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The effectiveness of backward contact tracing in networks

Sadamori Kojaku, Laurent Hébert-Dufresne, Enys Mones, Sune Lehmann, Yong-Yeol Ahn

arXiv:2005.02362v3q-bio.PEphysics.app-phphysics.soc-ph

TL;DR

Traditional contact tracing is labor-intensive and slow, with its efficacy and cost-benefit trade-offs questioned. This paper analyzes biases leveraged by backward tracing through theoretical and simulation analyses. It finds strategically performed tracing can be effective and efficient without massive scale, including more cost-efficient than case isolation per prevented case when infection detection is difficult.

  • Problem

    Traditional contact tracing is labor-intensive and slow, and its efficacy and cost-benefit trade-offs have been questioned.

    Source: §1 · p. 2

  • Method

    The paper analyzes network sampling biases leveraged by backward tracing using simulations on synthetic and empirical contact networks.

    Source: §1 · p. 2

  • Results

    Theoretical and simulation analyses indicate strategically performed tracing can be highly effective and efficient without massive scale, and more cost-efficient than case isolation in prevented cases per isolation, especially when detecting infection is difficult.

    Source: §3 · p. 10

  • Takeaways & Limitations

    The authors argue that strategic contact tracing protocols should reach infection sources because effectiveness hinges on reaching the source.

    Source: §3 · p. 11

    The simulations assume instantaneous diagnostic testing and isolation and equal infection and isolation probabilities, though these probabilities may vary by demographics.

    Source: §3 · p. 11

Abstract

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Discovering and isolating infected individuals is a cornerstone of epidemic control. Because many infectious diseases spread through close contacts, contact tracing is a key tool for case discovery and control. However, although contact tracing has been performed widely, the mathematical understanding of contact tracing has not been fully established and it has not been clearly understood what determines the efficacy of contact tracing. Here, we reveal that, compared with "forward" tracing---tracing to whom disease spreads, "backward" tracing---tracing from whom disease spreads---is profoundly more effective. The effectiveness of backward tracing is due to simple but overlooked biases arising from the heterogeneity in contacts. Using simulations on both synthetic and high-resolution empirical contact datasets, we show that even at a small probability of detecting infected individuals, strategically executed contact tracing can prevent a significant fraction of further transmissions. We also show that---in terms of the number of prevented transmissions per isolation---case isolation combined with a small amount of contact tracing is more efficient than case isolation alone. By demonstrating that backward contact tracing is highly effective at discovering super-spreading events, we argue that the potential effectiveness of contact tracing has been underestimated. Therefore, there is a critical need for revisiting current contact tracing strategies so that they leverage all forms of biases. Our results also have important consequences for digital contact tracing because it will be crucial to incorporate the capability for backward and deep tracing while adhering to the privacy-preserving requirements of these new platforms.

1 Introduction

Contact tracing can aid epidemic control, but traditional tracing is labor-intensive and slow, and its efficacy and cost-benefit trade-offs have been questioned. The paper argues backward tracing leverages network sampling biases and examines tracing across synthetic and empirical contact networks.

  • Contact tracing is potent because infections spread through close physical contacts, though traditional tracing is labor-intensive and slow and its efficacy and cost-benefit trade-offs have been questioned.Source: §1 · pp. 1–2
  • Super-spreading events are common and often discovered through contact tracing.Source: §1 · p. 2
  • Backward tracing can exploit an additional offspring-sampling bias alongside the friendship paradox, with additive effects that help identify super-spreaders and super-spreading events.Source: §1 · p. 2
  • Degree heterogeneity strengthens these sampling biases because contact counts vary across individuals.Source: §1 · p. 2
  • Simulations on synthetic and empirical contact networks examine whether strategically executed tracing can control epidemics effectively and efficiently.Source: §1 · p. 2

2 Results

The results explain how contact-network structure and backward tracing bias sampling toward high-contact parents, then show that tracing can reduce transmission across heterogeneous and relatively homogeneous networks. Analytical results further show that increased tracing lowers sustained transmission with diminishing returns.

2.1 Bias owing to the friendship paradox

  • Because contacts are sampled in proportion to degree, infected neighbors tend to have more contacts than randomly sampled people, producing the friendship paradox.The excess-degree distribution has mean at least as large as the degree distribution's mean.Source: §2.1 · p. 3

2.2 Bias owing to backward tracing

  • Backward tracing samples parents with probability proportional to k(k−1)pk, a stronger degree bias than the friendship-paradox weighting proportional to kpk.Frequency-based tracing can reach nodes with degrees similar to parents without knowing transmission direction.Source: §2.2 · pp. 3, 5, Figure 1 · p. 4

2.3 Effectiveness of contact tracing for heterogeneous networks

  • In heterogeneous networks, tracing with pt ≥ 0.5 reduces the infection peak by more than 70%, and tracing just 10 of 250,000 nodes still flattens the curve.Contact tracing also isolates fewer nodes while preventing more cases than case isolation, yielding more prevented cases per isolation.Source: §2.3 · pp. 5–6, Figure 2 · p. 7

