Hopf bifurcation and stability of periodic solutions for delay differential model of HIV infection of CD4 T-cells. (English) Zbl 1474.92060

Summary: This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of \(\text{CD} 4^+\) T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.


92C60 Medical epidemiology
92D30 Epidemiology
34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology


Full Text: DOI


[1] Liu, D.; Wang, B., A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis, Applied Mathematical Modelling, 37, 7, 4608-4625 (2013) · Zbl 1426.92079 · doi:10.1016/j.apm.2012.09.065
[2] Ncube, I., Absolute stability and Hopf bifurcation in a Plasmodium falciparum malaria model incorporating discrete immune response delay, Mathematical Biosciences, 243, 1, 131-135 (2013) · Zbl 1310.92034 · doi:10.1016/j.mbs.2013.02.010
[3] Song, X.; Wang, S.; Dong, J., Stability properties and Hopf bifurcation of a delayed viral infection model with lytic immune response, Journal of Mathematical Analysis and Applications, 373, 2, 345-355 (2011) · Zbl 1208.34128 · doi:10.1016/j.jmaa.2010.04.010
[4] Song, X.; Wang, S.; Zhou, X., Stability and Hopf bifurcation for a viral infection model with delayed non-lytic immune response, Journal of Applied Mathematics and Computing, 33, 1-2, 251-265 (2010) · Zbl 1203.34134 · doi:10.1007/s12190-009-0285-y
[5] Wang, T.; Hu, Z.; Liao, F., Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response, Journal of Mathematical Analysis and Applications, 411, 1, 63-74 (2014) · Zbl 1308.92104 · doi:10.1016/j.jmaa.2013.09.035
[6] Wang, T.; Hu, Z.; Liao, F.; Ma, W., Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity, Mathematics and Computers in Simulation, 89, 13-22 (2013) · Zbl 1490.92118 · doi:10.1016/j.matcom.2013.03.004
[7] Bai, Z.; Zhou, Y., Dynamics of a viral infection model with delayed CTL response and immune circadian rhythm, Chaos, Solitons & Fractals, 45, 9-10, 1133-1139 (2012) · Zbl 1258.92020 · doi:10.1016/j.chaos.2012.06.001
[8] Beretta, E.; Carletti, M.; Kirschner, D. E.; Marino, S., Stability analysis of a mathematical model of the immune response with delays, Mathematics for Life Science and Medicine, 177-206 (2007), Berlin, Germany: Springer, Berlin, Germany
[9] Hu, Z.; Zhang, J.; Wang, H.; Ma, W.; Liao, F., Dynamics analysis of a delayed viral infection model with logistic growth and immune impairment, Applied Mathematical Modelling, 38, 2, 524-534 (2014) · Zbl 1427.92087 · doi:10.1016/j.apm.2013.06.041
[10] Huang, G.; Yokoi, H.; Takeuchi, Y.; Sasaki, T., Impact of intracellular delay, immune activation delay and nonlinear incidence on viral dynamics, Japan Journal of Industrial and Applied Mathematics, 28, 3, 383-411 (2011) · Zbl 1226.92049 · doi:10.1007/s13160-011-0045-x
[11] Rihan, F. A.; Abdel Rahman, D. H., Delay differential model for tumor-immune dynamics with HIV infection of CD \(4^+\) T-cells, International Journal of Computer Mathematics, 90, 3, 594-614 (2013) · Zbl 1272.93020 · doi:10.1080/00207160.2012.726354
[12] Tam, J., Delay effect in a model for virus replication, Journal of Mathemathics Applied in Medicine and Biology, 16, 1, 29-37 (1999) · Zbl 0914.92012 · doi:10.1093/imammb/16.1.29
[13] Wang, Z.; Xu, R., Stability and Hopf bifurcation in a viral infection model with nonlinear incidence rate and delayed immune response, Communications in Nonlinear Science and Numerical Simulation, 17, 2, 964-978 (2012) · Zbl 1239.92071 · doi:10.1016/j.cnsns.2011.06.024
[14] Mittler, J. E.; Sulzer, B.; Neumann, A. U.; Perelson, A. S., Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Mathematical Biosciences, 152, 2, 143-163 (1998) · Zbl 0946.92011 · doi:10.1016/S0025-5564(98)10027-5
[15] Rihan, F. A.; Abdel Rahman, D. H.; Lakshmanan, S., A time delay model of tumour-immune system interactions: global dynamics, parameter estimation, sensitivity analysis, Applied Mathematics and Computation, 232, 606-623 (2014) · Zbl 1410.92053 · doi:10.1016/j.amc.2014.01.111
[16] Zhang, L.; Zhang, C.; Zhao, D., Hopf bifurcation analysis of integro-differential equation with unbounded delay, Applied Mathematics and Computation, 217, 10, 4972-4979 (2011) · Zbl 1219.37038 · doi:10.1016/j.amc.2010.11.046
[17] Perelson, A. S.; Neumann, A. U.; Markowitz, M.; Leonard, J. M.; Ho, D. D., HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271, 5255, 1582-1586 (1996) · doi:10.1126/science.271.5255.1582
