  With the hit of new pandemic threats, scientific frameworks are needed to
understand the unfolding of the epidemic. The use of mobile apps that are able
to trace contacts is of utmost importance in order to control new infected
cases and contain further propagation. Here we present a theoretical approach
using both percolation and message--passing techniques, to the role of contact
tracing, in mitigating an epidemic wave. We show how the increase of the app
adoption level raises the value of the epidemic threshold, which is eventually
maximized when high-degree nodes are preferentially targeted. Analytical
results are compared with extensive Monte Carlo simulations showing good
agreement for both homogeneous and heterogeneous networks. These results are
important to quantify the level of adoption needed for contact-tracing apps to
be effective in mitigating an epidemic.
