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Exact Equations for SIR Epidemics on Tree Graphs.

K J Sharkey1, I Z Kiss, R R Wilkinson

  • 1Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK, kjs@liv.ac.uk.

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Summary
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This study presents a novel method for modeling infectious disease spread on complex networks. The pair-based moment closure accurately predicts infectious dynamics in acyclic networks, simplifying analysis.

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Area of Science:

  • Epidemiology
  • Network Science
  • Mathematical Biology

Background:

  • Markovian susceptible-infectious-removed (SIR) models are crucial for understanding disease dynamics.
  • Analyzing SIR models on complex, weighted, and directed/undirected contact networks presents significant challenges.
  • Existing methods may struggle with the heterogeneity and finite nature of real-world contact networks.

Purpose of the Study:

  • To develop and validate a deterministic representation for expected infectious time series in Markovian SIR dynamics.
  • To investigate the applicability of a pair-based moment closure method on time-invariant weighted contact networks.
  • To provide a computationally efficient approach for analyzing disease spread on complex networks.

Main Methods:

  • Utilized Markovian SIR dynamics with Poisson infection and removal processes.
  • Employed a specific pair-based moment closure representation.
  • Focused on time-invariant weighted contact networks, considering both directed and undirected links.
  • Analyzed networks with no cycles in the underlying graph.

Main Results:

  • Proved that the pair-based moment closure accurately generates the expected infectious time series for acyclic networks.
  • Demonstrated that this deterministic representation simplifies the analysis of complex, heterogeneous, and finite Markovian systems.
  • Showcased the straightforward numerical evaluation of the proposed model.

Conclusions:

  • The pair-based moment closure offers a powerful and accurate tool for predicting infectious disease trajectories on specific network structures.
  • This deterministic approach provides a computationally tractable method for analyzing SIR dynamics in complex systems.
  • The findings facilitate a deeper understanding of epidemic spread in networked populations.