  A population can be immune to epidemics even if not all of its individual
members are immune to the disease, so long as sufficiently many are immune -
this is the traditional notion of herd immunity. In the smartphone era a
population can be immune to epidemics even if not a single one of its members
is immune to the disease - a notion we call "digital herd immunity", which is
similarly an emergent characteristic of the population. This immunity arises
because contact-tracing protocols based on smartphone capabilities can lead to
highly efficient quarantining of infected population members and thus the
extinguishing of nascent epidemics. When the disease characteristics are
favorable and smartphone usage is high enough, the population is in this immune
phase. As usage decreases there is a novel "contact-tracing phase transition"
to an epidemic phase. We present and study a simple branching-process model for
COVID-19 and show that digital immunity is possible regardless of the
proportion of non-symptomatic transmission.
