Virus spread versus contact tracing: Two competing contagion processes

After the blockade that many nations suffered to stop the growth of the incidence curve of COVID-19 during the first half of 2020, they face the challenge of resuming their social and economic activity. The rapid airborne transmissibility of SARS-CoV-2, and the absence of a vaccine, calls for active containment measures to avoid the propagation of transmission chains. The best strategy to date, popularly known as test-track-treat (TTT), consists in testing the population for diagnosis, tracking the contacts of those infected, and treating by quarantine all these cases. The dynamical process that better describes the combined action of the former mechanisms is that of a contagion process that competes with the spread of the pathogen, cutting off potential contagion pathways. Here we propose a compartmental model that couples the dynamics of the infection with the contact tracing and isolation of cases. We develop an analytical expression for the effective case reproduction number $R_c\left(t\right)$ that reveals the role of contact tracing in the mitigation and suppression of the epidemics. We show that there is a trade-off between the infection propagation and the isolation of cases. If the isolation is limited to symptomatic individuals only, the incidence curve can be flattened but not bent. However, if contact tracing is applied to asymptomatic individuals too, the strategy can bend the curve and suppress the epidemics. Quantitative results are dependent on the network topology. We quantify the most important indicator of the effectiveness of contact tracing, namely, its capacity to reverse the increasing tendency of the epidemic curve, causing its bending.

Figure 1

Flow diagram of the compartmental epidemic model. The epidemiological compartments are susceptible ($S$), exposed ($E$), presymptomatic ($P$), infectious asymptomatic ($I_A$), infectious symptomatic ($I_S$), detected ($D$), and recovered ($R$). Arrows represent the possible transitions between the different states.

Figure 2

CT versus symptomatic detection in three proximity networks (top). The color and size of the nodes is proportional to their connectivity, being the small (large) and yellow (blue) ones those vertices with the smallest (largest) degree. Panels (a)–(c) show the attack rate, $R^{\infty }$ as a function of the quality of symptomatic $\delta$ and CT detection $f$. Panels (d)–(f) show the fraction of detected cases by CT, $F^{\mathrm{CT}}$. We have highlighted the case $F^{\mathrm{CT}}=0.5$ (orange line) signaling that symptomatic and CT detections identify the same number of cases. The infectivity probability per contact (${\beta }_{\text{S}}={\beta }_{\text{A}}=\beta$) is chosen so that the attack rate in the absence of any kind of detection ($\delta =f=0$) is $R^{\infty }=0.9$ for the three networks.

Figure 3

Symptomatic detection and CT as a function of degree $k$ of the nodes. The panel shows the probability of being detected via (a) symptoms and (b) CT as a function of the degree of the nodes for the School proximity network. The curves correspond (from red to blue) to $f=0$, 0.1, 0.2, 0.3, 0.5, and 1.

Figure 4

Dynamical evolution of the epidemic trajectory under symptomatic detection and CT. Panels (a) and (b) show the evolution of the new contagions when different degrees of symptomatic detection and CT are implemented, respectively. In panel (b) we also show (impulses) the evolution for number of detected cases via CT when $f=1$. The bottom panels (c) and (d) show the evolution of ${\mathcal{R}}^c\left(t\right)$ corresponding to the different epidemic curves shown above. The calculations are performed using the School proximity network.

Figure 5

Evolution of the case reproduction number for the different degree classes. The panels show the evolutions of probability of ${\mathcal{R}}_k^c\left(t\right)$ for (a) symptomatic detection ($\delta =1, f=0$) and (b) CT ($\delta =0.05, f=1$) for all the degree classes present in the School proximity network. The curve in gray accounts for the points where ${\mathcal{R}}_k^c\left(t\right)=1$.

Figure 6

Validation of Markovian dynamics by stochastic simulations. Each panel shows the evolution of the occupation of each compartment as obtained from stochastic simulations [bands and points (median)] and the Markovian equations (lines). The network substrate is the School proximity network and the detection parameters are set to $\delta =0.1$ for symptomatic detection, and $f=0.4$ for CT.