Fig. 3: Finite tracing capacity makes the system vulnerable to large influx events.

A single large influx event (a total of 4000 hidden cases with 92% occurring in the 7 days around t = 0, normally distributed with standard deviation σ = 2 days) drives a metastable system with reduced tracing capacity (reached at \({N}_{\max }=470\)) to a new outbreak (d–f), whereas a metastable system with our default tracing capacity (reached at \({N}_{\max }=718\)) can compensate a sudden influx of this size (a–c). a, d The number of infections in the hidden pool (dotted) jump due to the influx event at t = 0, and return to stability for default capacity (a) or continue to grow in the system with reduced capacity (d). Correspondingly, the number of cases in the traced pool (solid line) either slowly increases after the event and absorbs most infections before returning to stability (inset in a, time axis prolonged to 1000 days), or proceeds to grow steeply (d). b, e The absolute number of new infections (dashed, yellow) jumps due to the large influx event (solid green line). The number of daily observed cases (solid brown line) slowly increases after the event, and relaxes back to baseline (a), or increases fast upon exceeding the maximum number of new observed cases \({N}_{\max }\) (solid gray line) for which tracing is effective. c, f The effective (dashed red line) and observed (solid dark red line) reproduction numbers change transiently due to the influx event before returning to 1 for the default tracing capacity. In the case of a reduced tracing capacity and a new outbreak, they slowly begin to grow afterward (f). All the curves plotted are obtained from numerical integration of Eqs. (1)–(5).