  Contacts between people are the absolute drivers of contagious respiratory
infections. For this reason, limiting and tracking contacts is a key strategy
for the control of the COVID-19 epidemic. Digital contact tracing has been
proposed as an automated solution to scale up traditional contact tracing.
However, the required penetration of contact tracing apps within a population
to achieve a desired target in the control of the epidemic is currently under
discussion within the research community. In order to understand the effects of
digital contact tracing, several mathematical models have been proposed. In
this article, we survey the main ones and we propose a compartmental SEIR model
with which it is possible, differently from the models in the related
literature, to derive closed-form conditions regarding the control of the
epidemic as a function of the contact tracing apps penetration and the testing
efficiency. Closed-form conditions are crucial for the understandability of
models, and thus for decision makers (including digital contact tracing
designers) to correctly assess the dependencies within the epidemic. With our
model, we find that digital contact tracing alone can rarely tame an epidemic:
for unrestrained COVID-19, this would require a testing turnaround of around 1
day and app uptake above 80% of the population, which are very difficult to
achieve in practice. However, digital contact tracing can still be effective if
complemented with other mitigation strategies, such as social distancing and
mask-wearing.
