Comprehensive testing schemes, followed by adequate contact tracing and isolation, represent the best public health interventions we can employ to reduce the impact of an ongoing epidemic when no or limited vaccine supplies are available and the implications of a full lockdown are to be avoided. However, the process of tracing can prove feckless for highly-contagious viruses such as SARS-CoV-2. The interview-based approaches often miss contacts and involve significant delays, while digital solutions can suffer from insufficient adoption rates or inadequate usage patterns. Here we present a novel way of modelling different contact tracing strategies, using a generalized multi-site mean-field model, which can naturally assess the impact of manual and digital approaches alike. Our methodology can readily be applied to any compartmental formulation, thus enabling the study of more complex pathogen dynamics. We use this technique to simulate a newly-defined epidemiological model, SEIR-T, and show that, given the right conditions, tracing in a COVID-19 epidemic can be effective even when digital uptakes are sub-optimal or interviewers miss a fair proportion of the contacts.