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

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 app ($Y$) or not ($N$). We denote by $f$ the fraction of $A$'s, by $y$ the fraction of $Y$'s and by $R_0$ the average number of secondary infections from a random infected individual. We investigate the effect of non-digital interventions and of digital interventions, depending on the willingness to quarantine, parameterized by four cooperating probabilities. For a given `effective' $R_0$ obtained with non-digital interventions, we use non-negative matrix theory and stopping line techniques to characterize mathematically the minimal fraction $y_0$ of app users needed to curb the epidemic. We show that under a wide range of scenarios, the threshold $y_0$ as a function of $R_0$ rises steeply from 0 at $R_0=1$ to prohibitively large values (of the order of $60-70\%$ up) whenever $R_0$ is above 1.3. Our results show that moderate rates of adoption of a contact tracing app can reduce $R_0$ but are by no means sufficient to reduce it below 1 unless it is already very close to 1 thanks to non-digital interventions.