Maximizing of Asymptomatic Stage of Fast Progressive HIV Infected Patient Using Embedding Method

ABSTRACT

A system of ordinary differential equations, which describe various aspects of the interaction of HIV with healthy cells in fast progressive patient, is utilized, and an optimal control problem is constructed to prolong survival and delay the progression to AIDS as far as possible, subject to drug costs. Optimal control problem is approximated by linear programming model using measure theoretical approach and suboptimal combinations of reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) drug efficacies are proposed. The Comparison of healthy CD4+ Tcells counts, virus particles and immune response, before and after the treatment is introduced.

A system of ordinary differential equations, which describe various aspects of the interaction of HIV with healthy cells in fast progressive patient, is utilized, and an optimal control problem is constructed to prolong survival and delay the progression to AIDS as far as possible, subject to drug costs. Optimal control problem is approximated by linear programming model using measure theoretical approach and suboptimal combinations of reverse transcriptase inhibitor (RTI) and protease inhibitor (PI) drug efficacies are proposed. The Comparison of healthy CD4+ Tcells counts, virus particles and immune response, before and after the treatment is introduced.

Cite this paper

nullH. Zarei, A. Kamyad and S. Effati, "Maximizing of Asymptomatic Stage of Fast Progressive HIV Infected Patient Using Embedding Method,"*Intelligent Control and Automation*, Vol. 1 No. 1, 2010, pp. 48-58. doi: 10.4236/ica.2010.11006.

nullH. Zarei, A. Kamyad and S. Effati, "Maximizing of Asymptomatic Stage of Fast Progressive HIV Infected Patient Using Embedding Method,"

References

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[10] W. Garira, D. S. Musekwa and T. Shiri, “Optimal Control of Combined Therapy in a Single Strain HIV1 Model,” Electronic Journal of Differential Equations, Vol. 2005, No. 52, 2005, pp. 122.

[11] J. Karrakchou, M. Rachik and S. Gourari, “Optimal Control and Infectiology: Application to an HIV/AIDS Model,” Journal of Applied Mathematics and Computing, Vol. 177, No. 2, 2006, pp. 807818.

[12] A. Heydari, M. H. Farahi and A. A. Heydari, “Chemotherapy in an HIV Model by a Pair of Optimal Control,” Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, 2007, pp. 5863.

[13] B. M. Adams, H. T. Banks, H. D. Kwon and H. T. Tran, “Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches,” Mathematical Biosciences and Engineering, Vol. 1, No. 2, 2004, pp. 223241.

[14] F. Neri, J. Toivanen and R. A. E. M?kinen, “An Adaptive Evolutionary Algorithm with Intelligent Mutation Local Searchers for Designing Multidrug Therapies for HIV,” Applied Intelligence, Vol. 27, No. 3, 2007, pp. 219235.

[15] R. Culshaw, S. Ruan and R. J. Spiteri, “Optimal HIV Treatment by Maximizing Immune Response,” Journal of Mathematical Biology, Vol. 48, No. 5, 2004, pp. 545562.

[16] O. Krakovska and L. M. Wahl, “Costs Versus Benefits: Best Possible and Best Practical Treatment Regimens for HIV,” Journal of Mathematical Biology, Vol. 54, No. 3, 2007, pp. 385406.

[17] C. D. Myburgh and K. H. Wong, “An Optimal Control Approach to Therapeutic Intervention in HIV Infected Individuals,” Proceedings of the Third International Conference on Control Theory and Applications, Pretoria, South Africa, 2001.

[18] B. M. Adams, H. T. Banks, M. Davidian, H. D. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, “HIV Dynamics: Modeling, Data Analysis, and Optimal Treatment Protocols,” Journal of Computational and Applied Math ematics, Vol. 184, No. 1, 2005, pp. 1049.

[19] J. AlvarezRamirez, M. Meraz and J. X. VelascoHernandez, “Feedback Control of the Chemotherapy of HIV,” International Journal of Bifurcation and Chaos, Vol. 10, No. 9, 2000, pp. 22072219.

[20] S. Butler, D. Kirschner and S. Lenhart, “Optimal Control of Chemotherapy Affecting the Infectivity of HIV,” In: O. Arino, D. Axelrod, M. Kimmel and M. Langlais, Eds., Advances in Mathematical Population Dynamics: Molecules, Cells, Man, World Scientific, Singapore, pp.104 120.

[21] H. Shim, S. J. Han, C. C. Chung, S. Nam and J. H. Seo, “Optimal Scheduling of Drug Treatment for HIV Infection: Continuous dose Control and Receding Horizon Control,” International Journal of Control, Automation, and Systems, Vol. 1, No. 3, 2003, pp. 401407.

