Fig. 6: A relaxation of restrictions can slowly overwhelm the finite tracing capacity and trigger a new outbreak.

a At t = 0, the hidden reproduction number increases from \({R}_{t}^{H}=1.8\) to \({R}_{t}^{H}=2.0\) (i.e., slightly above its critical value). This leads to a slow increase in traced active cases (solid blue line). b When the number of observed new cases (solid brown line) exceeds the tracing capacity limit \({N}_{\max }\) (solid gray line), the tracing system breaks down, and the outbreak starts to accelerate. c After an initial transient at the onset of the change in \({R}_{t}^{H}\), the observed reproduction number (solid red line) faithfully reflects both the slight increase of the hidden reproduction number due to relaxation of contact constraints, and the strong increase after the tracing capacity (solid gray line) is exceeded at t ≈ 100. In both cases, the observed reproduction number \({\hat{R}}_{t}^{\,\text{obs}\,}\) approaches two different limit values R_∞, which are derived from a linear stability analysis (further details in Supplementary Fig. 5). All the curves plotted are obtained from numerical integration of Eqs. (1)–(5).