Abstract and Applied Analysis

Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay

Yanhui Zhai, Ying Xiong, Xiaona Ma, and Haiyun Bai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more, the global existence of periodic solutions is established by using a global Hopf bifurcation result.

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 242410, 7 pages.

First available in Project Euclid: 3 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Zhai, Yanhui; Xiong, Ying; Ma, Xiaona; Bai, Haiyun. Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 242410, 7 pages. doi:10.1155/2014/242410. https://projecteuclid.org/euclid.aaa/1412360628

Export citation


  • J. Li, X. Yu, X. Pu et al., “Environmental connections of novel avian-origin H7N9 influenza virus infection and virus adaptation to the human,” Science China Life Sciences, vol. 56, no. 6, pp. 485–492, 2013.
  • J. He, L. Ning, and Y. Tong, “Origins and evolutionary genomics of the novel 2013 avian-origin H7N9 influenza A virus in China: early findingsčommentComment on ref. [16?]: Please update the information of this reference, if possible.,” http://arxiv.org/abs/1304.1985.
  • C. C. McCluskey, “Global stability for an SIR epidemic model with delay and nonlinear incidence,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 3106–3109, 2010.
  • K. L. Cooke, “Stability analysis for a vector disease model,” The Rocky Mountain Journal of Mathematics, vol. 9, no. 1, pp. 31–42, 1979.
  • R. Xu and Z. Ma, “Global stability of a SIR epidemic model with nonlinear incidence rate and time delay,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3175–3189, 2009.
  • F. Zhang and Z. Li, “Global stability of an SIR epidemic model with constant infectious period,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 285–291, 2008.
  • S. Ruan, “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays,” Quarterly of Applied Mathematics, vol. 59, no. 1, pp. 159–173, 2001.
  • J. Wei and C. Yu, “Hopf bifurcation analysis in a model of oscillatory gene expression with delay,” Proceedings of the Royal Society of Edinburgh A. Mathematics, vol. 139, no. 4, pp. 879–895, 2009.
  • X. Meng, H. Huo, and H. Xiang, “Hopf bifurcation in a three-species system with delays,” Journal of Applied Mathematics and Computing, vol. 35, no. 1-2, pp. 635–661, 2011.
  • J. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799–4838, 1998. \endinput