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[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 929-945. doi: 10.3934/mbe.2014.11.929
Related Papers:
| [1] |
Yicang Zhou, Zhien Ma .
Global stability of a class of discrete
age-structured SIS models with immigration. Mathematical Biosciences and Engineering, 2009, 6(2): 409-425.
doi: 10.3934/mbe.2009.6.409
|
| [2] |
Simone De Reggi, Francesca Scarabel, Rossana Vermiglio .
Approximating reproduction numbers: a general numerical method for age-structured models. Mathematical Biosciences and Engineering, 2024, 21(4): 5360-5393.
doi: 10.3934/mbe.2024236
|
| [3] |
Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva .
Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1159-1186.
doi: 10.3934/mbe.2017060
|
| [4] |
Joan Ponce, Horst R. Thieme .
Can infectious diseases eradicate host species? The effect of infection-age structure. Mathematical Biosciences and Engineering, 2023, 20(10): 18717-18760.
doi: 10.3934/mbe.2023830
|
| [5] |
Julien Arino, Fred Brauer, P. van den Driessche, James Watmough, Jianhong Wu .
A final size relation for epidemic models. Mathematical Biosciences and Engineering, 2007, 4(2): 159-175.
doi: 10.3934/mbe.2007.4.159
|
| [6] |
Christine K. Yang, Fred Brauer .
Calculation of $R_0$ for age-of-infection models. Mathematical Biosciences and Engineering, 2008, 5(3): 585-599.
doi: 10.3934/mbe.2008.5.585
|
| [7] |
Zhen Jin, Guiquan Sun, Huaiping Zhu .
Epidemic models for complex networks with demographics. Mathematical Biosciences and Engineering, 2014, 11(6): 1295-1317.
doi: 10.3934/mbe.2014.11.1295
|
| [8] |
Jianquan Li, Zhien Ma, Fred Brauer .
Global analysis of discrete-time SI and SIS epidemic models. Mathematical Biosciences and Engineering, 2007, 4(4): 699-710.
doi: 10.3934/mbe.2007.4.699
|
| [9] |
Lin Zhao, Haifeng Huo .
Spatial propagation for a reaction-diffusion SI epidemic model with vertical transmission. Mathematical Biosciences and Engineering, 2021, 18(5): 6012-6033.
doi: 10.3934/mbe.2021301
|
| [10] |
Tailei Zhang, Hui Li, Na Xie, Wenhui Fu, Kai Wang, Xiongjie Ding .
Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China. Mathematical Biosciences and Engineering, 2020, 17(2): 1757-1775.
doi: 10.3934/mbe.2020092
|
Abstract
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 $\mathcal{R}_0$.
References
|
[1]
|
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]
|
Bull. Math. Biol., 69 (2007), 1067-1091.
|
|
[3]
|
J. Math. Biol., 53 (2006), 421-436.
|
|
[4]
|
SIAM J. Math. Anal., 22 (1991), 1065-1080.
|
|
[5]
|
in Dynamical Systems, World Scientific, (1993), 1-19.
|
|
[6]
|
Springer-Verlag, Berlin-New York, 1993.
|
|
[7]
|
Dynam. Syst. Appl., 9 (2000), 361-376.
|
|
[8]
|
J. Math. Biol., 28 (1990), 365-382.
|
|
[9]
|
Nonlinear Anal., 35 (1999), 797-814.
|
|
[10]
|
J. Math. Biol., 28 (1990), 411-434.
|
|
[11]
|
Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 69-96.
|
|
[12]
|
J. Math. Biol., 65 (2012), 309-348.
|
|
[13]
|
J. Math. Anal. Appl., 402 (2013), 477-492.
|
|
[14]
|
Amer. Math. Soc. Translation, 1950 (1950), 128pp.
|
|
[15]
|
J. Math. Anal. Appl., 213 (1997), 511-533.
|
|
[16]
|
J. Math. Anal. Appl., 363 (2010), 230-237.
|
|
[17]
|
in Differential Equations Models in Biology, Epidemiology and Ecology (eds. S. Busenberg and M. Martelli), Springer, 92 (1991), 139-158.
|
|
[18]
|
SIAM J. Appl. Math., 70 (2009), 188-211.
|
|
[19]
|
J. Dyn. Diff. Equat., 20 (2008), 699-717.
|
|
[20]
|
$6^{th}$ edition, Springer-Verlag, Berlin-New York, 1980.
|
|
[21]
|
J. Math. Anal. Appl., 325 (2007), 496-516.
|
DownLoad: