evolution ofthe contagion process and the network topology, as well as incorporate realistic feedback-loop mechanisms related to individual behavioral changes to the contagion. Despite considerableprogress, the state-of-the-art does not allow for studying generaltime-varying networks, where links between individuals dynamically switch to reflect the complexity of social behavior. Here,we tackle this problem by considering a temporal network, inwhich reducible, associated with node-specific properties, andirreducible links, describing dyadic social ties, simultaneouslyvary over time. We develop a general mean field theory for theSusceptible-Infected-Susceptible model and conduct an extensivenumerical campaign to elucidate the role of network parameterson the average degree of the temporal network and the epidemicthreshold. Specifically, we describe how the interplay betweenreducible and irreducible links influences the disease dynamics,*Address all correspondence to these authors.offering insights towards the analysis of complex dynamical networks across science and engineering.