American Institute of Mathematical Sciences

Advanced
December  2017, 14(5&6): 1535-1563. doi: 10.3934/mbe.2017080

Threshold dynamics of a time periodic and two–group epidemic model with distributed delay

Lin Zhao a,b, , Zhi-Cheng Wang a,, and  Liang Zhang a,

a. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
b. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Z.-C. Wang

Received  May 14, 2016 Accepted  December 31, 2016 Published  May 2017

In this paper, a time periodic and two–group reaction–diffusion epidemic model with distributed delay is proposed and investigated. We firstly introduce the basic reproduction number $R_0$ for the model via the next generation operator method. We then establish the threshold dynamics of the model in terms of $R_0$, that is, the disease is uniformly persistent if $R_0 > 1$, while the disease goes to extinction if $R_0 < 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

Keywords: Two–group epidemic model, time periodic, distributed delay, threshold dynamics, Lyapunov functional.
Mathematics Subject Classification: Primary: 35K57; Secondary: 35B10, 35B35, 34B40, 92D30.
Citation: Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
References:
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show all references

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[20]

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[21]

W. Huang, K. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple–group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.  doi: 10.1137/0152047.  Google Scholar

[22]

G. Huang and A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013), 687-691.  doi: 10.1016/j.aml.2013.01.010.  Google Scholar

[23]

J. M. Hyman and J. Li, Differential susceptibility epidemic models, J. Math. Biol., 50 (2005), 626-644.  doi: 10.1007/s00285-004-0301-7.  Google Scholar

[24]

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[25]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516.  doi: 10.1137/070709761.  Google Scholar

[26]

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J. Li and X. Zou, Generalization of the Kermack–McKendrick SIR model to a patchy environment for a disease with latency, Math. Model. Nat. Phenom., 4 (2009), 92-118.  doi: 10.1051/mmnp/20094205.  Google Scholar

[28]

J. Li and X. Zou, Dynamics of an epidemic model with non–local infections for diseases with latency over a patchy environment, J. Math. Biol., 60 (2010), 645-686.  doi: 10.1007/s00285-009-0280-9.  Google Scholar

[29]

M. Li, Z. Shuai and C. Wang, Global stability of multi–group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47.  doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar

[30]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007), 1-40.  doi: 10.1002/cpa.20154.  Google Scholar

[31]

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[32]

Y. Lou and X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.  Google Scholar

[33]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with deasonal developmental durations, J Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.  Google Scholar

[34]

P. Magal and C. McCluskey, Two–group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013), 1058-1095.  doi: 10.1137/120882056.  Google Scholar

[35]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar

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M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, Springer, New York, 2015. doi: 10.1007/978-1-4899-7612-3.  Google Scholar

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R. Martain and H. L. Smith, Abstract functional differential equations and reaction–diffusion system, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[38]

C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anal. Real World Appl., 11 (2010), 55-59.  doi: 10.1016/j.nonrwa.2008.10.014.  Google Scholar

[39]

C. McCluskey and Y. Yang, Global stability of a diffusive virus dynamics model with general incidence function and time delay, Nonlinear Anal. Real World Appl., 25 (2015), 64-78.  doi: 10.1016/j.nonrwa.2015.05.003.  Google Scholar

[40]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

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B. Perthame, Parabolic Equations in Biology, Springer, Cham, 2015. doi: 10.1007/978-3-319-19500-1.  Google Scholar

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