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September  2018, 8(3): 299-314. doi: 10.3934/naco.2018019

A controlled treatment strategy applied to HIV immunology model

Shohel Ahmed 1,, , Abdul Alim 1, and  Sumaiya Rahman 2,

1. 

Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh
2. 

Department of Statistics, University of Dhaka, Dhaka-1000, Bangladesh

* Corresponding author: SHOHEL AHMED, E-mail: shohel2443@math.buet.ac.bd

Received  April 2017 Revised  November 2017 Published  June 2018

Optimal control can be helpful to test and compare different vaccination strategies of a certain disease. This study investigates a mathematical model of HIV infections in terms of a system of nonlinear ordinary differential equations (ODEs) which describes the interactions between the human immune systems and the HIV virus. We introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The aim is to obtain a new optimal chemotherapeutic strategy where an isoperimetric constraint on the chemotherapy supply plays a crucial role. We outline the steps in formulating an optimal control problem, derive optimality conditions and demonstrate numerical results of an optimal control for the model. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on cell-virus interactions.

Keywords: Optimal control involving ODEs, Chemotherapy of HIV, Optimality conditions, Pontryagin's maximum principle.
Mathematics Subject Classification: Primary: 37N35, 49J15, 58F17; Secondary: 58E25.
Citation: Shohel Ahmed, Abdul Alim, Sumaiya Rahman. A controlled treatment strategy applied to HIV immunology model. Numerical Algebra, Control and Optimization, 2018, 8 (3) : 299-314. doi: 10.3934/naco.2018019
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, Bioscience Engineering, 1 (2004), 223-242.  doi: 10.3934/mbe.2004.1.223.

[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, Journal of Computational Applied Mathematics, 184 (2005), 10-49.  doi: 10.1016/j.cam.2005.02.004.

[3]

S. Butler, D. Kirschner and S. Lenhart, Optimal control of chemotherapy affecting the infectivity of HIV, Advances in Mathematical Population Dynamics: Molecules, Cells and Man, 6 (1997), 104-120. 

[4]

D. R. Burton, R. C. Desrosiers, R. W. Doms, W. C. Koff, P. D. Kwong, J. P. Moore, G. J. Nabel, J. Sodroski, I. A. Wilson and R. T. Wyatt, HIV vaccine design and the neutralizing antibody problem, Nature Immunology, 5 (2004), 233-236.  doi: 10.1038/ni0304-233.

[5]

W. Cheney and D. Kincaid, Numerical Mathematics and Computing, Thomson, Belmont, California, 2004.

[6]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 32 (1998), 1-12. 

[7]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. doi: 10.1007/978-1-4612-6380-7.

[8]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.

[9]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications and Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710.

[10]

D. Kirschner, Using mathematics to understand HIV immune dynamics, AMS Notices, 43 (1996), 191-202. 

[11]

D. Kirschner and A. S. Perelson, A model for the immune system response to HIV: AZT treatment studies, Mathematical Population Dynamics: Analysis of Heterogeneity, 1 (1995), 295-310. 

[12]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bulletin of Mathematical Biology, 58 (1996), 367-390.  doi: 10.1007/BF02458312.

[13]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, Journal of Mathematical Biology, 35 (1997), 775-792.  doi: 10.1007/s002850050076.

[14]

U. Ledzewicz and H. Schattler, On optimal controls for a general mathematical model for chemotherapy of HIV, Proceedings of the American Control Conference, 5 (2002), 3454-3459.  doi: 10.1109/ACC.2002.1024461.

[15]

S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, 1$^{st}$ edition, Chapman & Hall/CRC Mathematical and Computational Biology, 2007.

[16]

D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, 162 (1982), Academic Press, New York. doi: 10.2307/2322889.

[17]

S. Merrill, AIDS: Background and the dynamics of the decline of immune competence, Theoretical Immunology, Addison-Wesley, New York, Part 2 (1987), 59-75. 

[18]

S. Merrill, Modeling the interaction of HIV with cells of the immune response, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer-Verlag, New York, 83 (1989), 371-385.  doi: 10.1007/978-3-642-93454-4_18.

