  We consider the Markovian SIR epidemic model, but where individuals either
recover naturally or are diagnosed, the latter implying isolation and subject
to being contact traced. We focus on the novel digital contact tracing based on
individuals using a tracing app and consider both the situation when this is
the only form of contact tracing, and when combined with manual contact
tracing. It is proven that the epidemic process converges to a limiting process
as the population size $n$ grows large. The limiting process is not a branching
process (as is common for epidemic processes) because infectees may be contact
traced "from" their infectors, thus creating dependencies. But by instead
treating to-be-traced components as macro-individuals allows a multi-type
branching process interpretation, thus giving an expression for the
reproduction number $R$. This reproduction number (for the digital-only case)
is then converted to an individual reproduction number $R^{(ind)}$ being easier
to interpret. It is proven that, contrary to $R$, $R^{(ind)}$ decays
monotonically with $\pi$ (the fraction of app-users), and the two reproduction
numbers are proven to have the same threshold at 1. The digital (only) contact
tracing is then compared with earlier results for manual (only) contact tracing
where a fraction $p$ of all contacts are traced. It is proven that the critical
fraction app-users $\pi_c$ for which digital contact tracing is reduced to
$R=1$, and the critical fraction manually contact traced $p_c$ to reach $R=1$,
satisfy $\pi_c>p_c$.

Probability;Populations and Evolution