# American Institute of Mathematical Sciences

2014, 19(1): 153-172. doi: 10.3934/dcdsb.2014.19.153

## An age-structured model with immune response of HIV infection: Modeling and optimal control approach

 1 Department of Mathematics, Inha University, 100 Inharo, Nam-gu, Incheon 402-751, South Korea 2 Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea 3 Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea

Received  July 2012 Revised  August 2013 Published  December 2013

This paper develops and analyzes an age-structured model of HIV infection with various compartments, including target cells, infected cells, viral loads and immune effector cells, to provide a better understanding of the interaction between HIV and the immune system. We show that the proposed model has one uninfected steady state and several infected steady states. We conduct a local stability analysis of these steady states by using a generalized Jacobian matrix method in conjunction with the Laplace transform. In addition, we consider various techniques and ideas from optimal control theory to derive optimal therapy protocols by using two types of dynamic treatment methods representing reverse transcriptase inhibitors and protease inhibitors. We derive the necessary conditions (an optimality system) for optimal control functions by considering the first variations of the Lagrangian. Further, we obtain optimal therapy protocols by solving a large optimality system of equations through the use of a difference scheme based on the Runge-Kutta method. The results of numerical simulations indicate that the optimal therapy protocols can facilitate long-term control of HIV through a strong immune response after the discontinuation of the therapy.
Citation: Hee-Dae Kwon, Jeehyun Lee, Myoungho Yoon. An age-structured model with immune response of HIV infection: Modeling and optimal control approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 153-172. doi: 10.3934/dcdsb.2014.19.153
##### References:
 [1] L. M. Abia and J. C. Lopez-Marcos, Runge-Kutta methods for age-structured population models,, Appl. Numer. Math., 17 (1995), 1. doi: 10.1016/0168-9274(95)00010-R. [2] 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,, J. Comput. Appl. Math., 184 (2005), 10. doi: 10.1016/j.cam.2005.02.004. [3] B. M. Adams, H. T. Banks, H. D. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches,, Math. Biosci. Eng., 1 (2004), 223. doi: 10.3934/mbe.2004.1.223. [4] J. Alvarez-Ramirez, M. Meraz and J. X. Velasco-Hernandez, Feedback control of the chemotherapy of HIV,, Int. J. Bifur. Chaos, 10 (2000), 2207. doi: 10.1142/S0218127400001377. [5] S. H. Bajaria, G. Webb and D. E. Kirschner, Predicting differential responses to structured treatment interruptions during HAART,, Bull. Math. Biol., 66 (2004), 1093. doi: 10.1016/j.bulm.2003.11.003. [6] H. T. Banks, T. Jang and H.-D. Kwon, Feedback control of HIV antiviral therapy with long measurement time,, Int. J. Pure Appl. Math., 66 (2011), 461. [7] H. T. Banks, H.-D. Kwon, J. A. Toivanen and H. T. Tran, A state-dependent Riccati equation-based estimator approach for HIV feedback control,, Optimal Control Appl. Methods, 27 (2006), 93. doi: 10.1002/oca.773. [8] M. E. Brandt and G. Chen, Feedback control of a biodynamical model of HIV-1,, IEEE Trans. on Biom. Engrg., 48 (2001), 754. doi: 10.1109/10.930900. [9] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bull, 64 (2002), 29. doi: 10.1006/bulm.2001.0266. [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model,, Electronic J. of Differential Equation, (1998), 1. [11] M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). [12] A. V. