  The challenges presented by the COVID-19 epidemic have created a renewed
interest in the development of new methods to combat infectious diseases. A
prominent property of the SARS-CoV-2 transmission is the significant fraction
of asymptomatic transmission. This may influence the effectiveness of the
standard contact tracing procedure for quarantining potentially infected
individuals. However, the effects of asymptomatic transmission on the epidemic
threshold of epidemic spreading on networks are largely unknown. Here we study
the critical percolation transition in a simple epidemic network model in the
presence of a recursive contact tracing algorithm for instant quarantining. We
find that, above a certain fraction of asymptomatic transmission, standard
contact tracing loses its ability to suppress spreading below the epidemic
threshold. However, we also find that recursive contact tracing opens a
possibility to contain epidemics with a large fraction of asymptomatic or
presymptomatic transmission. In particular, we calculate the required fraction
of network nodes participating in the contact tracing for networks with
arbitrary degree distributions and for varying recursion depths and discuss the
influence of recursion depth and asymptomatic rate on the epidemic percolation
phase transition. We test and illustrate our theoretical results using
numerical simulations on infection trees and networks. We anticipate recursive
contact tracing to provide a basis for digital, app-based contact tracing tools
that extend the efficiency of contact tracing to diseases with a large fraction
of asymptomatic transmission.
