JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (1): 81-87.doi: 10.6040/j.issn.1671-9352.0.2016.314

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Dynamical analysis of a class of periodic epidemic model with delay

WANG Shuang-ming   

  1. School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou 730020, Gansu, China
  • Received:2016-06-29 Online:2017-01-20 Published:2017-01-16

Abstract: A time-periodic and delayed epidemic system modeling the spread of bacteria is studied by method of dynamical systems. In terms of corresponding periodic eigenvalue problem, we establish the global dynamics of the system.

Key words: delay, positive periodic solutions, global attractivity, periodic epidemic model

CLC Number: 

  • O175
[1] CAPASSO V. Mathematical structures of epidemic systems(lecture notes in biomath)[M]. Heidelberg: Springer-Verlag, 1993: 59-75.
[2] KUANG Yang. Delay differential equations with applications in population dynamics[M]. Boston: Academic Press, 1993: 119-143.
[3] SMITH H L. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical surveys and monographs[M]. Providence: American Mathematical Society, 1995: 145-161.
[4] SMOLLER J. Shock waves and reaction-diffusiom equations[M]. Berlin: Springer-Verlag, 1983: 181-198.
[5] 王智诚, 王双明. 一类时间周期的时滞反应扩散模型的空间动力学研究[J]. 兰州大学学报(自然科学版), 2013(4): 535-540. WANG Zhicheng, WANG Shuangming. Spatial dynamics of a class of delayed nonlocal reaction-diffusion model with time period[J]. Journal of Lanzhou University(Natural Sciences), 2013(4):534-540.
[6] ZHAO Xiaoqiang. Dynamical systems in population biology[M]. New York: Springer-Verlag, 2003: 63-98.
[7] ZHANG Liang, WANG Zhicheng. Spatial dynamics of a diffusive predator-prey model with stage structure[J]. Discrete Continuous Dynamical Systems Ser B, 2015, 20(6):1831-1853.
[8] ZHANG Liang, WANG Zhicheng, ZHAO Xiaoqiang. Threshold dynamics of a time periodic reaction—diffusion epidemic model with latent period[J]. Journal of Differential Equations, 2015, 258(9):3011-3036.
[9] CAPASSO V, MADDALENA L. Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases[J]. J Math Biology, 1981, 13(2):173-184.
[10] WU Shiliang. Entire solutions in a bistable reaction-diffusion system modeling man-environment-man epidemics[J]. Nonlinear Analysis: Real World Applications, 2012, 13(5):1991-2005.
[11] WU Shiliang, WANG Haiyan. Front-like entire solutions for monostable reaction-diffusion systems[J]. Journal of Dynamics and Differential Equations, 2013, 25(2):505-533.
[12] YANG Yunrui, LI Wantong, WU Shiliang. Exponential stability of traveling fronts in a diffusion epidemic system with delay[J]. Nonlinear Analysis: Real World Applications, 2011, 12(2):1223-1234.
[13] HESS P. Periodic-parabolic boundary value problems and positivity[M]. Harlow, UK: Longman Scientific & Technical, 1991: 30-38.
[14] MARTIN R H, SMITH H L. Abstract functional differential equations and Reaction-Diffusion systems[J]. Trans Amer Math Soc, 1990, 321(1):1-44.
[15] MARTIN R H. Nonlinear operators and differential equations in banach spaces[M]. New York: John Wiley and Sons, 1976: 354-366.
[16] ZHAO Xiaoqiang. Global attractivity and stability in some monotonediscrete dynamical systems[J]. Bull Austral Math Soc, 1996, 53(2):305-324.
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