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, Proc. Natl. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. [13] T. Jang, H. D. Kwon and J. Lee, Free terminal time optimal control problem of an HIV model based on a conjugate gradient method,, Bull. Math. Biol., 73 (2011), 2408. doi: 10.1007/s11538-011-9630-z. [14] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Math. Biol., 35 (1997), 775. doi: 10.1007/s002850050076. [15] D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS,, Bull. Math. Biol., 58 (1996), 367. doi: 10.1007/BF02458312. [16] H. D. Kwon, J. Lee, and S.-D. Yang, Optimal control of an age-structured model of HIV infection,, Appl. Math. Comput., 219 (2012), 2766. doi: 10.1016/j.amc.2012.09.003. [17] J. Lisziewicz, E. Rosenberg, J. Lieberman, H. Jessen, L. Lopalco, R. Siliciano and F. Lori, Control of HIV despite the discontinuation of antiretroviral therapy,, New England J. Med., 340 (1999), 1683. doi: 10.1056/NEJM199905273402114. [18] M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size,, Math. Biosci., 182 (2003), 1. doi: 10.1016/S0025-5564(02)00184-0. [19] A. R. McLean and S. D. W. Frost, Zidovudine and HIV: Mathematical models of within-host population dynamics,, Reviews in Medical Virology, 5 (1995), 141. doi: 10.1002/rmv.1980050304. [20] F. A. Milner, M. Iannelli and Z. Feng, A two-strain tuberculosis model with age of infection,, SIAM J. Appl. Math., 62 (2002), 1634. doi: 10.1137/S003613990038205X. [21] H. Moore and W. Gu, A mathematical model for treatment-resistant mutations of HIV,, Math. Biosci. Eng., 2 (2005), 363. doi: 10.3934/mbe.2005.2.363. [22] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells,, Math. Biosci. Eng., 1 (2004), 267. doi: 10.3934/mbe.2004.1.267. [23] G. M. Ortiz, D. F. Nixon, A. Trkola, J. Binley, X. Jin, S. Bonhoeffer and M. Markowitz, HIV-1-specific immune responses in subjects who temporarily contain virus replication after discontinuation of highly active antiretroviral therapy,, J. Clin. Invest., 104 (1999), 13. doi: 10.1172/JCI7371. [24] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. [25] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. [26] D. D. Richman, D. Havlir, J. Corbeil, D. Looney, C. Ignacio, S. A. Spector, J. Sullivan, S. Cheeseman, K. Barringer and D. Pauletti, Nevirapine resistance mutations of human immunodeficiency virus type 1 selected during therapy,, J. Virol., 68 (1994), 1660. [27] 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 horixon control,, Int. J. Control Autom. Systems, 1 (2003), 282. [28] T. Shiri, W. Garira and S. D. Musekwa, A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters,, Math. Biosci. Eng., 2 (2005), 811. doi: 10.3934/mbe.2005.2.811. [29] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus condentration during primary HIV infection,, J. Theor. Biol., 203 (2000), 285. doi: 10.1006/jtbi.2000.1076. [30] H. R. Thieme and C. Castillo-Chavez, How may the infection-age-dependent infectivity affect the dynamics of HIV/AIDS,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. [31] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985).

