# 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, 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. doi: 10.1007/s11538-006-9166-9. [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [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. doi: 10.1137/0522069. [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. doi: 10.1007/978-3-642-75301-5. [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. doi: 10.1007/BF00178324. [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. doi: 10.1016/S0362-546X(97)00597-X. [10] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [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. doi: 10.3934/dcdsb.2006.6.69. [12] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z. [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. doi: 10.1016/j.jmaa.2013.01.044. [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. doi: 10.1006/jmaa.1997.5554. [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. doi: 10.1016/j.jmaa.2009.08.027. [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. doi: 10.1007/978-3-642-45692-3_10. [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. doi: 10.1137/080732870. [19] W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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. doi: 10.1016/j.jmaa.2006.01.085.

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##### References:
 [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. doi: 10.1007/s11538-006-9166-9. [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [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. doi: 10.1137/0522069. [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. doi: 10.1007/978-3-642-75301-5. [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. doi: 10.1007/BF00178324. [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. doi: 10.1016/S0362-546X(97)00597-X. [10] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [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. doi: 10.3934/dcdsb.2006.6.69. [12] H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348. doi: 10.1007/s00285-011-0463-z. [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. doi: 10.1016/j.jmaa.2013.01.044. [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. doi: 10.1006/jmaa.1997.5554. [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. doi: 10.1016/j.jmaa.2009.08.027. [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. doi: 10.1007/978-3-642-45692-3_10. [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. doi: 10.1137/080732870. [19] W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equat., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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. doi: 10.1016/j.jmaa.2006.01.085.
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