[19]

M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control, Computational Optimization and Applications, 53 (2012), 207-226.  doi: 10.1007/s10589-011-9454-7.

[20]

M. A. Nowak and R. M. May, Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Mathematical Bioscience, 106 (1991), 1-21.  doi: 10.1016/0025-5564(91)90037-J.

[21]

A. Perelson, D. Kirschner and R. DeBoer, The dynamics of HIV infection of CD4$^+$T cells, Mathematical Biosciences, 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.

[22]

A. S. Perelson, Modeling the interaction of the immune system with HIV, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer-Verlag, New York, 83 (1989), 350-370.  doi: 10.1007/978-3-642-93454-4_17.

[23]

L. S Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.

[24]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-5282-5.

[25]

R. M. Ribeiro and S. Bonhoeffer, A stochastic model for primary HIV infection: Optimal timing of therapy, AIDS, 13 (1999), 351-357.  doi: 10.1097/00002030-199902250-00007.

[26]

F. R. Stengel, Mutation and control of the human immunodeficiency virus, Mathematical Bioscience, 213 (2008), 93-102.  doi: 10.1016/j.mbs.2008.03.002.

[27]

D. G. Zill, Differential Equations with Boundary-Value Problems, Blue Kingfisher, 2017.

show all references

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, Bioscience Engineering, 1 (2004), 223-242.  doi: 10.3934/mbe.2004.1.223.

[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, Journal of Computational Applied Mathematics, 184 (2005), 10-49.  doi: 10.1016/j.cam.2005.02.004.

[3]

S. Butler, D. Kirschner and S. Lenhart, Optimal control of chemotherapy affecting the infectivity of HIV, Advances in Mathematical Population Dynamics: Molecules, Cells and Man, 6 (1997), 104-120. 

[4]

D. R. Burton, R. C. Desrosiers, R. W. Doms, W. C. Koff, P. D. Kwong, J. P. Moore, G. J. Nabel, J. Sodroski, I. A. Wilson and R. T. Wyatt, HIV vaccine design and the neutralizing antibody problem, Nature Immunology, 5 (2004), 233-236.  doi: 10.1038/ni0304-233.

[5]

W. Cheney and D. Kincaid, Numerical Mathematics and Computing, Thomson, Belmont, California, 2004.

[6]

K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 32 (1998), 1-12. 

[7]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975. doi: 10.1007/978-1-4612-6380-7.

[8]

W. Hackbush, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.

[9]

H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Applications and Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710.

[10]

D. Kirschner, Using mathematics to understand HIV immune dynamics, AMS Notices, 43 (1996), 191-202. 

[11]

D. Kirschner and A. S. Perelson, A model for the immune system response to HIV: AZT treatment studies, Mathematical Population Dynamics: Analysis of Heterogeneity, 1 (1995), 295-310. 

[12]

D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS, Bulletin of Mathematical Biology, 58 (1996), 367-390.  doi: 10.1007/BF02458312.

[13]

D. Kirschner, S. Lenhart and S. Serbin, Optimal control of the chemotherapy of HIV, Journal of Mathematical Biology, 35 (1997), 775-792.  doi: 10.1007/s002850050076.

[14]

U. Ledzewicz and H. Schattler, On optimal controls for a general mathematical model for chemotherapy of HIV, Proceedings of the American Control Conference, 5 (2002), 3454-3459.  doi: 10.1109/ACC.2002.1024461.

[15]

S. Lenhart and J. Workman, Optimal Control Applied to Biological Models, 1$^{st}$ edition, Chapman & Hall/CRC Mathematical and Computational Biology, 2007.

[16]

D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, 162 (1982), Academic Press, New York. doi: 10.2307/2322889.

[17]

S. Merrill, AIDS: Background and the dynamics of the decline of immune competence, Theoretical Immunology, Addison-Wesley, New York, Part 2 (1987), 59-75. 

[18]

S. Merrill, Modeling the interaction of HIV with cells of the immune response, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer-Verlag, New York, 83 (1989), 371-385.  doi: 10.1007/978-3-642-93454-4_18.

