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

  • 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.

    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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  • 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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    © 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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