[22] D. Kirschner, S. Lenhart and S. Serbin, “Optimal Control of the Chemotherapy of HIV Infection: Scheduling, Amounts and Initiation of Treatment,” Journal of Mathematical Biology, Vol. 35, No. 7, 1997, pp. 775792.

[23] U. Ledzewicz and H. Sch?ttler, “On Optimal Controls for a General Mathematical Model for Chemotherapy of HIV,” Proceedings of the 2002 American Control Conference, Anchorage, Vol. 5, 2002, pp. 34543459.

[24] R. Zurakowski, M. J. Messina, S. E. Tuna and A. R. Teel, “HIV Treatment Scheduling Via Robust Nonlinear Model Predictive Control,” Proceedings of the 5th Asian Control Conference, Melbourne, Vol. 1, 2004, pp. 2532.

[25] R. Zurakowski and A. R. Teel, “Enhancing Immune Response to HIV Infection Using MPCBased Treatment Scheduling,” Proceedings of the 2003 American Control Conference, Denver, Vol. 2, 2003, pp. 11821187.

[26] R. Zurakowski, A. R. Teel and D. Wodarz, “Utilizing Alternate Target Cells in Treating HIV Infection through Scheduled Treatment Interruptions,” Proceedings of the 2004 American Control Conference, Melbourne, Vol. 1, 2004, pp. 946951.

[27] R. Zurakowski and A. R. Teel, “A Model Predictive Control Based Scheduling Method for HIV Therapy,” Journal of Theoretical Biology, Vol. 238, No. 2, 2006, pp. 368382.

[28] H. T. Banks, H. D. Kwon, J. Toivanen and H. T. Tran, “An State Dependent Riccati Equation Based Estimator Approach for HIV Feedback Control,” Optimal Control Applications and Methods, Vol. 27, No. 2, 2006, pp. 93 121.

[29] M. A. L. Caetano and T. Yoneyama, “Short and Long Period Optimization of Drug Doses in the Treatment of AIDS,” Anais da Academia Brasileira de Ciências, Vol. 74, No. 3, 2002, pp. 379392.

[30] A. M. Jeffrey, X. Xia and I. K. Craig, “When to Initiate HIV Therapy: A Control Theoretic Approach,” IEEE Transactions on Biomedical Engineering, Vol. 50, No. 11, 2003, pp. 12131220.

[31] J. J. Kutch, P. Gurfil, “Optimal Control of HIV Infection with a ContinuouslyMutating Viral Population,” Proceedings of the 2002 American Control Conference, Anchorage, Vol. 5, 2002, pp. 40334038.

[32] P. Borrow, A. Tishon, S. Lee, J. Xu, I. S. Grewal, M. B. Oldstone and R. A. Flavell, “CD40LDeficient Mice Show Deficits in Antiviral Immunity and Have an ImPaired Memory CD8? CTL,” Journal of Experimental Medicine, Vol. 183, No. 5, 1996, pp. 21292142.

[33] C. Young, “Calculus of Variations and Optimal Control Theory,” Sunders, Philadelphia, 1969.

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[35] J. E. Rubio, “Control and Optimization: The Linear Treatment of Nonlinear Problems,” Manchester University Press, Manchester, 1986.

[36] H. H. Mehne, M. H. Farahi and A. V. Kamyad, “MILP Modelling for the Time Optimal Control Problem in The Case of Multiple Targets,” Optimal Control Applications and Methods, Vol. 27, No. 2, 2005, pp. 7791.

[37] X. Jin, et al., “An Antigenic Threshold for Maintaining Human Immunodeficiency Virus Type 1Specific Cytotoxic T Lymphocytes,” Molecular Medicine, Vol. 6, 2000, pp. 803809.

[1] T. W. Chun, L. Stuyver, S. B. Mizell, L. A. Ehler, J. A. Mican, M. Baseler, A. L. Lloyd, M. A. Nowak and A. S. Fauci, “Presence of an Inducible HIV1 Latent Reservoir During Highly Active Antiretroviral Therapy,” Proceedings of the National Academy of Sciences, Vol. 94, No. 24, 1997, pp. 1319313197.

