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Birth and death of links control disease spreading in empirical contact networks

Petter Holme, Fredrik Liljeros

arXiv:1307.6436v2q-bio.PEcs.SIphysics.soc-ph

TL;DR

The paper investigates how socially structured populations affect the spreading of infectious diseases. The authors compare SIS and SIR simulations on empirical networks with reference data that remove or alter interevent, beginning, and ending intervals. Interevent interval distributions are not important for outbreak sizes, while link birth and death and the number of contacts over links are essential.

  • Problem

    The paper investigates how socially structured populations affect the spreading of infectious diseases.

    Source: Introduction · p. 2

  • Method

    The authors compare SIS and SIR simulations on empirical networks with reference data that remove or alter interevent, beginning, and ending intervals.

    Source: Introduction · p. 2, Preliminaries · pp. 5–6

  • Results

    Interevent interval distributions are not important for outbreak sizes, while link birth and death and the number of contacts over links are essential.

    Source: Discussion · p. 9

  • Takeaways & Limitations

    For epidemiological purposes, these empirical networks can be represented as links with beginnings, endings, and contact counts.

    Source: Discussion · p. 8

    The relationship between the summary measure ϕ and Δ is imperfect for some data sets, including Facebook.

    Source: The link turnover picture and… · p. 7

Abstract

from arXiv · show

We investigate what structural aspects of a collection of twelve empirical temporal networks of human contacts are important to disease spreading. We scan the entire parameter spaces of the two canonical models of infectious disease epidemiology -- the Susceptible-Infectious-Susceptible (SIS) and Susceptible-Infectious-Removed (SIR) models. The results from these simulations are compared to reference data where we eliminate structures in the interevent intervals, the time to the first contact in the data, or the time from the last contact to the end of the sampling. The picture we find is that the birth and death of links, and the total number of contacts over a link, are essential to predict outbreaks. On the other hand, the exact times of contacts between the beginning and end, or the interevent interval distribution, do not matter much. In other words, a simplified picture of these empirical data sets that suffices for epidemiological purposes is that links are born, is active with some intensity, and die.

Introduction

This paper examines which temporal structures in empirical human-contact networks matter for disease spreading, using broad SIS and SIR simulations to compare ongoing-link and link-turnover pictures. It focuses on the epidemiological relevance of link beginnings, ends, and contact counts, while assessing the limits of static-network reductions and the dataset dependence of these conclusions.

Introduction

  • Network epidemiology improves realism by modeling contact structure, but ignoring contact timing can cause large errors in outbreak-dynamics predictions.The paper therefore seeks a simplification retaining important temporal and topological information without unnecessary detail.Source: Introduction · p. 2
  • The study contrasts an ongoing-link picture, where interevent intervals matter on a static network, with a link-turnover picture centered on link birth and death.The two pictures are presented as similarly complex alternatives for simplifying empirical temporal networks.Source: Introduction · p. 2
  • Using twelve temporal network data sets, the authors scan the full SIS and SIR parameter spaces and compare originals with versions altering one temporal feature at a time.The modified data target interevent-interval heterogeneity, beginning intervals, and end intervals.Source: Introduction · p. 3

Results

  • Across the available empirical data sets, many links have beginning and end intervals much shorter than expected from their interevent intervals, suggesting links are born and die during sampling.This motivates treating link turnover as a parsimonious description of the observed contact sequences.Source: Preliminaries · p. 5
  • At most a little over 0.5 of links differ significantly from ongoing-link predictions, versus 0.95 expected if that picture held perfectly.The comparison uses the fraction ϕ of links whose beginning and end intervals differ significantly from interevent-interval predictions.Source: Preliminaries · p. 5
  • Destroying beginning- and end-interval structure changes SIR outbreak sizes more than neutralizing interevent intervals across all data sets.Average deviations in Ω are larger for BIN and EIN than for IIN, supporting the link-turnover picture.Source: The link turnover picture and… · p. 7
  • The SIS model gives the same qualitative result: shifting links to the beginning or end affects spreading more than making interevent intervals equal.This conclusion holds despite SIS measuring different quantities from SIR.Source: SIS model · p. 8

