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Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1535-1563. doi: 10.3934/mbe.2017080
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Threshold dynamics of a time periodic and two–group epidemic model with distributed delay

  • . School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
  • Received: 14 May 2016 Accepted: 31 December 2016 Published: 01 October 2017

  • MSC : Primary: 35K57; Secondary: 35B10, 35B35, 34B40, 92D30

  • 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 $R_0 \gt 1$, while the disease goes to extinction if $R_0 \lt 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.

    Citation: Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1535-1563. doi: 10.3934/mbe.2017080

    Related Papers:

  • 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 $R_0 \gt 1$, while the disease goes to extinction if $R_0 \lt 1$. Finally, we study the global dynamics for the model in a special case when all the coefficients are independent of spatio–temporal variables.


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    [1] [ S. Altizer,A. Dobson,P. Hosseini,P. Hudson,M. Pascual,P. Rohani, Seasonality and the dynamics of infectious disease, Ecol. Lett., 9 (2006): 467-484.
    [2] [ R. M. Anderson, Discussion: the Kermack-McKendrick epidemic threshold theorem, Bull. Math. Biol., 53 (1991): 3-32.
    [3] [ R. M. Anderson and R. May, Infectious Diseases of Humanns: Dynamics and Control, Oxford University Press, Oxford, 1991.
    [4] [ N. Bacaër,D. Ait,H. El, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011): 741-762.
    [5] [ N. Bacaër,S. Guernaoui, The epidemic threshold of vector–borne disease with seasonality, J. Math. Biol., 53 (2006): 421-436.
    [6] [ E. Beretta,T. Hara,W. Ma,Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001): 4107-4115.
    [7] [ B. Bonzi,A. A. Fall,A. Iggidr,G. Sallet, Stability of differential susceptibility and infectivity epidemic models, J. Math. Biol., 62 (2011): 39-64.
    [8] [ F. Brauer, Compartmental models in epidemiology, Mathematical Epidemiology, Springer, 56 (2008): 19-79.
    [9] [ L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach space, Arch. Math., 56 (1991): 49-57.
    [10] [ L. Cai,M. Martcheva,X.-Z. Li, Competitive exclusion in a vector-host epidemic model with distributed delay, J. Biol. Dyn., 7 (2013): 47-67.
    [11] [ D. Dancer and P. Koch Medina, Abstract ecolution equations, Periodic problem and applications, Longman, Harlow, UK, 1992.
    [12] [ O. Diekmann,J. Heesterbeek,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.
    [13] [ W. E. Fitzgibbon,M. Langlais,M. E. Parrott,G. F. Webb, A diffusive system with age dependency modeling FIV, Nonlinear Anal., 25 (1995): 975-989.
    [14] [ W. E. Fitzgibbon,C. B. Martin,J. J. Morgan, A diffusive epidemic model with criss–cross dynamics, J. Math. Anal. Appl., 184 (1994): 399-414.
    [15] [ W. E. Fitzgibbon,M. E. Parrott,G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci., 128 (1995): 131-155.
    [16] [ D. Gao and S. Ruan, Malaria models with spatial effects, John Wiley & Sons. (in press)
    [17] [ I. Gudelj,K. A. J. White,N. F. Britton, The effects of spatial movement and group interactions on disease dynamics of social animals, Bull. Math. Biol., 66 (2004): 91-108.
    [18] [ Z. Guo,F.-B. Wang,X. Zou, Threshold dynamics of an infective disease model with a fixed latent period and non–local infections, J. Math. Biol., 65 (2012): 1387-1410.
    [19] [ P. Hess, Periodic–Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK, 1991.
    [20] [ H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000): 599-653.
    [21] [ W. Huang,K. Cooke,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.
    [22] [ G. Huang,A. Liu, A note on global stability for a heroin epidemic model with distributed delay, Appl. Math. Lett., 26 (2013): 687-691.
    [23] [ J. M. Hyman,J. Li, Differential susceptibility epidemic models, J. Math. Biol., 50 (2005): 626-644.
    [24] [ H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012): 309-348.
    [25] [ Y. Jin,X.-Q. Zhao, Spatial dynamics of a nonlocal periodic reaction–diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009): 2496-2516.
    [26] [ T. Kato, Peturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelerg, 1976.
    [27] [ J. Li,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.
    [28] [ J. Li,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.
    [29] [ M. Li,Z. Shuai,C. Wang, Global stability of multi–group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010): 38-47.
    [30] [ X. Liang,X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 60 (2007): 1-40.
    [31] [ Y. Lou,X.-Q. Zhao, Threshold dynamics in a time–delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009): 169-186.
    [32] [ Y. Lou,X.-Q. Zhao, A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011): 543-568.