[2] D. Finzi, M. Hermankova, T. Pierson, L. M. Carruth, C. Buck, R. E. Chaisson, T. C. Quinn, K. Chadwick, J. Margolick, R. Brookmeyer, et al., “Identification of a Reservoir for HIV1 in Patients on Highly Active Antiretroviral Therapy,” Science, Vol. 278, No. 5341, 1997, pp. 12951300

[3] J. K. Wong, M. Hezareh, H. F. Gunthard, D. V. Havlir, C. C. Ignacio, C. A. Spina and D. D. Richman, “Recovery of ReplicationCompetent HIV Despite Prolonged Suppression of Plasma Viremia,” Science, Vol. 278, No. 5341, 1997, pp. 12911295.

[4] M. M. Hadjiandreou, R. Conejeros and V. S. Vassiliadis, “Towards a LongTerm Model Construction for the Dynamic Simulation of HIV Infection,” Mathematical Biosciences and Engineering, Vol. 4, No. 3, 2007, pp. 489 504.

[5] A. S. Perelson, A. U. Neumann, M. Markowitz, et al., “HIV1 Dynamics in Vivo: Virion Clearance Rate, Infected Cell LifeSpan, and Viral Generation Time,” Science, Vol. 271, No. 5255, 1996, pp. 15821586.

[6] D. Wodarz and M. A. Nowak, “Specific Therapy Regimes Could Lead to LongTerm Immunological Control of HIV,” Proceedings of the National Academy of Sciences, Vol. 96, No. 6, 1999, pp. 1446414469.

[7] A. Landi, A. Mazzoldi, C. Andreoni, M. Bianchi, A. Cavallini, M. Laurino, L. Ricotti, R. Iuliano, B. Matteoli and L. CeccheriniNelli, “Modelling and Control of HIV Dynamics,” Computer Methods and Programs in Biomedicine, Vol. 89, No. 2, 2008, pp. 162168.

[8] K. R. Fister, S. Lenhart and J. S. McNally, “Optimizing Chemotherapy in an HIV Model,” Electronic Journal of Differential Equations, Vol. 1998, No. 32, 1998, pp. 1 12.

[9] M. M. Hadjiandreou, R. Conejeros and D. I. Wilson, “LongTerm HIV Dynamics Subject to Continuous The rapy and Structured Treatment Interruptions,” Chemical Engineering Science, Vol. 64, No. 7, 2009, pp. 1600 1617.

[10] W. Garira, D. S. Musekwa and T. Shiri, “Optimal Control of Combined Therapy in a Single Strain HIV1 Model,” Electronic Journal of Differential Equations, Vol. 2005, No. 52, 2005, pp. 122.

[11] J. Karrakchou, M. Rachik and S. Gourari, “Optimal Control and Infectiology: Application to an HIV/AIDS Model,” Journal of Applied Mathematics and Computing, Vol. 177, No. 2, 2006, pp. 807818.

[12] A. Heydari, M. H. Farahi and A. A. Heydari, “Chemotherapy in an HIV Model by a Pair of Optimal Control,” Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, 2007, pp. 5863.

[13] B. M. Adams, H. T. Banks, H. D. Kwon and H. T. Tran, “Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches,” Mathematical Biosciences and Engineering, Vol. 1, No. 2, 2004, pp. 223241.

[14] F. Neri, J. Toivanen and R. A. E. M?kinen, “An Adaptive Evolutionary Algorithm with Intelligent Mutation Local Searchers for Designing Multidrug Therapies for HIV,” Applied Intelligence, Vol. 27, No. 3, 2007, pp. 219235.

[15] R. Culshaw, S. Ruan and R. J. Spiteri, “Optimal HIV Treatment by Maximizing Immune Response,” Journal of Mathematical Biology, Vol. 48, No. 5, 2004, pp. 545562.

[16] O. Krakovska and L. M. Wahl, “Costs Versus Benefits: Best Possible and Best Practical Treatment Regimens for HIV,” Journal of Mathematical Biology, Vol. 54, No. 3, 2007, pp. 385406.

[17] C. D. Myburgh and K. H. Wong, “An Optimal Control Approach to Therapeutic Intervention in HIV Infected Individuals,” Proceedings of the Third International Conference on Control Theory and Applications, Pretoria, South Africa, 2001.

[18] B. M. Adams, H. T. Banks, M. Davidian, H. D. Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, “HIV Dynamics: Modeling, Data Analysis, and Optimal Treatment Protocols,” Journal of Computational and Applied Math ematics, Vol. 184, No. 1, 2005, pp. 1049.

[19] J. AlvarezRamirez, M. Meraz and J. X. VelascoHernandez, “Feedback Control of the Chemotherapy of HIV,” International Journal of Bifurcation and Chaos, Vol. 10, No. 9, 2000, pp. 22072219.

