A mathematical assessment of the efficiency of quarantining and contact tracing in curbing the COVID-19 epidemic

¹ Laboratoire de Probabilités, Statistique & Modélisation (LPSM), Sorbonne Université, CNRS UMR8001, Université de Paris, Paris, France.
² Center for Interdisciplinary Research in Biology (CIRB), Collège de France, CNRS UMR7241, INSERM U1050, PSL Research University, Paris, France.

^* Corresponding author: amaury.lambert@upmc.fr

Received: 7 September 2020
Accepted: 24 July 2021

Abstract

In our model of the COVID-19 epidemic, infected individuals can be of four types, according whether they are asymptomatic (A) or symptomatic (I), and use a contact tracing mobile phone application (Y ) or not (N). We denote by R₀ the average number of secondary infections from a random infected individual. We investigate the effect of non-digital interventions (voluntary isolation upon symptom onset, quarantining private contacts) and of digital interventions (contact tracing thanks to the app), depending on the willingness to quarantine, parameterized by four cooperating probabilities. For a given ‘effective’ R₀ obtained with non-digital interventions, we use non-negative matrix theory and stopping line techniques to characterize mathematically the minimal fraction y₀ of app users needed to curb the epidemic, i.e., for the epidemic to die out with probability 1. We show that under a wide range of scenarios, the threshold y₀ as a function of R₀ rises steeply from 0 at R₀ = 1 to prohibitively large values (of the order of 60−70% up) whenever R₀ is above 1.3. Our results show that moderate rates of adoption of a contact tracing app can reduce R₀ but are by no means sufficient to reduce it below 1 unless it is already very close to 1 thanks to non-digital interventions.