2014, 11(4): 929-945. doi: 10.3934/mbe.2014.11.929

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

1. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan

2. 

Dipartimento di Mathematica, Università di Trento, 38050 Povo (Trento), Italy

Received  January 2013 Revised  May 2013 Published  March 2014

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. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
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show all references

References:
[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. Google Scholar

[2]

Bull. Math. Biol., 69 (2007), 1067-1091. doi: 10.1007/s11538-006-9166-9.  Google Scholar

[3]

J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0.  Google Scholar

[4]

SIAM J. Math. Anal., 22 (1991), 1065-1080. doi: 10.1137/0522069.  Google Scholar

[5]

in Dynamical Systems, World Scientific, (1993), 1-19.  Google Scholar

[6]

Springer-Verlag, Berlin-New York, 1993. doi: 10.1007/978-3-642-75301-5.  Google Scholar

[7]

Dynam. Syst. Appl., 9 (2000), 361-376.  Google Scholar

[8]

J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[9]

Nonlinear Anal., 35 (1999), 797-814. doi: 10.1016/S0362-546X(97)00597-X.  Google Scholar

[10]

J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326.  Google Scholar

[11]

Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 69-96. doi: 10.3934/dcdsb.2006.6.69.  Google Scholar

[12]

J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z.  Google Scholar

[13]

J. Math. Anal. Appl., 402 (2013), 477-492. doi: 10.1016/j.jmaa.2013.01.044.  Google Scholar

[14]

Amer. Math. Soc. Translation, 1950 (1950), 128pp.  Google Scholar

[15]

J. Math. Anal. Appl., 213 (1997), 511-533. doi: 10.1006/jmaa.1997.5554.  Google Scholar

[16]

J. Math. Anal. Appl., 363 (2010), 230-237. doi: 10.1016/j.jmaa.2009.08.027.  Google Scholar

[17]

in Differential Equations Models in Biology, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158. doi: 10.1007/978-3-642-45692-3_10.  Google Scholar

[18]

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[19]

J. Dyn. Diff. Equat., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8.  Google Scholar

[20]

$6^{th}$ edition, Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[21]

J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

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