Epidemic models with digital and manual contact tracing

We analyze a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and combination with manual tracing. We prove that as the population size $n$ grows large, the epidemic process converges to a limiting process, which, unlike typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number $R$. This is then converted to an individual reproduction number $R^{(ind)}$, which, contrary to $R$, decays monotonically with the fraction of app-users while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction app-users $\pi_c$ required for $R=1$ is higher than the critical fraction manually contact traced $p_c$ for manual tracing.