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

show all references

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

[1]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69

[2]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[3]

C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819

[4]

Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77

[5]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[6]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37

[7]

Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109

[8]

Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291

[9]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

[10]

Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455

[11]

Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013

[12]

Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291

[13]

David Greenhalgh, Yanfeng Liang, Xuerong Mao. Demographic stochasticity in the SDE SIS epidemic model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2859-2884. doi: 10.3934/dcdsb.2015.20.2859

[14]

Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure & Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037

[15]

Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97

[16]

Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239

[17]

John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

[18]

Jing Ge, Ling Lin, Lai Zhang. A diffusive SIS epidemic model incorporating the media coverage impact in the heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2763-2776. doi: 10.3934/dcdsb.2017134

[19]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1

[20]

Eleonora Messina. Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation. Conference Publications, 2015, 2015 (special) : 826-834. doi: 10.3934/proc.2015.0826

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]