# American Institute of Mathematical Sciences

2013, 2013(special): 311-322. doi: 10.3934/proc.2013.2013.311

## An optimal control problem in HIV treatment

 1 Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204 2 Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992 3 Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona

Received  September 2012 Revised  February 2013 Published  November 2013

We consider a three-dimensional nonlinear control model, which describes the dynamics of HIV infection with nonlytic immune response and possible effects of controllable medication intake on HIV-infected patients. This model has the following phase variables: populations of the infected and uninfected cells and the concentration of an antiviral drug. The medication intake rate is chosen to be a bounded control function. The optimal control problem of minimizing the infected cells population at the terminal time is stated and solved. The types of the optimal control for different model parameters are obtained analytically. This allowed us to reduce the two-point boundary value problem for the Pontryagin Maximum Principle to one of the finite dimensional optimization. Numerical results are presented to demonstrate the optimal solution.
Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
##### References:
 [1] 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, 1, N 2, (2004), 223-241. [2] S. Anita, V. Arnaŭtu and V. Capasso, Introduction to Optimal Control Problems in Life Sciences and Economics, Birkhäuser, USA, 2011. [3] S. Butler, D. Kirschner and S. Lenhart, Optimal Control of the Chemotherapy Affecting the Infectivity of HIV, in Advances in Mathematical Population Dynamics - Molecules, Cells and Man, 6, (eds. O. Arino, D. Axelrod and M. Kimmel), World Scientific, Singapore, 1997, 557-569. [4] R.V. Culshaw, S. Ruan and R.J. Spiteri, Optimal HIV treatment by maximising immune response, Journal of Mathematical Biology, 48, N 5, (2004), 545-562. [5] A.V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM Journal on Control and Optimization, 30, N 5, (1992), 1087-1091. [6] K.R. Fister, S. Lenhart and J.S. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 1998, N 32, (1998), 1-12. [7] E.V. Grigorieva and E.N. Khailov, Attainable set of a nonlinear controlled microeconomic model, Journal of Dynamical and Control Systems, 11, N 2, (2005), 157-176. [8] E.V. Grigorieva, N.V. Bondarenko, E.N. Khailov and A. Korobeinikov, Three-dimensional nonlinear control model of wastewater biotreatment, Neural, Parallel, and Scientific Computations, 20, (2012), 23-36. [9] E.V. Grigorieva and E.N. Khailov, Attainable Set of a Nonlinear Controlled System Describing the Process of Production and Sales of a Consumer Good, in Problems of Dynamical Control, issue 1, (eds. Yu.S. Osipov and A.V. Kryazhimskii), MAX Press, Moscow, 2005, 312-331. [10] E.V. Grigorieva, N.V. Bondarenko, E.N. Khailov and A. Korobeinikov, Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion, in Industrial Waste, (eds. K.-Y. Show and X. Guo), InTech, Croatia, 2012, 91-120. [11] G. Huang, Y. Takeuchi and A. Korobeinikov, HIV evolution and progression of the infection to AIDS, Journal of Theoretical Biology, 307, N 5, (2012), 149-159. [12] S. Iwami, T. Miura, S. Nakaoka and Y. Takeuchi, Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds, Journal of Theoretical Biology, 260, N 4, (2009), 490-501. [13] H.R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23, (2002), 199-213. [14] D. Kirschner, S. Lenhart and S. Serbin, Optimizing chemotherapy of HIV infection: scheduling, ammounts and initiation of treatment, Journal of Mathematical Biology, 35, (1997), 775-792. [15] J.J. Kutch and P. Gurfil, Optimal control of HIV infection with a continuously-mutating viral population, in Proceedings of American Control Conference, Anchorage, Alaska, (2002), 4033-4038. [16] U. Ledzewicz and H. Schättler, On optimal controls for a general mathematical model for chemotherapy of HIV, in Proceedings of American Control Conference, Anchorage, Alaska, (2002), 3454-3459. [17] E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [18] S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models, CRC Press, Taylor & Francis Group, London, 2007. [19] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [20] G. Sansone, Equazioni Differenziali nel Campo Reale, Parte Prima, Nicola Zanichelli, Bologna, 1948. [21] E. Shudo and Y. Iwasa, Dynamic optimization of host defence, immune memory, and post-infection pathogen levels in mammals, Journal of Theoretical Biology, 228, (2004), 17-29. [22] M.A. Stafford, L. Corey, Y. Cao, E.S. Daar, D.D. Ho and A.S. Perelson, Modeling plasma virus concentration during primary HIV infection, Journal of Theoretical Biology, 203, N 3, (2000), 285-301. [23] R.F. Stengel, R. Ghigliazza, N. Rulkarni and O. Laplace, Optimal control of innate immune response, Optimal Control Applications & Methods, 23, (2002), 91-104. [24] C. Vargas-De-Leon and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Mathematical Medicine and Biology, doi: 10.1093/imammb/dqr027, (2011), 1-8. [25] F.P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [26] V.V. Velichenko and D.A. Pritykin, Control of the medical treatment of AIDS, Automation and Remote Control, 67, N 3, (2006), 493-511. [27] D. Wodarz, J. Christensen and A. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends in Immunology, 23, N 4, (2002), 194-210. [28] D. Wodarz and M.A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24, (2002), 1178-1187. [29] H.G. Zadeh, H.C. Nejad, M.M. Abadi and H.M. Sani, A new fast optimal control for HIV-infection dynamics based on AVK method and fuzzy estimator, American Journal of Scientific Research, 32, (2011), 11-16.

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##### References:
 [1] 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, 1, N 2, (2004), 223-241. [2] S. Anita, V. Arnaŭtu and V. Capasso, Introduction to Optimal Control Problems in Life Sciences and Economics, Birkhäuser, USA, 2011. [3] S. Butler, D. Kirschner and S. Lenhart, Optimal Control of the Chemotherapy Affecting the Infectivity of HIV, in Advances in Mathematical Population Dynamics - Molecules, Cells and Man, 6, (eds. O. Arino, D. Axelrod and M. Kimmel), World Scientific, Singapore, 1997, 557-569. [4] R.V. Culshaw, S. Ruan and R.J. Spiteri, Optimal HIV treatment by maximising immune response, Journal of Mathematical Biology, 48, N 5, (2004), 545-562. [5] A.V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM Journal on Control and Optimization, 30, N 5, (1992), 1087-1091. [6] K.R. Fister, S. Lenhart and J.S. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 1998, N 32, (1998), 1-12. [7] E.V. Grigorieva and E.N. Khailov, Attainable set of a nonlinear controlled microeconomic model, Journal of Dynamical and Control Systems, 11, N 2, (2005), 157-176. [8] E.V. Grigorieva, N.V. Bondarenko, E.N. Khailov and A. Korobeinikov, Three-dimensional nonlinear control model of wastewater biotreatment, Neural, Parallel, and Scientific Computations, 20, (2012), 23-36. [9] E.V. Grigorieva and E.N. Khailov, Attainable Set of a Nonlinear Controlled System Describing the Process of Production and Sales of a Consumer Good, in Problems of Dynamical Control, issue 1, (eds. Yu.S. Osipov and A.V. Kryazhimskii), MAX Press, Moscow, 2005, 312-331. [10] E.V. Grigorieva, N.V. Bondarenko, E.N. Khailov and A. Korobeinikov, Finite-Dimensional Methods for Optimal Control of Autothermal Thermophilic Aerobic Digestion, in Industrial Waste, (eds. K.-Y. Show and X. Guo), InTech, Croatia, 2012, 91-120. [11] G. Huang, Y. Takeuchi and A. Korobeinikov, HIV evolution and progression of the infection to AIDS, Journal of Theoretical Biology, 307, N 5, (2012), 149-159. [12] S. Iwami, T. Miura, S. Nakaoka and Y. Takeuchi, Immune impairment in HIV infection: existence of risky and immunodeficiency thresholds, Journal of Theoretical Biology, 260, N 4, (2009), 490-501. [13] H.R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications & Methods, 23, (2002), 199-213. [14] D. Kirschner, S. Lenhart and S. Serbin, Optimizing chemotherapy of HIV infection: scheduling, ammounts and initiation of treatment, Journal of Mathematical Biology, 35, (1997), 775-792. [15] J.J. Kutch and P. Gurfil, Optimal control of HIV infection with a continuously-mutating viral population, in Proceedings of American Control Conference, Anchorage, Alaska, (2002), 4033-4038. [16] U. Ledzewicz and H. Schättler, On optimal controls for a general mathematical model for chemotherapy of HIV, in Proceedings of American Control Conference, Anchorage, Alaska, (2002), 3454-3459. [17] E.B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [18] S. Lenhart and J.T. Workman, Optimal Control Applied to Biological Models, CRC Press, Taylor & Francis Group, London, 2007. [19] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. [20] G. Sansone, Equazioni Differenziali nel Campo Reale, Parte Prima, Nicola Zanichelli, Bologna, 1948. [21] E. Shudo and Y. Iwasa, Dynamic optimization of host defence, immune memory, and post-infection pathogen levels in mammals, Journal of Theoretical Biology, 228, (2004), 17-29. [22] M.A. Stafford, L. Corey, Y. Cao, E.S. Daar, D.D. Ho and A.S. Perelson, Modeling plasma virus concentration during primary HIV infection, Journal of Theoretical Biology, 203, N 3, (2000), 285-301. [23] R.F. Stengel, R. Ghigliazza, N. Rulkarni and O. Laplace, Optimal control of innate immune response, Optimal Control Applications & Methods, 23, (2002), 91-104. [24] C. Vargas-De-Leon and A. Korobeinikov, Global stability of a population dynamics model with inhibition and negative feedback, Mathematical Medicine and Biology, doi: 10.1093/imammb/dqr027, (2011), 1-8. [25] F.P. Vasil'ev, Optimization Methods, Factorial Press, Moscow, 2002. [26] V.V. Velichenko and D.A. Pritykin, Control of the medical treatment of AIDS, Automation and Remote Control, 67, N 3, (2006), 493-511. [27] D. Wodarz, J. Christensen and A. Thomsen, The importance of lytic and nonlytic immune responses in viral infections, Trends in Immunology, 23, N 4, (2002), 194-210. [28] D. Wodarz and M.A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24, (2002), 1178-1187. [29] H.G. Zadeh, H.C. Nejad, M.M. Abadi and H.M. Sani, A new fast optimal control for HIV-infection dynamics based on AVK method and fuzzy estimator, American Journal of Scientific Research, 32, (2011), 11-16.
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