To quarantine, or not to quarantine: A theoretical framework for disease control via contact tracing

Abstract

Contact tracing via smartphone applications is expected to be of major importance for maintaining control of the COVID-19 pandemic. However, viable deployment demands a minimal quarantine burden on the general public. That is, consideration must be given to unnecessary quarantining imposed by a contact tracing policy. Previous studies have modeled the role of contact tracing, but have not addressed how to balance these two competing needs. We propose a modeling framework that captures contact heterogeneity. This allows contact prioritization: contacts are only notified if they were acutely exposed to individuals who eventually tested positive. The framework thus allows us to address the delicate balance of preventing disease spread while minimizing the social and economic burdens of quarantine. This optimal contact tracing strategy is studied as a function of limitations in testing resources, partial technology adoption, and other intervention methods such as social distancing and lockdown measures. The framework is globally applicable, as the distribution describing contact heterogeneity is directly adaptable to any digital tracing implementation.

Keywords: Asymptotic analysis; Basic reproduction number; Contact tracing; Epidemiological modeling.

Fig. 1

Disease dynamics and contact tracing framework.(left) Upon infection, susceptible individuals ($S$) enter a latent exposed stage ($E$) when the disease incubates before infectiousness. Upon becoming infectious, a proportion $p_a$ of the population remain asymptomatic ($A$), while the remainder pass through a presymptomatic stage ($P$) before becoming symptomatic ($I$). Infectious individuals are removed ($R$) through recovery/isolation (rates ${\gamma }_A$ and ${\gamma }_I$), isolation after testing (dotted arrow, rate $\tau$), or quarantining as a consequence of contact tracing (dashed arrows, rate $\alpha (1-\Theta )I+\alpha \Theta I({\mu }_K∕K)$ for $K\in \{E,A,P,I\}$). The total contact tracing rate $\alpha$ depends on the proportion of the population who adopt the contact tracing $u_a$, the testing rate $\tau$, the factor $(1-u_s\left(t\right))$, representing the reduction of transmission rates due to social intervention measures and the notification threshold, $u_n$, representing the minimal exposure required to notify traced contacts. Removal via contact tracing is partitioned into the fraction of contacts who were infected by the tested case $\Theta$, and those who were not $(1-\Theta )$, where $\Theta$ represents the tracing precision and depends on the susceptible proportion $S$ and the notification threshold $u_n$. Contact tracing also causes quarantine of susceptible individuals ($Q$) that had non-infectious contact with an infected case (dashed arrow at rate $\alpha (1-\Theta )I$, return rate $q$). The force of infection, $F$ depends on transmission rates ${\beta }_K$ and infection densities, and social intervention measures. (right) Contact heterogeneity is incorporated via a distribution of exposure levels: contact are encountered at exposure $e$ with density $\rho \left(e\right)$ and result in infection with probability $p_i\left(e\right)$. The notification threshold (illustrated for $u_n=1$) affects the contact tracing rate and precision via the integral $f_c$ (blue and green shaded region) expressing the fraction of contacts notified, and the integral $f_i$ (green shaded region), which captures the proportion of contacts who had infectious contact with the tested case. (Color version online.)

Fig. 2

Flattening the curve. (left) Early-phase disease outcomes as a function of the maximum testing capacity ${\tau }_{\infty }$. Colored curves correspond to an adoption fraction $u_a\in \{0,0.2,0.3,0.4,0.5\}$ and show the time $t$ for 0.1% of the population to have been infected. Black dashed lines are asymptotic approximations (detailed in Supplementary Materials Section S2 B). Increasing testing capacity and the adoption fraction decelerates the initial disease outbreak. (right) Long-term disease dynamics for $u_a=0.4$. Colored curves correspond to ${\tau }_{\infty }\in \{1,2,5,10,20\}\times 10^{-4}$ and show the total fraction of infected individuals $E+A+P+I$. If present, markers along the curves (of corresponding color) show the time when the testing capacity saturates (smaller $t$) and desaturates (larger $t$). Increased testing capacity delays testing saturation, leading to smaller peak infection proportions. [inset] Susceptible proportion $S$ corresponding to the same simulation as the main plot. As testing desaturates the susceptible proportion rises. This is a result of a large number of healthy quarantined individuals $Q$ returning to the susceptible pool.

Fig. 3

Conditions for outbreak prevention via${\mathcal{R}}_0$analysis. (left) Basic reproduction number ${\mathcal{R}}_0$ for $S^{\ast }=1$ and $u_n=0$ as a function of adoption fraction $u_a$ and social interventions $u_s$. The dashed curve shows the ${\mathcal{R}}_0=1$ level set, which is the intervention threshold separating an outbreak from no outbreak. Increasing social intervention $u_s$ or contact tracing adoption $u_a$ increases disease control. (right) Intervention thresholds for $S^{\ast }\in \{1,0.925,0.85,0.775,0.7\}$ and $u_n=0$. Dotted curves show the level sets ${\mathcal{R}}_0{|}_{S^{\ast }=1}{S}^{\ast }=1$ and correspond to neglecting the dependence of the contact tracing efficiency on $S^{\ast }$. Previous contact tracing descriptions do not account for the susceptible proportion of the population, and thus underestimate the necessary disease control.

Fig. 4

To quarantine or not to quarantine. (left) Quarantine cost ${\int }_0^{T}Q\mathrm{d}t$ as a function of notification threshold $u_n$, for $u_a=0.5$, $u_s=0.35$, and $T\in \{1,1.5,2,3,4\}$ years. The notification threshold axis is mapped from the interval $[0,u_n^{\ast }]$ to $[1,\infty ]$ so as to spread out the region near criticality $u_n\approx u_n^{\ast }$ to illustrate the minimum cost (with $u_n^{\ast }$ denoting the notification threshold for which ${\mathcal{R}}_0=1$). Each marked $u_n$ value corresponds to an identical mark in the inset. As $u_n$ increases from $u_n=0$ to the value at which the minimum is obtained, the quarantine cost improves by nearly two orders of magnitude while there is little variation in ${\mathcal{R}}_0$. Black dashed curves show the asymptotic approximation derived in Supplementary Materials Section S2 B. [inset] Basic reproduction number ${\mathcal{R}}_0$ for $u_s=0.35$ as a function of $u_a$ and $u_n$. The dashed line shows the ${\mathcal{R}}_0=1$ level set. (right) Optimal contact tracing precision $\Theta$ (i.e. the precision associated with the optimal notification threshold), as a function of social intervention measures $u_s$ and the adoption faction $u_a$, within the region where an outbreak is controllable for sufficiently small $u_n$ but not controllable for arbitrarily large $u_n$. At the lower ${\mathcal{R}}_0=1$ boundary only $u_n=0$ prevents the epidemic, which corresponds to the minimal precision ${\Theta }_{\text{min}}$. Beyond the boundary there is a rapid increase in the optimal $u_n$ and thus a rapid increase in precision. In the vicinity of the upper boundary the optimal $u_n$ diverges, corresponding to the precision converging to ${\Theta }_{\text{max}}$. [inset] One-dimensional slices of the tracing precision for fixed $u_a$ (green) and fixed $u_s$ (orange). We denote by $u_a^{\ast }$ and $u_s^{\ast }$ the critical parameter values for which ${\mathcal{R}}_0=1$ when all else is fixed, while $u_s^{\text{max}}$ denotes the upper boundary of the region, beyond which there is no outbreak even in the absence of contact tracing. (Color version online.)