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

Previous Article
Global dynamics for two-species competition in patchy environment
MBE Home
This Issue
Next Article
A continuous phenotype space model of RNA virus evolution within a host

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

Toshikazu Kuniya ^1, and Mimmo Iannelli ^2,

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

Abstract
Related Papers

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 $\mathcal{R}_0$.

Keywords: periodicity, age-structure, vertical transmission, basic reproduction number., SIS epidemic model.

Mathematics Subject Classification: Primary: 92D30, 35Q92; Secondary: 45P0.

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.

show all references

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.

[1]	Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete and 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 and Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77
[5]	Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170
[6]	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
[7]	Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[8]	Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba. A multi-group SIR epidemic model with age structure. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3515-3550. doi: 10.3934/dcdsb.2016109
[9]	Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
[10]	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
[11]	Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170
[12]	Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220
[13]	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
[14]	Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure and Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97
[15]	Jia-Feng Cao, Wan-Tong Li, Fei-Ying Yang. Dynamics of a nonlocal SIS epidemic model with free boundary. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 247-266. doi: 10.3934/dcdsb.2017013
[16]	Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291
[17]	David Greenhalgh, Yanfeng Liang, Xuerong Mao. Demographic stochasticity in the SDE SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2859-2884. doi: 10.3934/dcdsb.2015.20.2859
[18]	Fei-Ying Yang, Wan-Tong Li. Dynamics of a nonlocal dispersal SIS epidemic model. Communications on Pure and Applied Analysis, 2017, 16 (3) : 781-798. doi: 10.3934/cpaa.2017037
[19]	Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239
[20]	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

2018 Impact Factor: 1.313

Tools

Metrics

Other articles
by authors

on AIMS
Toshikazu Kuniya Mimmo Iannelli
on Google Scholar
Toshikazu Kuniya Mimmo Iannelli

[Back to Top]