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Stochastic modeling of the dynamics of \(CD4^ +\) \(T\)-cell infection by HIV and some Monte Carlo studies. (English) Zbl 0887.92021
Summary: We develop a stochastic model for the interaction between \(\text{CD}4^+ \text{T}\) cells and the human immunodeficiency virus (HIV) by taking into account the basic biological mechanism as described, e.g., by A. S. Perelson et al. [Math. Biosci. 114, No. 1, 81-125 (1993; Zbl 0796.92016)], D. Schenzle [Stat. Med. 13, 2067-2079 (1994)]. We studied this stochastic model through extensive Monte Carlo simulations. Our results show that, in some cases, there is a positive probability that the virus will be eliminated by the process. We have also shown that, at the earlier stage of the infection, the probability distributions of the \(\text{CD}4^+ \text{T}\) cells and free HIV are skewed; however, these distributions will eventually converge to the Gaussian distributions after several years. A real-data example is given to illustrate the application of our model.

MSC:
92C50 Medical applications (general)
92C60 Medical epidemiology
65C05 Monte Carlo methods
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[1] Perelson, A.S.; Kirschner, D.E.; Boer, R.D., Dynamics of HIV infection of CD4^{+} T cells, Math. biosci., 114, 81-125, (1993) · Zbl 0796.92016
[2] Schenzle, D., A model for AIDS pathogenesis, Stat. med., 13, 2067-2079, (1994)
[3] Kirschner, D.E.; Perelson, A.S., A model for the immune system response to HIV: AZT treatment studies, (), 295-310
[4] Phillips, A.N., Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271, 497-499, (1996)
[5] Wu, H.; Tan, W.Y., Modeling and monitoring the progression of HIV infection using nonlinear Kalman filter, (), 4-8
[6] Merrill, S.J., Modeling the interaction of HIV with cells of the immune system, Lect. notes biomath., 83, 371-385, (1989)
[7] Bolognesi, D.P., Do antibodies enhance the infection of cells by HIV?, Nature, 340, 431-432, (1989)
[8] Tan, W.Y.; Piantadosi, S., On stochastic growth process with application to stochastic logistic growth, Stat. sin., 1, 527-540, (1991) · Zbl 0822.60078
[9] Ho, D.D.; Neumann, A.U.; Perelson, A.S.; Chen, W.; Leonard, J.M.; Markowitz, M., Rapid turnover of plasma virus and CD4 lymphocytes in HIV-1 infection, Nature, 373, 123-126, (1995)
[10] 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, 1582-1586, (1996)
[11] Wei, X.; Ghosh, S.K.; Taylor, M.E.; Johnson, V.A.; Emini, E.A.; Deutsch, P.; Lifson, J.D.; Bonhoeffer, S.; Nowak, M.A.; Hahn, B.H.; Saag, M.S.; Shaw, G.M., Viral dynamics in human immunodeficiency virus type 1 infection, Nature, 373, 117-122, (1995)
[12] Tan, W.Y.; Ye, Z., Assessment of effects of different types of HIV-1 and macrophage by a stochastic model of HIV pathogenesis in HIV-infected individuals, (1997), Department of Mathematical Sciences, the University of Memphis Memphis, TN, Paper in preparation
[13] Tan, W.Y.; Hsu, H., Some stochastic models of AIDS spread, Stat. med., 8, 121-136, (1989)
[14] IMSL, MATH/LIBRARY User’s manual, (1989), IMSL Houston, Texas
[15] Deacon, N.J.; Tsykin, A.; Soloman, A.; Smith, K.; Ludford-Menting, M.; Hooker, D.J.; McPhee, D.A.; Greenway, A.L.; Ellett, A.; Chatfield, C.; Lawson, V.A.; Crowe, S.; Moerz, A.; Sonza, S.; Learmont, J.; Sullivan, J.S.; Cunningham, A.; Dwyer, D.; Dowton, D.; Mills, J., Genomic structure of an attenuated quasi species of HIV-1 from a blood transfusion donor and recipients, Science, 270, 988-991, (1995)
[16] Ratkowsky, D.A., Nonlinear regression modeling, (1983), Marcel Dekker New York and Basel · Zbl 0572.62054
[17] Perelson, A.S., Modeling the interaction of the immune system with HIV, (), 350-370
[18] Ioannidis, J.P.A.; Cappelleri, J.C.; Lau, J.; Sacks, H.S.; Skolnik, P.R., Predictive value of viral load measurements in asymptomatic untreated HIV-1 infection: a mathematical model, Aids, 10, 3, 255-262, (1996)
[19] Lange, N.; Carlin, B.P.; Gelfand, A.E., Hierarchical Bayes models for the progression of HIV infection using longitudinal CD4 T-cell numbers (with discussion), J. am. stat. assoc., 87, 419, 615-632, (1992) · Zbl 0850.62838
[20] Kiuchi, A.S.; Hartigan, J.A.; Holford, T.R.; Rubinstein, P.; Stevens, C.E., Change points in the series of T4 counts prior to AIDS, Biometrics, 51, 236-248, (1995) · Zbl 0825.62800
[21] Mellors, J.W.; Rinaldo, C.R.; Gupta, P.; White, R.M.; Todd, J.A.; Kingsley, L.A., Prognosis in HIV-1 infection predicted by the quantity of virus inplasma, Science, 272, 1167-1170, (1996)
[22] Struchiner, C.J.; Halloran, M.E., Modeling AIDS vaccines: the celluar level, Mem. inst. oswaldo cruz Rio de Janeiro, 87, 1, 103-113, (1992)
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