2.4 Contact tracing on temporal contact network of students

  • On the student contact network, contact tracing lowers the peak by about 50%, and tracing a few contacts is comparably effective to tracing all close contacts.The network has 567 students and a fairly homogeneous degree distribution; tracing has diminishing returns as tracing probability rises, yet prevents at least roughly five cases per isolation.Source: §2.4 · pp. 6, 8, Figure 3 · p. 8

2.5 Analytical analysis of contact tracing on networks with arbitrary degree sequence

  • Increasing tracing probability P can control epidemics and stop sustained transmission, with diminishing returns and a smooth rather than sharp epidemic threshold.The analysis attributes this to tracing preferentially targeting transmission events from individuals with probability proportional to k^2, localizing remaining spread around lower-degree nodes.Source: Figure 4 · p. 9, §2.5 · p. 10

3 Discussion

Contact tracing can be highly effective and efficient when protocols exploit sampling biases associated with heterogeneous contact networks. The discussion highlights backward and deep tracing for digital systems while noting assumptions that constrain the simulations.

3 Discussion

  • Theoretical and simulation analyses indicate that strategically bias-leveraging tracing can be effective at small scale and more cost-efficient than case isolation alone in prevented cases per isolation.The claimed efficiency advantage is especially relevant when detecting infection is difficult.Source: §3 · p. 10
  • Digital tracing should identify likely infection parents and notify other recent contacts of traced nodes, since multiple notifications may indicate a parent.The proposed multi-step notifications aim to leverage contact-network sampling biases.Source: §3 · p. 10
  • The simulations assume instantaneous testing and isolation and equal infection and isolation probabilities, although delays and demographic variation may affect intervention effectiveness.The authors note that heterogeneous probabilities may hinder opt-in tracing, including when sources refuse tracing or data are not shared across borders.Source: §3 · p. 11
  • The authors argue contact tracing has greater potential than commonly considered, while emphasizing that effectiveness depends on protocols reaching infection sources.They state that a model implementation need not compromise privacy or decentralization, and also describe a possible centralized hybrid for deep tracing.Source: §3 · pp. 10–11

4 Methods

The methods combine empirical student-contact data with epidemic simulations and a people–gathering network model. The gathering model accounts for the greater likelihood that larger gatherings are traced.

4.1 Data

  • Contact data from 567 students covered 28 days at 5-minute resolution, with contacts inferred from Bluetooth scans stronger than −75 dB after excluding people with under 60% valid scans.The source dataset included more than 700 students; the −75 dB threshold corresponds to approximately 1 m.Source: §4.1 · p. 11

4.2 Simulating epidemic spreading

  • Simulations used SIR on static and people-gathering networks and SEIR on the student network, with student parameters chosen to produce outbreaks reaching about 50% of the population.The static/gathering SIR setup used T = 0.25, γ = 0.25, and ρ = 10^-3; student SEIR used expected infectious and incubation periods of 5 and 1 days, respectively. Student simulations seeded 1% of the population and continued contacts cyclically beyond the 28-day record.Source: §4.2 · p. 11

4.3 People–gathering networks

  • In the people–gathering model, tracing samples a gathering of k participants roughly in proportion to k^2, making larger gatherings more likely to be identified.A person’s degree represents mobility across gatherings, while a gathering’s degree is its participant count.Source: §4.3 · pp. 11–12

5 Epidemic probability

The section derives epidemic probability under contact tracing from offspring distributions, assuming an infected node can notify its parent and prevent transmission to the parent’s grandchildren. It uses generating functions and a self-consistency condition to characterize sustained spread.

5 Epidemic probability

  • Each newly infected node has probability P of notifying its parent, which prevents infections to all of that parent’s grandchildren.The analysis calculates the probability that contact tracing stops disease spread.Source: §5 · p. 12
  • Without tracing, offspring distributions differ by sampling: infected nodes are sampled at random or by following a random transmission.T is the probability of transmission through an edge; r_k and q_k are offspring-count probabilities, respectively.Source: §5 · p. 12
  • Under tracing, a parent’s offspring continue spreading only if tracing fails for all of them, with probability (1−P)^k.This modifies the offspring distribution for nodes sampled by following a random transmission.Source: §5 · p. 12
  • R0 distinguishes standard transmissions from transmissions quickly isolated by tracing, enabling calculation of successful branching-event probabilities.The variables x and y count standard transmissions and transmissions isolated quickly enough to stop the transmission tree, respectively.Source: §5 · p. 13
  • The probability of an epidemic is calculated from whether at least one transmission around the patient leads to sustained transmission.The probability u that transmission without tracing around the parent does not lead to sustained transmission is determined by a self-consistency condition.Source: §5 · p. 13

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