# SMILE

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

### 2020

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 (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' $R_0$ 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 the effective $R_0$ 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.

# Bibtex

@article {Lambert2020mathematical,
author = {Lambert, Amaury},
title = {A mathematical assessment of the efficiency of quarantining and contact tracing in curbing the {COVID}-19 epidemic},
elocation-id = {2020.05.04.20091009},
year = {2020},
doi = {10.1101/2020.05.04.20091009},
publisher = {Cold Spring Harbor Laboratory Press},
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 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 (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' $R_0$ 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 the effective $R_0$ 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.},
URL = {https://www.medrxiv.org/content/early/2020/05/08/2020.05.04.20091009},
eprint = {https://www.medrxiv.org/content/early/2020/05/08/2020.05.04.20091009.full.pdf},
journal = {medRxiv}
}