How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology

Alessia Andò; Dimitri Breda; Giulia Gava; Alessia Andò; Dimitri Breda; Giulia Gava

doi:10.3934/mbe.2020273

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5059-5084. doi: 10.3934/mbe.2020273

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How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology

Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics University of Udine, via delle scienze 206, 33100 Udine, Italy

Received: 05 May 2020 Accepted: 16 July 2020 Published: 24 July 2020

A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parentso decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.
- linear chain trick,
- stability analysis,
- distributed delay,
- SIR models with vaccination,
- behavioral epidemiology
Citation: Alessia Andò, Dimitri Breda, Giulia Gava. How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5059-5084. doi: 10.3934/mbe.2020273

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Abstract

A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parentso decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.

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