January  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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model,, Electronic J. of Differential Equation, (1998), 1. Google Scholar

[11]

M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985). Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model,, Electronic J. of Differential Equation, (1998), 1. Google Scholar

[11]

M. D. Gunzburger, Perspectives in Flow Control and Optimization,, SIAM, (2003). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics,, Marcel Dekker, (1985). Google Scholar

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