    [33] [ Y. Lou,X.-Q. Zhao, A theoretical approach to understanding population dynamics with deasonal developmental durations, J Nonlinear Sci., 27 (2017): 573-603.
    [34] [ P. Magal,C. McCluskey, Two–group infection age model including an application to nosocomial infection, SIAM J. Appl. Math., 73 (2013): 1058-1095.
    [35] [ P. Magal,X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005): 251-275.
    [36] [ M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, Springer, New York, 2015.
    [37] [ R. Martain,H. L. Smith, Abstract functional differential equations and reaction–diffusion system, Trans. Amer. Math. Soc., 321 (1990): 1-44.
    [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.
    [39] [ C. McCluskey,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.
    [40] [ J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.
    [41] [ R. Peng,X.-Q. Zhao, A reaction–diffusion SIS epidemic model in a time–periodic environment, Nonlinearity, 25 (2012): 1451-1471.
    [42] [ B. Perthame, Parabolic Equations in Biology, Springer, Cham, 2015.
    [43] [ L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, Mathematical Surveys and Monographs, 102. American Mathematical Society, Providence, RI, 2003.
    [44] [ R. Ross, An application of the theory of probabilities to the study of a priori pathometry: Ⅰ, Proc. R. Soc. Lond., 92 (1916): 204-230.
    [45] [ S. Ruan, Spatial−temporal dynamics in nonlocal epidemiological models, Mathematics for Life Science and Medicine, Springer−Verlag, Berlin, (2007), 99–122.
    [46] [ S. Ruan and J. Wu, Modeling Spatial Spread of Communicable Diseases Involving Animal Hosts, Chapman & Hall/CRC, Boca Raton, FL, (2009), 293–316.
    [47] [ H. L. Smith, Monotone Dynamical System: An Introduction to the Theorey of Competitive and Cooperative Systems, Math. Surveys and Monogr. vol 41, American Mathematical Society, Providence, 1995.
    [48] [ R. Sun, Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence, Comput. Math. Appl., 60 (2010): 2286-2291.
    [49] [ Y. Takeuchi,W. Ma,E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000): 931-947.
    [50] [ H. R. Thieme, Mathematics in population biology, Princeton University Press, Princeton, NJ, 2003.
    [51] [ H. R. Thieme, Spectral bound and reproduction number for infinite–dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009): 188-211.
    [52] [ H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, J. Differential Equations, 250 (2011): 3772-3801.
    [53] [ P. van den Driessche,X. Zou, Modeling relapse in infectious diseases, Math. Biosci., 207 (2007): 89-103.
    [54] [ B.-G. Wang,W.-T. Li,Z.-C. Wang, A reaction–diffusion SIS epidemic model in an almost periodic environment, Z. Angew. Math. Phys., 66 (2015): 3085-3108.
    [55] [ B.-G. Wang,X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dynam. Differential Equations, 25 (2013): 535-562.
    [56] [ L. Wang,Z. Liu,X. Zhang, Global dynamics of an SVEIR epidemic model with distributed delay and nonlinear incidence, Appl. Math. Comput., 284 (2016): 47-65.
    [57] [ W. Wang,X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008): 699-717.
    [58] [ W. Wang,X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011): 147-168.
    [59] [ W. Wang,X.-Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012): 1652-1673.
    [60] [ W. Wang,X.-Q. Zhao, Spatial invasion threshold of Lyme disease, SIAM J. Appl. Math., 75 (2015): 1142-1170.
    [61] [ J. Wu, Spatial structure: Partial differential equations models, Mathematical Epidemiology, Springer, Berlin, 1945 (2008): 191-203.
    [62] [ D. Xu,X.-Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl., 311 (2005): 417-438.
    [63] [ Z. Xu,X.-Q. Zhao, A vector–bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012): 2615-2634.
    [64] [ L. Zhang,J.-W. Sun, Global stability of a nonlocal epidemic model with delay, Taiwanese J. Math., 20 (2016): 577-587.
    [65] [ L. Zhang and Z. -C. Wang, A time-periodic reaction-diffusion epidemic model with infection period, Z. Angew. Math. Phys. , 67 (2016), Art. 117, 14 pp.
    [66] [ L. Zhang,Z.-C. Wang,Y. Zhang, Dynamics of a reaction-diffusion waterborne pathogen model with direct and indirect transmission, Comput. Math. Appl., 72 (2016): 202-215.
    [67] [ L. Zhang,Z.-C. Wang,X.-Q. Zhao, Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period, J. Differential Equations, 258 (2015): 3011-3036.
    [68] [ Y. Zhang,X.-Q. Zhao, A reaction–diffusion Lyme disease model with seasonality, SIAM J. Appl. Math., 73 (2013): 2077-2099.
    [69] [ X. -Q. Zhao, Dynamical System in Population Biology, Spring-Verlag, New York. 2003.
    [70] [ X.-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012): 787-808.
    [71] [ X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyman. Differential Equations, 29 (2017): 67-82.
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