  After the blockade that many nations suffered to stop the growth of the
incidence curve of COVID-19 during the first half of 2020, they face the
challenge of resuming their social and economic activity. The rapid airborne
transmissibility of SARS-CoV-2, and the absence of a vaccine, calls for active
containment measures to avoid the propagation of transmission chains. The best
strategy up to date, popularly known as Test-Track-Treat (TTT), consist in
testing the population for diagnosis, track the contacts of those infected, and
treat by quarantine all these cases. The dynamical process that better
describes the combined action of the former mechanisms is that of a contagion
process that competes with the spread of the pathogen, cutting off potential
contagion pathways. Here we propose a compartmental model that couples the
dynamics of the infection with the contact tracing and isolation of cases. We
develop an analytical expression for the effective case reproduction number
$R_c(t)$ that reveals the role of contact tracing in the mitigation and
suppression of the epidemics. We show that there is a trade off between the
infection propagation and the isolation of cases. If the isolation is limited
to symptomatic individuals only, the incidence curve can be flattened but not
bended. However, if contact tracing is applied to asymptomatic individuals too,
the strategy can bend the curve and suppress the epidemics. Quantitative
results are dependent on the network topology. We quantify, the most important
indicator of the effectiveness of contact tracing, namely its capacity to
reverse the increasing tendency of the epidemic curve, causing its bending.
