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A delay-differential equation model of HIV infection of \(\text{CD}4^+\) T-cells. (English) Zbl 0981.92009

Summary: A.S. Perelson, D.E. Kirschner and R. De Boer [Math. Biosci. 114, No. 1, 81-125 (1993; Zbl 0796.92016)] proposed an ODE model of cell-free viral spread of human immunodeficiency virus (HIV) in a well-mixed compartment such as the bloodstream. Their model consists of four components: uninfected healthy \(\text{CD4}^+\) T-cells, latently infected \(\text{CD4}^+\) T-cells, actively infected \(\text{CD4}^+\) T-cells, and free virus. This model has been important in the field of mathematical modeling of HIV infection and many other models have been proposed which take the model of Perelson, Kirschner and De Boer as their inspiration, so to speak.
We first simplify their model into one consisting of only three components: the healthy \(\text{CD4}^+\) T-cells, infected \(\text{CD4}^+\) T-cells, and free virus and discuss the existence and stability of the infected steady state. Then, we introduce a discrete time delay to the model to describe the time between infection of a \(\text{CD4}^+\) T-cell and the emission of viral particles on a cellular level. We study the effect of the time delay on the stability of the endemically infected equilibrium, and criteria are given to ensure that the infected equilibrium is asymptotically stable for all delays. Numerical simulations are presented to illustrate the results.

MSC:

92C50 Medical applications (general)
34K20 Stability theory of functional-differential equations
92C60 Medical epidemiology
34K25 Asymptotic theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations

Citations:

Zbl 0796.92016
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References:

