Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

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

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.

  Article Metrics

Keywords Two–group epidemic model; time periodic; distributed delay; threshold dynamics; Lyapunov functional

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


  • [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.


This article has been cited by

  • 1. Lin Zhao, Zhi-Cheng Wang, Shigui Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, Nonlinear Analysis: Real World Applications, 2020, 51, 102966, 10.1016/j.nonrwa.2019.102966
  • 2. Xinchang Wang, Xiaosong Tang, Zhiwei Wang, Xiaolin Li, Global dynamics of a diffusive viral infection model with general incidence function and distributed delays, Ricerche di Matematica, 2020, 10.1007/s11587-020-00481-0

Reader Comments

your name: *   your email: *  

Copyright Info: 2017, Zhi-Cheng Wang, licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved