Back
 IJMNTA  Vol.10 No.2 , June 2021
Bifurcation and Stability Analysis of HIV Infectious Model with Two Time Delays
Abstract: The HIV problem is studied by version of delay mathematical models which consider the apoptosis of uninfected CD4+ T cells which cultured with infected T cells in big volume. The opportunistic infection and the apoptosis of uninfected CD4+ T cells are caused directly or indirectly by a toxic substance produced from HIV genes. Ubiquitously, the nonlinear incidence rate brings forth the increasing number of infected CD4+ T cells with introduction of small time delay, and in addition, there also exists a natural time delay factor during the process of virus replication. With state feedback control of time delay, the bifurcating periodical oscillating phenomena is induced via Hopf bifurcation. Mathematically, with the geometrical criterion applied in the stability analysis of delay model, the critical threshold of Hopf bifurcation in multiple delay differential equations which satisfy the transversal condition is derived. By applying reduction dimensional method combined with the center manifold theory, the stability of the bifurcating periodical solution is analyzed by the perturbation near Hopf point.
Cite this paper: Ma, S. (2021) Bifurcation and Stability Analysis of HIV Infectious Model with Two Time Delays. International Journal of Modern Nonlinear Theory and Application, 10, 49-64. doi: 10.4236/ijmnta.2021.102004.
References

[1]   Korobeinikov, A. (2004) Global Properties of Basic Virus Dynamics Models. Bulletin of Mathematical Biology, 66, 879-883.
https://doi.org/10.1016/j.bulm.2004.02.001

[2]   Korobeinikov, A. (2007) Global Properties of Infectious Disease Models with Nonlinear Incidence. Bulletin of Mathematical Biology, 69, 1871-1886.
https://doi.org/10.1007/s11538-007-9196-y

[3]   Li, M.Y. and Shu, H. (2010) Global Dynamics of an In-Host Viral Model with Intracellular Delay. Bulletin of Mathematical Biology, 72, 1492-1505.
https://doi.org/10.1007/s11538-010-9503-x

[4]   Wang, K.F., Fan, A.J. and Torres, A. (2010) Global Properties of an Improved Hepatitis B Virus Model. Nonlinear Analysis: Real World Applications, 11, 3131-3138.
https://doi.org/10.1016/j.nonrwa.2009.11.008

[5]   Perelson, A.S., Kirschner, D.E. and De Boer, R. (1993) Dynamics of HIV Infection of CD4+ T-Cells. Mathematical Biosciences, 114, 81-125.
https://doi.org/10.1016/0025-5564(93)90043-A

[6]   Herz, A.V.M., Bonhoeffer, S., Anderson,R.M., May,R.M. and Nowak,M.A. (1996) 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, 7247-7251.
https://doi.org/10.1073/pnas.93.14.7247

[7]   Huang, G., Ma, W. and Takeuchi, Y. (2009) Global Properties for Virus Dynamics Model with Beddington-DeAngelis Functional Response. Applied Mathematics Letters, 22, 1690-1693.
https://doi.org/10.1016/j.aml.2009.06.004

[8]   Srivastava, P.K. and Chandra, P. (2010) Modeling the Dynamics of HIV and CD4+ T Cells during Primary Infection. Nonlinear Analysis: Real World Applications, 11, 612-618.
https://doi.org/10.1016/j.nonrwa.2008.10.037

[9]   Guo, S.B. and Ma, W.B. (2016) Global Behavior of Delay Differential Equations Model of HIV Infection with Apoptosis. Discrete and Continuous Dynamical Systems: Series B, 21, 103-119. https://doi.org/10.3934/dcdsb.2016.21.103

[10]   Song, X. and Neumann, A.U. (2007) Global Stability and Periodic Solution of the Viral Dynamics. Journal of Mathematical Analysis and Applications, 329, 281-297.
https://doi.org/10.1016/j.jmaa.2006.06.064

[11]   Huang, G., Yokoi, H., Takeuchi, Y., Kajiwara, T. and Sasaki, T. (2011) Impact of Intracellular Delay, Immune Activation Delay and Nonlinear Incidence on Viral Dynamics. Japan Journal of Industrial and Applied Mathematics, 28, 383-411.
https://doi.org/10.1007/s13160-011-0045-x

[12]   Hirsch, M.W., Smith, H.L. and Zhao, X.-Q. (2001) Chain Transitivity, Attractivity, and Strong Repellors for Semidynamical Systems. Journal of Dynamics and Differential Equations, 13, 107-131. https://doi.org/10.1023/A:1009044515567

[13]   De Boer, R.J. and Perelson, A.S. (1998) Target Cell Limited and Immune Control Models of HIV Infection: A Comparison. Journal of Theoretical Biology, 19, 201-214.
https://doi.org/10.1006/jtbi.1997.0548

[14]   Kepler, T.B. and Perelson, A.S. (1993) Cyclic Re-Entry of Germinal Center B Cells and the Efficiency of Affinity Maturation. Immunology Today, 14, 412-415.
https://doi.org/10.1016/0167-5699(93)90145-B

[15]   Kuang, Y. (1993) Delay Differential Equations with Applications in Population Dynamics. Academic Press, Inc., Boston.

[16]   Engelborghs, K., Luzynina, T. and Roose, D. (2002) Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL. ACM Transactions on Mathematical Software, 28, 1-21. https://doi.org/10.1145/513001.513002

[17]   Beretta, E. and Kuang, Y. (2002) Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependant Parameters. SIAM Journal on Mathematical Analysis, 33, 1144-1165. https://doi.org/10.1137/S0036141000376086

[18]   Ma, S.Q., Lu, Q.S. and Mei, S.L. (2005) Dynamics of a Logistic Population Model with Maturation Delay and Nonlinear Birth Rate. Discrete and Continuous Dynamical Systems: Series B, 5, 736-752. https://doi.org/10.3934/dcdsb.2005.5.735

[19]   Ma, S.Q. (2018) Application of Extended Geometrical Criterion to Population Model with Two Time Delays. Journal of Mathematical Researches, 10, 63-76.
https://doi.org/10.5539/jmr.v10n3p63

[20]   Ma, S.Q., Feng, Z.S. and Lu, Q.S. (2008) A Two-Parameter Geometrical Criteria for Delay Differential Equations. Discrete and Continuous Dynamical Systems: Series B, 9, 397-413.
https://doi.org/10.3934/dcdsb.2008.9.397

[21]   Hale, J.K. and Verduyn Lunel, S.M. (1993) Introduction to Functional Differential Equations. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4612-4342-7_1

[22]   Song, Z.-G. and Xu, J. (2013) Stability Switches and Double Hopf Bifurcation in a Two Neural Network System with Multiple Delays. Cognitive Neurodynamics, 7, 505-521.
https://doi.org/10.1007/s11571-013-9254-0

 
 
Top