Mass testing is a promising strategy to control future respiratory virus pandemics, however, it is limited by the availability of testing capacity. The aim of this work is to predict the effectiveness of contact tracing when used with mass testing, and to what extent contact tracing can reduce the required number of tests. We estimate the effect of contact tracing and mass testing for controlling a variety of respiratory virus pandemics. To do this, we use a branching model with individuals whose infectiousness and probability of testing positive depend on simulated viral load trajectories. We find that the addition of contact tracing is most useful when mass testing is done frequently enough to detect most infections, but cannot be done frequently enough to reliably isolate cases before they infect others. Our results show that the addition of effective contact tracing can prevent the same number of transmissions as doubling the mass testing frequency.