[19]

M. McAsey, L. Mou and W. Han, Convergence of the forward-backward sweep method in optimal control, Computational Optimization and Applications, 53 (2012), 207-226.  doi: 10.1007/s10589-011-9454-7.

[20]

M. A. Nowak and R. M. May, Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Mathematical Bioscience, 106 (1991), 1-21.  doi: 10.1016/0025-5564(91)90037-J.

[21]

A. Perelson, D. Kirschner and R. DeBoer, The dynamics of HIV infection of CD4$^+$T cells, Mathematical Biosciences, 114 (1993), 81-125.  doi: 10.1016/0025-5564(93)90043-A.

[22]

A. S. Perelson, Modeling the interaction of the immune system with HIV, Mathematical and Statistical Approaches to AIDS Epidemiology, Springer-Verlag, New York, 83 (1989), 350-370.  doi: 10.1007/978-3-642-93454-4_17.

[23]

L. S Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962.

[24]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-5282-5.

[25]

R. M. Ribeiro and S. Bonhoeffer, A stochastic model for primary HIV infection: Optimal timing of therapy, AIDS, 13 (1999), 351-357.  doi: 10.1097/00002030-199902250-00007.

[26]

F. R. Stengel, Mutation and control of the human immunodeficiency virus, Mathematical Bioscience, 213 (2008), 93-102.  doi: 10.1016/j.mbs.2008.03.002.

[27]

D. G. Zill, Differential Equations with Boundary-Value Problems, Blue Kingfisher, 2017.

Figure 1.  The battle between HIV and the immune system begins in the earnest after the virus replicates in the infected cells and the new viral particles escape.[26]
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Figure 2.  Particles of HIV (green spheres), the virus that causes AIDS, bud from an infected white blood cell before moving on to infect other cells. [27]
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Figure 3.  Global number of AIDS-related deaths, new HIV-infections and people living with HIV (1990-2015)
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Figure 4.  HIV immunology model with chemotherapy function $u(t)$.
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Figure 5.  HIV model with control and without control
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Figure 6.  HIV model with different percentage of chemotherapy
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Figure 7.  HIV model with different value of weight parameter $a$
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Figure 8.  HIV model with different value of k
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Figure 9.  HIV model with and without constraints.
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Figure 10.  Total Chemotherapy amount during treatment period.
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Figure 11.  HIV model for different chemotherapy amount
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Figure 12.  Total Chemotherapy amount during treatment period
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Table 1.  Description of parameter and values of the HIV model [15]
Parameters Description Value
$m_1$ Death rate of Uninfected CD4$^+$T cell population $ 0.02/d$
$m_2$ Death rate of infected CD4$^+$T cell population $ 0.5/d$
$m_3$ Death rate of free virus $ 4.4/d$
$k$ Rate of CD4$^+$T cell become infected by free virus $ 2..4\times10^{-5}mm^{3} /d$
$r$ Rate of growth for the CD4$^+$T cell population $ 0.03/d$
$N$ Number of free virus produced by $T_i$ cells 300
$T_{max}$ Maximum CD4$^+$T cell population level $ 1.5\times10^{3}/mm^{3}$
$s$ Source term for Uninfected CD4$^+$T cells $ 10 d^{-1} mm^{-3}$
$a$ Weight parameter 0.05
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Parameters Description Value
$m_1$ Death rate of Uninfected CD4$^+$T cell population $ 0.02/d$
$m_2$ Death rate of infected CD4$^+$T cell population $ 0.5/d$
$m_3$ Death rate of free virus $ 4.4/d$
$k$ Rate of CD4$^+$T cell become infected by free virus $ 2..4\times10^{-5}mm^{3} /d$
$r$ Rate of growth for the CD4$^+$T cell population $ 0.03/d$
$N$ Number of free virus produced by $T_i$ cells 300
$T_{max}$ Maximum CD4$^+$T cell population level $ 1.5\times10^{3}/mm^{3}$
$s$ Source term for Uninfected CD4$^+$T cells $ 10 d^{-1} mm^{-3}$
$a$ Weight parameter 0.05
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