show all references

##### References:
 [1] L. M. Abia and J. C. Lopez-Marcos, Runge-Kutta methods for age-structured population models,, Appl. Numer. Math., 17 (1995), 1. doi: 10.1016/0168-9274(95)00010-R. [2] 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,, J. Comput. Appl. Math., 184 (2005), 10. doi: 10.1016/j.cam.2005.02.004. [3] B. M. Adams, H. T. Banks, H. D. Kwon and H. T. Tran, Dynamic multidrug therapies for HIV: Optimal and STI control approaches,, Math. Biosci. Eng., 1 (2004), 223. doi: 10.3934/mbe.2004.1.223. [4] J. Alvarez-Ramirez, M. Meraz and J. X. Velasco-Hernandez, Feedback control of the chemotherapy of HIV,, Int. J. Bifur. Chaos, 10 (2000), 2207. doi: 10.1142/S0218127400001377. [5] S. H. Bajaria, G. Webb and D. E. Kirschner, Predicting differential responses to structured treatment interruptions during HAART,, Bull. Math. Biol., 66 (2004), 1093. doi: 10.1016/j.bulm.2003.11.003. [6] H. T. Banks, T. Jang and H.-D. Kwon, Feedback control of HIV antiviral therapy with long measurement time,, Int. J. Pure Appl. Math., 66 (2011), 461. [7] H. T. Banks, H.-D. Kwon, J. A. Toivanen and H. T. Tran, A state-dependent Riccati equation-based estimator approach for HIV feedback control,, Optimal Control Appl. Methods, 27 (2006), 93. doi: 10.1002/oca.773. [8] M. E. Brandt and G. Chen, Feedback control of a biodynamical model of HIV-1,, IEEE Trans. on Biom. Engrg., 48 (2001), 754. doi: 10.1109/10.930900. [9] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads,, Bull, 64 (2002), 29. doi: 10.1006/bulm.2001.0266. [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model,, Electronic J. of Differential Equation, (1998), 1. [11] M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). [12] A. V. Herz, S. Bonhoeffer, R. M. Anderson, R. M. May and M. A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay,, Proc. Natl. Acad. Sci., 93 (1996), 7247. doi: 10.1073/pnas.93.14.7247. [13] T. Jang, H. D. Kwon and J. Lee, Free terminal time optimal control problem of an HIV model based on a conjugate gradient method,, Bull. Math. Biol., 73 (2011), 2408. doi: 10.1007/s11538-011-9630-z. [14] D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV,, J. Math. Biol., 35 (1997), 775. doi: 10.1007/s002850050076. [15] D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS,, Bull. Math. Biol., 58 (1996), 367. doi: 10.1007/BF02458312. [16] H. D. Kwon, J. Lee, and S.-D. Yang, Optimal control of an age-structured model of HIV infection,, Appl. Math. Comput., 219 (2012), 2766. doi: 10.1016/j.amc.2012.09.003. [17] J. Lisziewicz, E. Rosenberg, J. Lieberman, H. Jessen, L. Lopalco, R. Siliciano and F. Lori, Control of HIV despite the discontinuation of antiretroviral therapy,, New England J. Med., 340 (1999), 1683. doi: 10.1056/NEJM199905273402114. [18] M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size,, Math. Biosci., 182 (2003), 1. doi: 10.1016/S0025-5564(02)00184-0. [19] A. R. McLean and S. D. W. Frost, Zidovudine and HIV: Mathematical models of within-host population dynamics,, Reviews in Medical Virology, 5 (1995), 141. doi: 10.1002/rmv.1980050304. [20] F. A. Milner, M. Iannelli and Z. Feng, A two-strain tuberculosis model with age of infection,, SIAM J. Appl. Math., 62 (2002), 1634. doi: 10.1137/S003613990038205X. [21] H. Moore and W. Gu, A mathematical model for treatment-resistant mutations of HIV,, Math. Biosci. Eng., 2 (2005), 363. doi: 10.3934/mbe.2005.2.363. [22] P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells,, Math. Biosci. Eng., 1 (2004), 267. doi: 10.3934/mbe.2004.1.267. [23] G. M. Ortiz, D. F. Nixon, A. Trkola, J. Binley, X. Jin, S. Bonhoeffer and M. Markowitz, HIV-1-specific immune responses in subjects who temporarily contain virus replication after discontinuation of highly active antiretroviral therapy,, J. Clin. Invest., 104 (1999), 13. doi: 10.1172/JCI7371. [24] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Research, 65 (2005), 7950. [25] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. [26] D. D. Richman, D. Havlir, J. Corbeil, D. Looney, C. Ignacio, S. A. Spector, J. Sullivan, S. Cheeseman, K. Barringer and D. Pauletti, Nevirapine resistance mutations of human immunodeficiency virus type 1 selected during therapy,, J. Virol., 68 (1994), 1660. [27] 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 horixon control,, Int. J. Control Autom. Systems, 1 (2003), 282. [28] T. Shiri, W. Garira and S. D. Musekwa, A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters,, Math. Biosci. Eng., 2 (2005), 811. doi: 10.3934/mbe.2005.2.811. [29] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus condentration during primary HIV infection,, J. Theor. Biol., 203 (2000), 285. doi: 10.1006/jtbi.2000.1076. [30] H. R. Thieme and C. Castillo-Chavez, How may the infection-age-dependent infectivity affect the dynamics of HIV/AIDS,, SIAM J. Appl. Math., 53 (1993), 1447. doi: 10.1137/0153068. [31] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985).
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