Virus spread versus contact tracing: two competing contagion processes

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.