Predicting the diffusion of real-world contagion processes requires a simplified description of human-to-human interactions. Temporal networks offer a powerful means to develop such a mathematically-transparent description. Through temporal networks, one may analytically study the co-evolution of the contagion process and the network topology, as well as incorporate realistic feedback-loop mechanisms related to individual behavioral changes to the contagion. Despite considerable progress, the state-of-the-art does not allow for studying general time-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, in which reducible, associated with node-specific properties, and irreducible links, describing dyadic social ties, simultaneously vary over time. We develop a general mean field theory for the Susceptible-Infected-Susceptible model and conduct an extensive numerical campaign to elucidate the role of network parameters on the average degree of the temporal network and the epidemic threshold. Specifically, we describe how the interplay between reducible and irreducible links influences the disease dynamics, offering insights towards the analysis of complex dynamical networks across science and engineering.