Effectiveness of Contact Tracing on Networks with Cliques

TL;DR

This analysis shows that contact tracing in networks with groups is more efficient the larger the groups are, and illustrates how contract tracing in real-world settings can be more efficient than predicted by models that treat the system as fully mixed or the network structure as locally tree-like.

Abstract

Contact tracing, the practice of isolating individuals who have been in contact with infected individuals, is an effective and practical way of containing disease spread. Here, we show that this strategy is particularly effective in the presence of social groups: Once the disease enters a group, contact tracing not only cuts direct infection paths but can also pre-emptively quarantine group members such that it will cut indirect spreading routes. We show these results by using a deliberately stylized model that allows us to isolate the effect of contact tracing within the clique structure of the network where the contagion is spreading. This will enable us to derive mean-field approximations and epidemic thresholds to demonstrate the efficiency of contact tracing in social networks with small groups. This analysis shows that contact tracing in networks with groups is more efficient the larger the groups are. We show how these results can be understood by approximating the combination of disease spreading and contact tracing with a complex contagion process where every failed infection attempt will lead to a lower infection probability in the following attempts. Our results illustrate how contact tracing in real-world settings can be more efficient than predicted by models that treat the system as fully mixed or the network structure as locally tree-like.

Figures (18)

• Figure 1: Illustration of $r$-regular $c$-clique network structures. Panels (a-c) highlight the immediate network vicinity of a focal (red) node within networks formed by 4, 3, and 2-cliques, respectively, where each node consistently has a degree of 6. These configurations are representative of the local topology repeated throughout the entire network. Panel (d) provides an example of a $4$-regular $3$-clique network, with each node having a degree of 4 and being part of two 3-cliques. Displayed are link stubs indicating connections to other nodes, demonstrating the typical local structure one would encounter in an extensive clique-based network. The shaded circular regions signify the proximity to a central node, which is marked in red. This shows the connectivity structure we examine using our $r$-regular clique-type networks.
• Figure 2: Diagram of the SIRQ model showing the flow between compartments based on transition probabilities based on the stochastic dynamics introduced in Sec. \ref{['sec:SIRQ']}. Susceptible individuals become infected with probability $p$ and enter quarantine with probability $\alpha$. The $\mathrm{Q}$ compartment includes people in quarantine, either infected or susceptible. Those who are both infected and quarantined move to the $\mathrm{Q_I}$ sub-compartment, while those who are only quarantined go to the $\mathrm{Q_S}$ sub-compartment of $\mathrm{Q}$. Fig. \ref{['fig:demo']} depicts these two situations. Infected individuals who are not quarantined go to the $\mathrm{I}$ compartment and will recover deterministically in the subsequent time step.
• Figure 3: Schematic of contact tracing and spreading without loops (a) and with local loops (b). Infections that would be successful are marked with solid red links, and successful contact tracing with dashed black links. After each exposure, a susceptible node isolates itself with probability $\alpha$ and becomes infected with probability $p$ independently. If no loops are considered, the combination of infections and contact tracing can be reduced to a single link. There are four possible scenarios: nothing happens; the infection spreads to the neighbor, but contact tracing fails; the infection fails to spread, but contact tracing succeeds; or (a) both infections spread, and the contact tracing succeeds so that the node will be in sub-compartment $\mathrm{Q_I}$. The last case is where we can benefit from contact tracing cutting indirect spreading paths thanks to the presence of clustering. (b) With local loops, an infection through a common neighbor of both nodes can be avoided. As the quarantine takes place close to the infection, it can prevent the infection from arriving at the neighbor through a local loop as the node is in sub-compartment $\mathrm{Q_S}$.
• Figure 4: Phase transitions from a disease-free equilibrium to an endemic state for 2, 3, and 4-clique networks with degree 6 as introduced in Sec. \ref{['sec:Networks']}. (a-c) The outbreak size $E$, normalized to the network size, is shown on the vertical axis for when (a) $\alpha = 0$ (no contact tracing), (b) $\alpha = 0.25$, and (c) $\alpha = 0.5$, from top to bottom respectively. Note that the transition points are shifted slightly to the right for larger clique sizes, $c$, even when there is no contact tracing ($\alpha=0$), but this difference is substantially amplified for larger $\alpha$ values. (d) The coefficient of variation of outbreak sizes in an ensemble, $\chi$ normalized to unity for $\alpha=0.5$. We use $\chi$ to numerically detect the transition point as it peaks at the epidemic threshold. Results are based on Monte Carlo simulations introduced in Sec. \ref{['sec:SIR']}.
• Figure 5: Each 3-clique can have four life stages or diffusion patterns with at least one infected node. Section \ref{['sec:m-field']} considers both recovered and quarantined nodes in the $\mathrm{R}$ compartment. Using a 6-regular 3-clique network, we observe that a $Z_1$ node can form a $Z_2$ motif with two $Z_1$ nodes and a $Z_3$ motif with four $Z_1$ nodes. A $Z_2$ node can also create a $Z_4$ motif with two $Z_1$ nodes. Nodes in the $Z_1$, $Z_2$, and $Z_4$ motifs can transition to an infection-annihilated states such as $\{\mathrm{R}, \mathrm{S}, \mathrm{S}\}$ which are not shown here. In Section \ref{['sec:Complex_contagion']}, we assume that both quarantined and susceptible nodes are in the $\mathrm{S}$ compartment, while only recovered individuals are in the $\mathrm{R}$ compartment.