Fig. 1: Illustration of interactions between the hidden H and traced T pools in our model.

a In our model, we distinguish two different infected population groups: the one that contains the infected individuals that remain undetected until tested (hidden pool H), and the one with infected individuals that we already follow and isolate (traced pool T). Super indexes s and a in both variables account for symptomatic and asymptomatic individuals. Until noticed, an outbreak will fully occur in the hidden pool, where case numbers increase according to this pool’s reproduction number \({R}_{t}^{H}\). Testing and tracing of hidden infections transfers them to the traced pool and helps to empty the hidden pool; this prevents offspring infections and reduces the overall growth of the outbreak. Due to the self-isolation imposed in the traced pool, its reproduction number \({R}_{t}^{T}\) is expected to be considerably smaller than \({R}_{t}^{H}\), and typically smaller than 1. Once an individual is tested positive, all the contacts since the infection are traced with some efficiency (η). Two external events further increase the number of infections in the hidden pool, namely, the new contagions occurring in the traced pool that leak to the hidden pool and an influx of externally acquired infections (Φ). In the absence of new infections, pool sizes are naturally reduced due to recovery (or removal), proportional to the recovery rate Γ. b Simplified depiction of the model showing the interactions of the two pools. New infections generated in the traced pool can remain there (ν) or leak to the hidden pool (ϵ). Note that the central epidemiological observables are highlighted in color: The \({\hat{N}}^{\text{obs}}\) (brown) and \({\hat{R}}_{t}^{\,\text{obs}\,}\) (dark red) can be inferred from the traced pool, but the effective reproduction number \({\hat{R}}_{t}^{\,\text{eff}\,}\) (light red) that governs the stability of the whole system remains hidden.