[18] Herz, A. V. M.; Bonhoeffer, S.; Anderson, R. M.; May, R. M.; Nowak, M. A., Viral dynamics in vivo: limitations on estimates of intracellular delay and virus decay, Proceedings of the National Academy of Sciences of the United States of America, 93, 14, 7247-7251 (1996) · doi:10.1073/pnas.93.14.7247
[19] Nowak, M. A.; Bangham, C. R. M., Population dynamics of immune responses to persistent viruses, Science, 272, 5258, 74-79 (1996) · doi:10.1126/science.272.5258.74
[20] Anderson, R. M.; May, R. M., Population biology of infectious diseases: part I, Nature, 280, 5721, 361-367 (1979) · doi:10.1038/280361a0
[21] de Boer, R. J.; Perelson, A. S., Target cell limited and immune control models of HIV infection: a comparison, Journal of Theoretical Biology, 190, 3, 201-214 (1998) · doi:10.1006/jtbi.1997.0548
[22] Wodarz, D.; Nowak, M. A., Specific therapy regimes could lead to long-term immunological control of HIV, Proceedings of the National Academy of Sciences of the United States of America, 96, 25, 14464-14469 (1999) · doi:10.1073/pnas.96.25.14464
[23] Wodarz, D.; Page, K. M.; Arnaout, R. A.; Thomsen, A. R.; Lifson, J. D.; Nowak, M. A., A new theory of cytotoxic T-lymphocyte memory: implications for HIV treatment, Philosophical Transactions of the Royal Society B: Biological Sciences, 355, 1395, 329-343 (2000) · doi:10.1098/rstb.2000.0570
[24] Lv, C.; Huang, L.; Yuan, Z., Global stability for an {HIV}-1 infection model with Beddington-DeAngelis incidence rate and {CTL} immune response, Communications in Nonlinear Science and Numerical Simulation, 19, 1, 121-127 (2014) · Zbl 1344.92173 · doi:10.1016/j.cnsns.2013.06.025
[25] Perelson, A. S.; Nelson, P. W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41, 1, 3-44 (1999) · Zbl 1078.92502 · doi:10.1137/S0036144598335107
[26] Wang, Y.; Zhou, Y.; Brauer, F.; Heffernan, J. M., Viral dynamics model with {CTL} immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67, 4, 901-934 (2013) · Zbl 1279.34059 · doi:10.1007/s00285-012-0580-3
[27] Pitchaimani, M.; Monica, C.; Divya, M., Stability analysis for hiv infection delay model with protease inhibitor, Biosystems, 114, 2, 118-124 (2013)
[28] Shu, H.; Wang, L., Role of cd \(4^+\) t-cell proliferation in HIV infection u nder antiretroviral therapy, Journal of Mathematical Analysis and Applications, 394, 2, 529-544 (2012) · Zbl 1275.92033 · doi:10.1016/j.jmaa.2012.05.027
[29] Wang, S.; Zhou, Y., Global dynamics of an in-host {HIV} 1-infection model with the long-lived infected cells and four intracellular delays, International Journal of Biomathematics, 5, 6 (2012) · Zbl 1297.92048 · doi:10.1142/S1793524512500581
[30] Fenton, A.; Lello, J.; Bonsall, M. B., Pathogen responses to host immunity: the impact of time delays and memory on the evolution of virulence, Proceedings of the Royal Society B: Biological Sciences, 273, 1597, 2083-2090 (2006) · doi:10.1098/rspb.2006.3552
[31] Nowak, M. A.; May, R. M.; Sigmund, K., Immune responses against multiple epitopes, Journal of Theoretical Biology, 175, 3, 325-353 (1995) · doi:10.1006/jtbi.1995.0146
[32] Hale, J., Theory of Functional Differential Equations (1997), New York, NY, USA: Springer, New York, NY, USA
[33] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics. Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191 (1993), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0777.34002
[34] Hassard, B. D., Theory and Applications of Hopf Bifurcation, 41 (1981), CUP Archive · Zbl 0474.34002
[35] Baker, C. T. H.; Bocharov, G.; Filiz, A.; Ford, N.; Paul, C.; Rihan, F. A.; A Tang, R. T.; Tian, H.; Wille, D.
[36] Rihan, F. A., Numerical treatment of delay differential equations in bioscience [Ph.D. thesis] (2000), Manchester, UK: The University of Manchester, Manchester, UK
[37] Shampine, L. F.; Thompson, S., Solving DDEs in Matlab, Applied Numerical Mathematics, 37, 4, 441-458 (2001) · Zbl 0983.65079 · doi:10.1016/S0168-9274(00)00055-6
[38] Paul, C., A user-guide to archi: an explicit runge-kutta code for so lving delay and neutral differential equations and parameter estimation problems, MCCM Techical Report, 283 (1997), University of Manchester
[39] Rihan, F. A.; Doha, E. H.; Hassan, M. I.; Kamel, N. M., Numerical treatments for Volterra delay integro-differential equations, Computational Methods in Applied Mathematics, 9, 3, 292-308 (2009) · Zbl 1184.65122 · doi:10.2478/cmam-2009-0018
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