Approximating reproduction numbers: a general numerical method for age-structured models

Simone De Reggi; Francesca Scarabel; Rossana Vermiglio; Simone De Reggi; Francesca Scarabel; Rossana Vermiglio

doi:10.3934/mbe.2024236

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 4: 5360-5393. doi: 10.3934/mbe.2024236

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Approximating reproduction numbers: a general numerical method for age-structured models

1.
Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy
2.
Department of Applied Mathematics, University of Leeds, Woodhouse, Leeds LS2 9JT, United Kingdom
3.
CDLab – Computational Dynamics Laboratory, University of Udine, Italy

Received: 19 December 2023 Revised: 15 February 2024 Accepted: 23 February 2024 Published: 07 March 2024

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.
- basic reproduction number,
- structured population dynamics,
- epidemic models,
- next generation operator,
- pseudospectral collocation,
- eigenvalue approximation
Citation: Simone De Reggi, Francesca Scarabel, Rossana Vermiglio. Approximating reproduction numbers: a general numerical method for age-structured models[J]. Mathematical Biosciences and Engineering, 2024, 21(4): 5360-5393. doi: 10.3934/mbe.2024236

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Abstract

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

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