Fig. 2. Stochastic vs. prearranged sampling in contact tracing.

a Plot, as a function of ϵ = f², of the ratio between the epidemic threshold r_C in the presence of CT protocols and the epidemic threshold of the non-adaptive case $r_{\mathrm{C}}^{\mathrm{NA}}$. In the inset we plot the ratio between the epidemic threshold of the manual CT $r_{\mathrm{C}}^{\mathrm{MANUAL}}$ and that of the app-based CT $r_{\mathrm{C}}^{\mathrm{APP}}$, as a function of ϵ = f². We consider homogeneous activity and attractiveness, setting ρ(a_S, b_S) = δ(a_S − a)δ(b_S − b). b If an asymptomatic node A infects a single susceptible node S (which subsequently becomes symptomatic, I), the infector is traced with probability ϵ in the manual CT and with probability f² in the digital CT. Thus the probability is the same if we impose f² = ϵ. However, if an asymptomatic node infects two susceptible nodes (which subsequently become symptomatic), the probability of tracing the infector with the manual protocol is 1 − (1−ϵ)² = 2f² − f⁴ (still considering ϵ = f²). This value is always larger than that of the digital protocol f(1 − (1−f )²) = 2f² − f³.