A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon

Wenjuan Guo; Ming Ye; Xining Li; Anke Meyer-Baese; Qimin Zhang; Wenjuan Guo; Ming Ye; Xining Li; Anke Meyer-Baese; Qimin Zhang

doi:10.3934/mbe.2019204

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4107-4121. doi: 10.3934/mbe.2019204

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A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, P.R. China
2.
Department of Earth, Ocean, and Atmospheric Science and Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, United States
3.
Department of Scientific Computing, Florida State University, Tallahassee, FL 32306-4120, United States

Received: 25 January 2019 Accepted: 29 April 2019 Published: 10 May 2019

This paper focuses on numerical approximation of the basic reproduction number $\mathcal{R}_0$, which is the threshold defined by the spectral radius of the next-generation operator in epidemiology. Generally speaking, $\mathcal{R}_0$ cannot be explicitly calculated for most age-structured epidemic systems. In this paper, for a deterministic age-structured epidemic system and its stochastic version, we discretize a linear operator produced by the infective population with a theta scheme in a finite horizon, which transforms the abstract problem into the problem of solving the positive dominant eigenvalue of the next-generation matrix. This leads to a corresponding threshold $\mathcal{R}_{0, n}$. Using the spectral approximation theory, we obtain that $\mathcal{R}_{0, n}\rightarrow\mathcal{R}_0$ as $n\rightarrow+\infty$. Some numerical simulations are provided to certify the theoretical results.
- Numerical approximation,
- basic reproduction number,
- age-structure epidemic system,
- spectral radius,
- theta scheme
Citation: Wenjuan Guo, Ming Ye, Xining Li, Anke Meyer-Baese, Qimin Zhang. A theta-scheme approximation of basic reproduction number for an age-structured epidemic system in a finite horizon[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4107-4121. doi: 10.3934/mbe.2019204

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Abstract

This paper focuses on numerical approximation of the basic reproduction number $\mathcal{R}_0$, which is the threshold defined by the spectral radius of the next-generation operator in epidemiology. Generally speaking, $\mathcal{R}_0$ cannot be explicitly calculated for most age-structured epidemic systems. In this paper, for a deterministic age-structured epidemic system and its stochastic version, we discretize a linear operator produced by the infective population with a theta scheme in a finite horizon, which transforms the abstract problem into the problem of solving the positive dominant eigenvalue of the next-generation matrix. This leads to a corresponding threshold $\mathcal{R}_{0, n}$. Using the spectral approximation theory, we obtain that $\mathcal{R}_{0, n}\rightarrow\mathcal{R}_0$ as $n\rightarrow+\infty$. Some numerical simulations are provided to certify the theoretical results.

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