Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response. (English) Zbl 1208.34128

The authors are concerned with a mathematical model describing the dynamics of interactions between susceptible host cells, a virus population and a lytic immune response. A delayed argument, corresponding to the time elapsed between the moments when the cell becomes infected and starts emitting virus particles, is introduced in the model. Under different conditions, the system consisting of one delay and two ordinary differential equations has three equilibrium points whose stability is analyzed. Numerical simulations are provided in the final part of the paper.


34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D30 Epidemiology
34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
92C60 Medical epidemiology
Full Text: DOI


[1] Bartholdy, C.; Christensen, J. P.; Wodarz, D.; Thomsen, A. R., Persistent virus infection despite chronic cytotoxic T-lymphocyte activation in Gamma interferon-deficient mice infection with lymphocytic choriomeningitis virus, J. Virology, 74, 1034-10311 (2000)
[2] Nowak, M. A.; Bangham, C. R.M., Population dynamics of immune response to persistent viruses, Science, 272, 74-79 (1996)
[3] Wodarz, D., Hepatitis C virus dynamics and pathology: The role of CTL and antibody response, J. General Virology, 84, 1743-1750 (2003)
[4] Wodarz, D.; Christensen, J. P.; Thomsen, A. R., The importance of lytic and nonlytic immune response in viral infections, Trends in Immunology, 23, 194-200 (2002)
[5] Wang, X.; Tao, Y., Lyapunov function and global properties of virus dynamics with immune response, Int. J. Biomath., 1, 4, 443-448 (2008) · Zbl 1156.92322
[6] Wang, K.; Wang, W.; Pang, H.; Liu, X., Complex dynamic behavior in a viral model with delayed immune response, Phys. D, 226, 197-208 (2007) · Zbl 1117.34081
[7] Song, X.; Neumann, A., Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329, 281-297 (2007) · Zbl 1105.92011
[8] Zhou, X.; Song, X.; Shi, X., Analysis of stability and Hopf bifurcation for an HIV infection model with time delay, Appl. Math. Comput., 199, 1, 23-38 (2008) · Zbl 1136.92027
[9] Culshaw, R. V.; Ruan, S., A delay-differential equation model of HIV infection of CD \(4^+\) T-cells, Math. Biosci., 165, 27-39 (2000) · Zbl 0981.92009
[10] Nelson, P. W.; Perelson, A. S., Mathematical analysis of a delay differential equation models of HIV-1 infection, Math. Biosci., 179, 73-94 (2002) · Zbl 0992.92035
[11] Tam, J., Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16, 1, 29-37 (1999) · Zbl 0914.92012
[12] Song, X.; Cheng, S., A delay-differential equation model of HIV infection of CD \(4^+T\)-cells, J. Korean Math. Soc., 42, 5, 1071-1086 (2005) · Zbl 1078.92042
[13] Beretta, E.; Kuang, Y., Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33, 1144-1165 (2002) · Zbl 1013.92034
[14] Hassard, B. D.; Kazariniff, N. D.; Wan, Y. H., Theory and Application of Hopf Bifurcation, London Math. Soc. Lecture Note Ser., vol. 41 (1981), Cambridge University Press · Zbl 0474.34002
[15] Burić, N.; Mudrinic, M.; Vasović, N., Time delay in a basic model of the immune response, Chaos Solitons Fractals, 12, 483-489 (2001) · Zbl 1026.92015
[16] Canabarro, A. A.; Gléria, I. M.; Lyra, M. L., Periodic solutions and chaos in a non-linear model for the delayed cellular immune response, Phys. A, 342, 234-241 (2004)
[17] Hale, J.; Verduyn Lunel, S. M., Introduction to Functional Differential Equations (1993), Springer-Verlag · 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.