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On the early epidemic dynamics for pairwise models.

Carlos Llensa1, David Juher1, Joan Saldaña1

  • 1Departament d׳Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, 17071 Girona, Catalonia, Spain.

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This study derives a relationship between the basic reproduction number (R0) and early growth rate in epidemic models. This finding allows for transmission potential measurement using early case growth, even in complex networks.

Keywords:
Basic reproduction numberInitial epidemic growth ratePairwise epidemic modelsSEIR model

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Network Science

Background:

  • Deterministic epidemic models like SIS, SIR, and SEIR are foundational for understanding disease spread.
  • Homogeneous mixing models assume uniform interaction, which may not reflect real-world contact structures.
  • Pair approximation models offer a more refined approach by considering interactions between individuals.

Purpose of the Study:

  • To derive and analyze the relationship between the basic reproduction number (R0) and the exponential growth rate in SIS, SIR, and SEIR models using pair approximation.
  • To investigate the impact of random rewiring in contact networks on this relationship.
  • To compare findings from pairwise models with those from homogeneous mixing models.

Main Methods:

  • Derivation of the relationship between R0 and exponential growth rate for SIS, SIR, and SEIR models under pair approximation.
  • Extension of models to include random rewiring of susceptible individuals' contacts.
  • Numerical simulations on complex contact networks to validate analytical predictions.

Main Results:

  • A formally consistent relationship between exponential growth rate and R0 was found, aligning with homogeneous mixing models.
  • This consistency enables the estimation of transmission potential from early case growth rates.
  • The SEIR pairwise model's R0 is influenced by the latent period duration, unlike the homogeneous mixing SEIR model.

Conclusions:

  • The derived relationship provides a robust method for assessing epidemic transmission potential, applicable even in complex network structures.
  • Pair approximation models offer distinct insights, particularly regarding the influence of the latent period on R0 in SEIR dynamics.
  • Numerical simulations confirm the analytical predictions, supporting the utility of these models for epidemiological analysis.