We study the effectiveness of tracking and testing in mitigating or suppressing epidemic outbreaks, in combination with or as an alternative to quarantines and global lockdowns. We study these intervention methods on a network-based SEIR model, augmented with an additional probability to model symptomatic, asymptomatic and pre-symptomatic cases. Our focus is on the basic trade-offs between economic costs and human lives lost, and how these trade-offs change under different lockdown, quarantine, tracking and testing policies. Our main findings are as follows: (i) Tests combined with patient quarantines reduce both economic costs and mortality, but require a large-scale testing capacity to achieve a significant improvement; (ii) Tracking significantly reduces both economic costs and mortality; (iii) Tracking combined with a limited number of tests can achieve containment without lockdowns; (iv) If there is a small flow of new incoming infections, dynamic On-Off lockdowns are more efficient than fixed lockdowns. Our simulation results underline the extreme effectiveness of tracking and testing policies in reducing both economic costs and mortality and their potential to contain epidemic outbreaks without imposing social distancing restrictions. This highlights the difficult social question of trading-off these gains with the privacy loss that tracking necessarily entails.