## Stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection.(English)Zbl 1266.92059

Summary: The stability and bifurcation analysis for a delay differential equation of hepatitis B virus infection is investigated. We show the existence of nonnegative equilibria under some appropriate conditions. The existence of a Hopf bifurcation with delay $$\tau$$ at the endemic equilibria is established by analyzing the distribution of the characteristic values. The explicit formulae which determine the direction of the bifurcations, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and the center manifold theorem. Numerical simulation verifies the theoretical results.

### MSC:

 92C60 Medical epidemiology 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional-differential equations 92-08 Computational methods for problems pertaining to biology
Full Text:

### References:

 [1] S. A. Gourley, Y. Kuang, and J. D. Nagy, “Dynamics of a delay differential equation model of hepatitis B virus infection,” Journal of Biological Dynamics, vol. 2, no. 2, pp. 140-153, 2008. · Zbl 1140.92014 [2] Y. Zheng, L. Min, Y. Ji, Y. Su, and Y. Kuang, “Global stability of endemic equilibrium point of basic virus infection model with application to HBV infection,” Journal of Systems Science & Complexity, vol. 23, no. 6, pp. 1221-1230, 2010. · Zbl 1402.92416 [3] S. Hews, S. Eikenberry, J. D. Nagy, and Y. Kuang, “Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth,” Journal of Mathematical Biology, vol. 60, no. 4, pp. 573-590, 2010. · Zbl 1311.92117 [4] B.-Z. Guo and L.-M. Cai, “A note for the global stability of a delay differential equation of hepatitis B virus infection,” Mathematical Biosciences and Engineering, vol. 8, no. 3, pp. 689-694, 2011. · Zbl 1260.92051 [5] J. K. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977. · Zbl 0352.34001 [6] J. Wei and S. Ruan, “Stability and bifurcation in a neural network model with two delays,” Physica D, vol. 130, no. 3-4, pp. 255-272, 1999. · Zbl 1066.34511 [7] X.-P. Yan, “Hopf bifurcation and stability for a delayed tri-neuron network model,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 579-595, 2006. · Zbl 1175.37086 [8] B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, UK, 1981. · Zbl 0474.34002 [9] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002 [10] Y. Song, M. Han, and J. Wei, “Stability and Hopf bifurcation analysis on a simplified BAM neural network with delays,” Physica D, vol. 200, no. 3-4, pp. 185-204, 2005. · Zbl 1062.34079 [11] J. K. Hale and S. Lunel, Introduction to Functional-Differential Equations, vol. 99, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.