# American Institute of Mathematical Sciences

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
##### References:
 [1] V. Andreasen, Instability in an SIR-model with age-dependent susceptibility,, in Mathematical Population Dynamics: Analysis of Heterogeneity, (1995), 3.   Google Scholar [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.  doi: 10.1007/s11538-006-9166-9.  Google Scholar [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421.  doi: 10.1007/s00285-006-0015-0.  Google Scholar [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.  doi: 10.1137/0522069.  Google Scholar [5] S. N. Busenberg, M. Iannelli and H. R. Thieme, Dynamics of an age structured epidemic model,, in Dynamical Systems, (1993), 1.   Google Scholar [6] S. N. Busenberg and K. Cooke, Vertically Transmitted Diseases,, Springer-Verlag, (1993).  doi: 10.1007/978-3-642-75301-5.  Google Scholar [7] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361.   Google Scholar [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.  doi: 10.1007/BF00178324.  Google Scholar [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.  doi: 10.1016/S0362-546X(97)00597-X.  Google Scholar [10] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411.  doi: 10.1007/BF00178326.  Google Scholar [11] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission,, Discrete Contin. Dyn. Syst., 6 (2006), 69.  doi: 10.3934/dcdsb.2006.6.69.  Google Scholar [12] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309.  doi: 10.1007/s00285-011-0463-z.  Google Scholar [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.  doi: 10.1016/j.jmaa.2013.01.044.  Google Scholar [14] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Amer. Math. Soc. Translation, 1950 (1950).   Google Scholar [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.  doi: 10.1006/jmaa.1997.5554.  Google Scholar [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.  doi: 10.1016/j.jmaa.2009.08.027.  Google Scholar [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, 92 (1991), 139.  doi: 10.1007/978-3-642-45692-3_10.  Google Scholar [18] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar [19] W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699.  doi: 10.1007/s10884-008-9111-8.  Google Scholar [20] K. Yosida, Functional Analysis,, $6^{th}$ edition, (1980).   Google Scholar [21] F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

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##### References:
 [1] V. Andreasen, Instability in an SIR-model with age-dependent susceptibility,, in Mathematical Population Dynamics: Analysis of Heterogeneity, (1995), 3.   Google Scholar [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.  doi: 10.1007/s11538-006-9166-9.  Google Scholar [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421.  doi: 10.1007/s00285-006-0015-0.  Google Scholar [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.  doi: 10.1137/0522069.  Google Scholar [5] S. N. Busenberg, M. Iannelli and H. R. Thieme, Dynamics of an age structured epidemic model,, in Dynamical Systems, (1993), 1.   Google Scholar [6] S. N. Busenberg and K. Cooke, Vertically Transmitted Diseases,, Springer-Verlag, (1993).  doi: 10.1007/978-3-642-75301-5.  Google Scholar [7] Y. Cha, M. Iannelli and F. A. Milner, Stability change of an epidemic model,, Dynam. Syst. Appl., 9 (2000), 361.   Google Scholar [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.  doi: 10.1007/BF00178324.  Google Scholar [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.  doi: 10.1016/S0362-546X(97)00597-X.  Google Scholar [10] H. Inaba, Threshold and stability results for an age-structured epidemic model,, J. Math. Biol., 28 (1990), 411.  doi: 10.1007/BF00178326.  Google Scholar [11] H. Inaba, Mathematical analysis of an age-structured SIR epidemic model with vertical transmission,, Discrete Contin. Dyn. Syst., 6 (2006), 69.  doi: 10.3934/dcdsb.2006.6.69.  Google Scholar [12] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments,, J. Math. Biol., 65 (2012), 309.  doi: 10.1007/s00285-011-0463-z.  Google Scholar [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.  doi: 10.1016/j.jmaa.2013.01.044.  Google Scholar [14] M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space,, Amer. Math. Soc. Translation, 1950 (1950).   Google Scholar [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.  doi: 10.1006/jmaa.1997.5554.  Google Scholar [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.  doi: 10.1016/j.jmaa.2009.08.027.  Google Scholar [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, 92 (1991), 139.  doi: 10.1007/978-3-642-45692-3_10.  Google Scholar [18] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity,, SIAM J. Appl. Math., 70 (2009), 188.  doi: 10.1137/080732870.  Google Scholar [19] W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equat., 20 (2008), 699.  doi: 10.1007/s10884-008-9111-8.  Google Scholar [20] K. Yosida, Functional Analysis,, $6^{th}$ edition, (1980).   Google Scholar [21] F. Zhang and X. Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar
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