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

[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.   Google Scholar

[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.  Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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

[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.   Google Scholar

[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.  Google Scholar

[5]

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

[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.   Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

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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