$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission

  • Received: 01 January 2013 Accepted: 29 June 2018 Published: 01 March 2014
  • MSC : Primary: 92D30, 35Q92; Secondary: 45P05.

  • 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

    Related Papers:

  • 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 $\mathcal{R}_0$.
    加载中
    [1] in Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Wuerz Publ., (1995), 3-14.
    [2] Bull. Math. Biol., 69 (2007), 1067-1091.
    [3] J. Math. Biol., 53 (2006), 421-436.
    [4] SIAM J. Math. Anal., 22 (1991), 1065-1080.
    [5] in Dynamical Systems, World Scientific, (1993), 1-19.
    [6] Springer-Verlag, Berlin-New York, 1993.
    [7] Dynam. Syst. Appl., 9 (2000), 361-376.
    [8] J. Math. Biol., 28 (1990), 365-382.
    [9] Nonlinear Anal., 35 (1999), 797-814.
    [10] J. Math. Biol., 28 (1990), 411-434.
    [11] Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 69-96.
    [12] J. Math. Biol., 65 (2012), 309-348.
    [13] J. Math. Anal. Appl., 402 (2013), 477-492.
    [14] Amer. Math. Soc. Translation, 1950 (1950), 128pp.
    [15] J. Math. Anal. Appl., 213 (1997), 511-533.
    [16] J. Math. Anal. Appl., 363 (2010), 230-237.
    [17] in Differential Equations Models in Biology, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158.
    [18] SIAM J. Appl. Math., 70 (2009), 188-211.
    [19] J. Dyn. Diff. Equat., 20 (2008), 699-717.
    [20] $6^{th}$ edition, Springer-Verlag, Berlin-New York, 1980.
    [21] J. Math. Anal. Appl., 325 (2007), 496-516.

    © 2014 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)
  • Reader Comments
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(15) PDF downloads(470) Cited by(3)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog