
-
Previous Article
On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints
- NACO Home
- This Issue
-
Next Article
Pricing down-and-out power options with exponentially curved barrier
A controlled treatment strategy applied to HIV immunology model
1. | Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh |
2. | Department of Statistics, University of Dhaka, Dhaka-1000, Bangladesh |
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.
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. |










Parameters | Description | Value |
Death rate of Uninfected CD4 |
||
Death rate of infected CD4 |
||
Death rate of free virus | ||
Rate of CD4 |
||
Rate of growth for the CD4 |
||
Number of free virus produced by |
300 | |
Maximum CD4 |
||
Source term for Uninfected CD4 |
||
Weight parameter | 0.05 |
Parameters | Description | Value |
Death rate of Uninfected CD4 |
||
Death rate of infected CD4 |
||
Death rate of free virus | ||
Rate of CD4 |
||
Rate of growth for the CD4 |
||
Number of free virus produced by |
300 | |
Maximum CD4 |
||
Source term for Uninfected CD4 |
||
Weight parameter | 0.05 |
[1] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110 |
[2] |
Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 |
[3] |
Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control and Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021 |
[4] |
Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161 |
[5] |
Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021059 |
[6] |
Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control and Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195 |
[7] |
Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 |
[8] |
Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001 |
[9] |
Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559 |
[10] |
Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control and Related Fields, 2021, 11 (3) : 555-578. doi: 10.3934/mcrf.2021012 |
[11] |
Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 |
[12] |
Md. Haider Ali Biswas, Maria do Rosário de Pinho. A nonsmooth maximum principle for optimal control problems with state and mixed constraints - convex case. Conference Publications, 2011, 2011 (Special) : 174-183. doi: 10.3934/proc.2011.2011.174 |
[13] |
Zhen Wu, Feng Zhang. Maximum principle for discrete-time stochastic optimal control problem and stochastic game. Mathematical Control and Related Fields, 2022, 12 (2) : 475-493. doi: 10.3934/mcrf.2021031 |
[14] |
Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101 |
[15] |
Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 |
[16] |
Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657 |
[17] |
Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436 |
[18] |
Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks and Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027 |
[19] |
Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 |
[20] |
Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]