$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission
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1.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914
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2.
Dipartimento di Mathematica, Università di Trento, 38050 Povo (Trento)
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Received:
01 January 2013
Accepted:
29 June 2018
Published:
01 March 2014
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MSC :
Primary: 92D30, 35Q92; Secondary: 45P05.
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In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fréchet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$.
Citation: Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 929-945. doi: 10.3934/mbe.2014.11.929
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Abstract
In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fréchet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$.
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