We discuss a continuous-time extension of the contact-based (CB) model, as proposed in [Koher et al. Phys. Rev. X 9, 031017 (2019)], for infections with permanent immunity on temporal networks. At the core of our methodology is a fundamental change to an edge-centered perspective, which allows for an accurate model on temporal networks, where the underlying time-aggregated graph has a tree structure. From the continuous-time CB model, we derive the infection propagator for the low prevalence limit and propose a novel spectral criterion to estimate the epidemic threshold. In addition, we explore the relation between the continuous-time CB model and the previously proposed edge-based compartmental model, as well as the message-passing framework.