`a`
Mathematical Biosciences and Engineering (MBE)
 

$R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission
Pages: 929 - 945, Issue 4, August 2014

doi:10.3934/mbe.2014.11.929      Abstract        References        Full text (414.9K)           Related Articles

Toshikazu Kuniya - Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan (email)
Mimmo Iannelli - Dipartimento di Mathematica, Università di Trento, 38050 Povo (Trento), Italy (email)

1 V. Andreasen, Instability in an SIR-model with age-dependent susceptibility, 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 N. Bacaër, Approximation of the basic reproduction number $R_{0}$ for vector-borne diseases with a periodic vector population, Bull. Math. Biol., 69 (2007), 1067-1091.       
3 N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.       
4 S. N. Busenberg, M. Iannelli and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), 1065-1080.       
5 S. N. Busenberg, M. Iannelli and H. R. Thieme, Dynamics of an age structured epidemic model, in Dynamical Systems, World Scientific, (1993), 1-19.       
6 S. N. Busenberg and K. Cooke, Vertically Transmitted Diseases, Springer-Verlag, Berlin-New York, 1993.       
7 Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model, Dynam. Syst. Appl., 9 (2000), 361-376.       
8 O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.       
9 M. Iannelli, M. Y. Kim and E. J. Park, Asymptotic behavior for an SIS epidemic model and its approximation, Nonlinear Anal., 35 (1999), 797-814.       
10 H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434.       
11 H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission, Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 69-96.       
12 H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.       
13 T. Kuniya and H. Inaba, Endemic threshold results for an age-structured SIS epidemic model with periodic parameters, J. Math. Anal. Appl., 402 (2013), 477-492.       
14 M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Amer. Math. Soc. Translation, 1950 (1950), 128pp.       
15 M. Langlais and S. N. Busenberg, Global behaviour in age structured S.I.S. models with seasonal periodicities and vertical transmission, J. Math. Anal. Appl., 213 (1997), 511-533.       
16 Y. Nakata and T. Kuniya, Global dynamics of a class of SEIRS epidemic models in a periodic environment, J. Math. Anal. Appl., 363 (2010), 230-237.       
17 H. R. Thieme, Stability change of the endemic equilibrium in age-structured models for the spread of S-I-R type infectious diseases, in Differential Equations Models in Biology, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158.       
18 H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.       
19 W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717.       
20 K. Yosida, Functional Analysis, $6^{th}$ edition, Springer-Verlag, Berlin-New York, 1980.       
21 F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.       

Go to top