Discussion

  • For disease-spreading purposes, empirical contacts can be represented as links with a beginning, an end, and a number of contacts.This link-turnover representation is contrasted with links remaining active before and after the sampling period.Source: Discussion · pp. 8–9
  • 0.1% average outbreak size in original E-mail 1 data rose to 79% in BIN data at δ = 0.01 and λ = 1.The example demonstrates that altering beginning intervals can strongly change outbreak size.Source: Discussion · p. 9
  • The link turnover picture explains how empirical temporal-network structure affects disease spreading, emphasizing the birth and death of links.The authors propose this concept as a contribution to mathematical epidemiology and suggest further research on link birth and death.Source: Discussion · pp. 9–10
  • Compared with the original data (Fig. 4a), BIN and EIN reference data produce much larger outbreak sizes, with contact rates concentrated toward the beginning or end, especially for short-duration infections and high transmission probabilities.Source: Discussion · pp. 9–10
  • The conclusions hold for the studied datasets and may generalize to many, but not all, relevant contact patterns, with sampling duration determining whether links show turnover or mainly single contacts.The discussion frames real disease-spreading contact patterns as lying between short-sampling and long-sampling extremes.Source: Discussion · p. 10

Methods

  • The methods explain beginning- and end-interval distributions using uncorrelated interevent-interval samples, with observation probability proportional to T + 1 – t and beginning-interval probability proportional to 2t.Source: Predicted cumulative distribution of… · p. 10
  • The simulations average 1000 infection starts for every individual, scan 400 exponential-grid coordinates in δ×λ, and use fixed infectious duration δ with per-contact transmission probability λ.The grid spans 0.001 to 1 in both dimensions, while the infectious individual is assumed infectious at its first contact.Source: SIR and SIS simulations · p. 11

Contributions

The study was conceived by P.H. and F.L., with P.H. developing the methods and analyzing the data; both authors wrote and approved the paper.

Contributions

  • P.H. and F.L. conceived the study, while P.H. developed the methods and analyzed the data.Source: Contributions · p. 11
  • P.H. and F.L. wrote the paper and all authors read and approved its final version.Source: Contributions · p. 11

Competing financial interests

Figure captions

The figures define competing temporal-network representations, reference-network manipulations, and outbreak-deviation measures for SIR and SIS analyses.

Figure captions

  • Figure 1 contrasts continuously active ongoing links with links active only between their first and last contacts.Source: Figure 1
  • Figure 3 tests ongoing-link consistency by comparing beginning and end intervals with interevent intervals, using p > 0.05 and ϕ = 0.95 as reference criteria.Source: Figure 3
  • Figure 4 constructs IIN, BIN, and EIN reference networks by altering interevent intervals or first- and last-contact timing while preserving selected link properties.Source: Figure 4 · p. 16
  • Figure 5 maps average SIR outbreak size over infectious duration δ and transmission probability λ for empirical and reference networks.Source: Figure 5
  • Figures 6 and 7 quantify deviations from empirical outbreak sizes for the reference data under SIR and SIS models, respectively.Source: Figure captions · p. 18, Figure 6, Figure 7

Table caption

  • Table 1 reports dataset statistics, including individuals, contacts, sampling duration, and time resolution, while the EIN/BIN notation distinguishes timing manipulations.Source: Table caption · pp. 18, 20, EIN BIN · p. 21, Table 1

EIN c

The supplied material lists dataset categories associated with EIN, BIN, and IIN manipulations.

EIN c

  • The labels associate EIN, BIN, and IIN with datasets including Prostitution, Hospital, Gallery, Conference, electronic communication, and physical proximity.Source: EIN BIN IIN · pp. 24–25
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