[20] S. Butler, D. Kirschner and S. Lenhart, “Optimal Control of Chemotherapy Affecting the Infectivity of HIV,” In: O. Arino, D. Axelrod, M. Kimmel and M. Langlais, Eds., Advances in Mathematical Population Dynamics: Molecules, Cells, Man, World Scientific, Singapore, pp.104 120.

[21] H. Shim, S. J. Han, C. C. Chung, S. Nam and J. H. Seo, “Optimal Scheduling of Drug Treatment for HIV Infection: Continuous dose Control and Receding Horizon Control,” International Journal of Control, Automation, and Systems, Vol. 1, No. 3, 2003, pp. 401407.

[22] D. Kirschner, S. Lenhart and S. Serbin, “Optimal Control of the Chemotherapy of HIV Infection: Scheduling, Amounts and Initiation of Treatment,” Journal of Mathematical Biology, Vol. 35, No. 7, 1997, pp. 775792.

[23] U. Ledzewicz and H. Sch?ttler, “On Optimal Controls for a General Mathematical Model for Chemotherapy of HIV,” Proceedings of the 2002 American Control Conference, Anchorage, Vol. 5, 2002, pp. 34543459.

[24] R. Zurakowski, M. J. Messina, S. E. Tuna and A. R. Teel, “HIV Treatment Scheduling Via Robust Nonlinear Model Predictive Control,” Proceedings of the 5th Asian Control Conference, Melbourne, Vol. 1, 2004, pp. 2532.

[25] R. Zurakowski and A. R. Teel, “Enhancing Immune Response to HIV Infection Using MPCBased Treatment Scheduling,” Proceedings of the 2003 American Control Conference, Denver, Vol. 2, 2003, pp. 11821187.

[26] R. Zurakowski, A. R. Teel and D. Wodarz, “Utilizing Alternate Target Cells in Treating HIV Infection through Scheduled Treatment Interruptions,” Proceedings of the 2004 American Control Conference, Melbourne, Vol. 1, 2004, pp. 946951.

[27] R. Zurakowski and A. R. Teel, “A Model Predictive Control Based Scheduling Method for HIV Therapy,” Journal of Theoretical Biology, Vol. 238, No. 2, 2006, pp. 368382.

[28] H. T. Banks, H. D. Kwon, J. Toivanen and H. T. Tran, “An State Dependent Riccati Equation Based Estimator Approach for HIV Feedback Control,” Optimal Control Applications and Methods, Vol. 27, No. 2, 2006, pp. 93 121.

[29] M. A. L. Caetano and T. Yoneyama, “Short and Long Period Optimization of Drug Doses in the Treatment of AIDS,” Anais da Academia Brasileira de Ciências, Vol. 74, No. 3, 2002, pp. 379392.

[30] A. M. Jeffrey, X. Xia and I. K. Craig, “When to Initiate HIV Therapy: A Control Theoretic Approach,” IEEE Transactions on Biomedical Engineering, Vol. 50, No. 11, 2003, pp. 12131220.

[31] J. J. Kutch, P. Gurfil, “Optimal Control of HIV Infection with a ContinuouslyMutating Viral Population,” Proceedings of the 2002 American Control Conference, Anchorage, Vol. 5, 2002, pp. 40334038.

[32] P. Borrow, A. Tishon, S. Lee, J. Xu, I. S. Grewal, M. B. Oldstone and R. A. Flavell, “CD40LDeficient Mice Show Deficits in Antiviral Immunity and Have an ImPaired Memory CD8? CTL,” Journal of Experimental Medicine, Vol. 183, No. 5, 1996, pp. 21292142.

[33] C. Young, “Calculus of Variations and Optimal Control Theory,” Sunders, Philadelphia, 1969.

[34] D. A. Wilson and J. E. Rubio, “Existence of Optimal Controls for the Diffusion Equation,” Journal of Optimization Theory and Applications, Vol. 22, 1977, pp. 91 101.

[35] J. E. Rubio, “Control and Optimization: The Linear Treatment of Nonlinear Problems,” Manchester University Press, Manchester, 1986.

[36] H. H. Mehne, M. H. Farahi and A. V. Kamyad, “MILP Modelling for the Time Optimal Control Problem in The Case of Multiple Targets,” Optimal Control Applications and Methods, Vol. 27, No. 2, 2005, pp. 7791.

[37] X. Jin, et al., “An Antigenic Threshold for Maintaining Human Immunodeficiency Virus Type 1Specific Cytotoxic T Lymphocytes,” Molecular Medicine, Vol. 6, 2000, pp. 803809.