[1] Anderson, R. M., Mathematical and statistical studies of the epidemiology of HIV, AIDS, 4, 107 (1990)
[2] R.M. Anderson, R.M. May, Complex dynamical behavior in the interaction between HIV and the immune system, in: A. Goldbeter ( Ed.), Cell to Cell Signalling: From Experiments to Theoretical Models, Academic Press, New York, 1989, p. 335; R.M. Anderson, R.M. May, Complex dynamical behavior in the interaction between HIV and the immune system, in: A. Goldbeter ( Ed.), Cell to Cell Signalling: From Experiments to Theoretical Models, Academic Press, New York, 1989, p. 335
[3] Bailey, J. J.; Fletcher, J. E.; Chuck, E. T.; Shrager, R. I., A kinetic model of CD \(4^+\) lymphocytes with the human immunodeficiency virus (HIV), BioSystems, 26, 177 (1992)
[4] Bonhoeffer, S.; May, R. M.; Shaw, G. M.; Nowak, M. A., Virus dynamics and drug therapy, Proc. Nat. Acad. Sci. USA, 94, 6971 (1997)
[5] De Boer, R. J.; Perelson, A. S., Target cell limited and immune control models of HIV infection: a comparison, J. Theor. Biol., 190, 201 (1998)
[6] Hraba, T.; Doležal, J.; Čelikovsky, S., Model-based analysis of CD \(4^+\) lymphocyte dynamics in HIV infected individuals, Immunobiology, 181, 108 (1990)
[7] N. Intrator, G.P. Deocampo, L.N. Cooper, Analysis of immune system retrovirus equations, in: A.S. Perelson (Ed.), Theoretical Immunology , II, Addison-Wesley, Redwood City, CA, 1988, p. 85; N. Intrator, G.P. Deocampo, L.N. Cooper, Analysis of immune system retrovirus equations, in: A.S. Perelson (Ed.), Theoretical Immunology , II, Addison-Wesley, Redwood City, CA, 1988, p. 85
[8] D.E. Kirschner, A.S. Perelson, A model for the immune system response to HIV: AZT treatment studies, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, vol. 1, Theory of Epidemics, Wuerz, Winnipeg, Canada, 1995, p. 295; D.E. Kirschner, A.S. Perelson, A model for the immune system response to HIV: AZT treatment studies, in: O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, vol. 1, Theory of Epidemics, Wuerz, Winnipeg, Canada, 1995, p. 295
[9] Kirschner, D. E.; Lenhart, S.; Serbin, S., Optimal control of the chemotherapy of HIV, J. Math. Biol., 35, 775 (1997) · Zbl 0876.92016
[10] Kirschner, D. E.; Webb, G. F., A model for the treatment strategy in the chemotherapy of AIDS, Bull. Math. Biol., 58, 367 (1996) · Zbl 0853.92009
[11] Kirschner, D. E.; Webb, G. F., Understanding drug resistance for monotherapy treatment of HIV infection, Bull. Math. Biol., 59, 763 (1997) · Zbl 0922.92011
[12] McLean, A. R.; Kirkwood, T. B.L., A model of human immunodeficiency virus infection in T-helper cell clones, J. Theor. Biol., 147, 177 (1990)
[13] McLean, A. R.; Nowak, M. A., Models of interaction between HIV and other pathogens, J. Theor. Biol., 155, 69 (1992)
[14] Nowak, M. A.; Bangham, R. M., Population dynamics of immune responses to persistent viruses, Science, 272, 74 (1996)
[15] Nowak, M. A.; May, R. M., Mathematical biology of HIV infection: antigenic variation and diversity threshold, Math. Biosci., 106, 1 (1991) · Zbl 0738.92008
[16] A.S. Perelson, Modelling the interaction of the immune system with HIV, in: C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, 1989, p. 350; A.S. Perelson, Modelling the interaction of the immune system with HIV, in: C. Castillo-Chavez (Ed.), Mathematical and Statistical Approaches to AIDS Epidemiology, Springer, Berlin, 1989, p. 350 · Zbl 0683.92001
[17] Perelson, A. S.; Kirschner, D. E.; De Boer, R., Dynamics of HIV Infection of CD \(4^+\) T-cells, Math. Biosci., 114, 81 (1993) · Zbl 0796.92016
[18] A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271 1996, p. 1582; A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time, Science, 271 1996, p. 1582
[19] Root-Bernstein, R. S.; Merrill, S. J., The necessity of cofactors in the pathogenesis of AIDS: a mathematical model, J. Theor. Biol., 187, 135 (1997)
[20] Spouge, J. L.; Shrager, R. I.; Dimitrov, D. S., HIV-1 infection kinetics in tissue culture, Math. Biosci., 138, 1 (1996) · Zbl 0873.92023
[21] Stilianakis, N. I.; Dietz, K.; Schenzle, D., Analysis of a model for the pathogenesis of AIDS, Math. Biosci., 145, 27 (1997) · Zbl 0896.92016
[22] Kirschner, D. E., Using mathematics to understand HIV immune dynamics, Notices Am. Math. Soc., 43, 191 (1996) · Zbl 1044.92503
[23] Perelson, A. S.; Nelson, P. W., Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41, 3 (1999) · Zbl 1078.92502
[24] Busenberg, S.; Cooke, K., Vertically Transmitted Diseases (1993), Springer: Springer Berlin · Zbl 0837.92021
[25] Cushing, J. M., Integrodifferential Equations and Delay Models in Population Dynamics (1977), Spring: Spring Heidelberg · Zbl 0363.92014
[26] Gopalsamy, K., Stability and Oscillations in Delay-Differential Equations of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039
[27] Kuang, Y., Delay-Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002
[28] MacDonald, N., Time Delays in Biological Models (1978), Spring: Spring Heidelberg
[29] G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman, UK, 1989; G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman, UK, 1989
[30] 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, Proc. Nat. Acad. Sci. USA, 93, 7247 (1996)
[31] Mittler, J. E.; Sulzer, B.; Neumann, A. U.; Perelson, A. S., Influence of delayed viral production on viral dynamics in HIV-1 infected patients, Math. Biosci., 152, 143 (1998) · Zbl 0946.92011
[32] Mittler, J. E.; Markowitz, M.; Ho, D. D.; Perelson, A. S., Improved estimates for HIV-1 clearance rate and intracellular delay, AIDS, 13, 1415 (1999)
[33] Tam, J., Delay effect in a model for virus replication, IMA J. Math. Appl. Med. Biol., 16, 29 (1999) · Zbl 0914.92012
[34] Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York · Zbl 0118.08201
[35] R.V. Culshaw, Mathematical Models of Cell-to-Cell and Cell-Free Viral Spread of HIV Infection, MSc Thesis, Dalhousie University, Halifax, Canada, 1997; R.V. Culshaw, Mathematical Models of Cell-to-Cell and Cell-Free Viral Spread of HIV Infection, MSc Thesis, Dalhousie University, Halifax, Canada, 1997
[36] Dieudonné, J., Foundations of Modern Analysis (1960), Academic Press: Academic Press New York · Zbl 0100.04201
[37] Haase, A. T., Quantitative image analysis of HIV-1 infection in lymphoid tissue, Science, 274, 985 (1996)
[38] Cavert, W., Kinetics of response in lymphoid tissues to antiretroviral therapy of HIV-1 infection, Science, 276, 960 (1997)
[39] Hockett, R. D., Constant mean viral copy number per infected cell in tissues regardless of high, low, or undetectable plasma HIV RNA, J. Exp. Med., 189, 1545 (1999)
[40] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge University: Cambridge University Cambridge · Zbl 0474.34002
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