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June  2020, 10(2): 207-225. doi: 10.3934/naco.2019048

## Optimal control of an HIV model with CTL cells and latently infected cells

 Laboratory of Mathematics and Applications, Faculty of Sciences and, Techniques, Hassan II University of Casablanca, PO Box 146, Mohammedia, Morocco

* Corresponding author: Jaouad Danane. Email: jaouaddanane@gmail.com

Received  February 2019 Revised  July 2019 Published  September 2019

This paper deals with an optimal control problem for an human immunodeficiency virus (HIV) infection model with cytotoxic T-lymphocytes (CTL) immune response and latently infected cells. The model under consideration describes the interaction between the uninfected cells, the latently infected cells, the productively infected cells, the free viruses and the CTL cells. The two treatments represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to show the role of optimal therapy in controlling the infection severity.

Citation: Jaouad Danane, Karam Allali. Optimal control of an HIV model with CTL cells and latently infected cells. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 207-225. doi: 10.3934/naco.2019048
##### 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, Math. Biosci. Eng., 1 (2004), 223–241. doi: 10.3934/mbe.2004.1.223. [2] K. Allali, J. Danane and Y. Kuang, Global analysis for an HIV infection model with CTL immune response and infected cells in eclipse phase, Applied Sciences, 7 (2017), 861. doi: 10.3390/app7080861. [3] W. Blattner, R. C. Gallo and H. M. Temin, HIV causes AIDS, Science, 241 (1988), 515-516. [4] E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient highlevels of viremia in patients with primary human immunodeficiency virus type 1, New Engl. J. Med., 324 (1991), 961-964. [5] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12. [6] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975. [7] J. O. Kahn and B. D. Walker, Acute human immunodeficiency virus type 1 infection, New Engl. J. Med., 339 (1998), 33-39. [8] G. R. Kaufmann, P. Cunningham, A. D. Kelleher, J. Zauders, A. Carr, J. Vizzard, M. Law and D. A. Cooper, Patterns of viral dynamics during primary human immunodeficiency virus type 1 infection, J. Infec. Dis., 178 (1998), 1812-1815. [9] C. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems and Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001. [10] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2018), 170-182.  doi: 10.1016/j.jprocont.2018.10.001. [11] J. M. Orellana, Optimal drug scheduling for HIV therapy effciency improvement, Biomed. Signal Process, 6 (2011), 379-386. [12] G. Pachpute and S. P. Chakrabarty, Dynamics of hepatitis C under optimal therapy and sampling based analysis, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2202-2212.  doi: 10.1016/j.cnsns.2012.12.032. [13] L. Pontryagin, V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962. [14] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [15] T. Schacker, A. Collier, J. Hughes, T. Shea and L. Corey, Clinical and epidemiologic features of primary HIV infection, Ann. Int. Med., 125 (1996), 257-264. [16] Q. Sun, L. Min and Y. Kuang, Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model, IET Systems Biology, 9 (2015), 95-103. [17] Q. Sun and L. Min, Dynamics analysis and simulation of a modified HIV infection model with a saturated infection rate, Computational and Mathematical Methods in Medicine, (2014), Article ID 145162, 14 pages. doi: 10.1155/2014/145162. [18] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284. [19] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. [20] X. Wang, A. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Applied Mathematics and Computation, 218 (2012), 9405-9414.  doi: 10.1016/j.amc.2012.03.024. [21] X. Wang, Y. Tao and X. Song, Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response, Nonlinear Dynamics, 66 (2011), 825-830.  doi: 10.1007/s11071-011-9954-0. [22] World Health Organization, HIV/AIDS key facts, Available from http://www.who.int/mediacentre/factsheets/fs360/en/index.html. [23] H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Computers and Mathematics with Applications, 62 (2011), 3091-3102.  doi: 10.1016/j.camwa.2011.08.022.

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, Math. Biosci. Eng., 1 (2004), 223–241. doi: 10.3934/mbe.2004.1.223. [2] K. Allali, J. Danane and Y. Kuang, Global analysis for an HIV infection model with CTL immune response and infected cells in eclipse phase, Applied Sciences, 7 (2017), 861. doi: 10.3390/app7080861. [3] W. Blattner, R. C. Gallo and H. M. Temin, HIV causes AIDS, Science, 241 (1988), 515-516. [4] E. S. Daar, T. Moudgil, R. D. Meyer and D. D. Ho, Transient highlevels of viremia in patients with primary human immunodeficiency virus type 1, New Engl. J. Med., 324 (1991), 961-964. [5] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12. [6] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975. [7] J. O. Kahn and B. D. Walker, Acute human immunodeficiency virus type 1 infection, New Engl. J. Med., 339 (1998), 33-39. [8] G. R. Kaufmann, P. Cunningham, A. D. Kelleher, J. Zauders, A. Carr, J. Vizzard, M. Law and D. A. Cooper, Patterns of viral dynamics during primary human immunodeficiency virus type 1 infection, J. Infec. Dis., 178 (1998), 1812-1815. [9] C. Liu, R. Loxton and K. L. Teo, A computational method for solving time-delay optimal control problems with free terminal time, Systems and Control Letters, 72 (2014), 53-60.  doi: 10.1016/j.sysconle.2014.07.001. [10] C. Liu, Z. Gong, H. W. J. Lee and K. L. Teo, Robust bi-objective optimal control of 1, 3-propanediol microbial batch production process, Journal of Process Control, 78 (2018), 170-182.  doi: 10.1016/j.jprocont.2018.10.001. [11] J. M. Orellana, Optimal drug scheduling for HIV therapy effciency improvement, Biomed. Signal Process, 6 (2011), 379-386. [12] G. Pachpute and S. P. Chakrabarty, Dynamics of hepatitis C under optimal therapy and sampling based analysis, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2202-2212.  doi: 10.1016/j.cnsns.2012.12.032. [13] L. Pontryagin, V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962. [14] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79. [15] T. Schacker, A. Collier, J. Hughes, T. Shea and L. Corey, Clinical and epidemiologic features of primary HIV infection, Ann. Int. Med., 125 (1996), 257-264. [16] Q. Sun, L. Min and Y. Kuang, Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model, IET Systems Biology, 9 (2015), 95-103. [17] Q. Sun and L. Min, Dynamics analysis and simulation of a modified HIV infection model with a saturated infection rate, Computational and Mathematical Methods in Medicine, (2014), Article ID 145162, 14 pages. doi: 10.1155/2014/145162. [18] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284. [19] K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991. [20] X. Wang, A. Elaiw and X. Song, Global properties of a delayed HIV infection model with CTL immune response, Applied Mathematics and Computation, 218 (2012), 9405-9414.  doi: 10.1016/j.amc.2012.03.024. [21] X. Wang, Y. Tao and X. Song, Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response, Nonlinear Dynamics, 66 (2011), 825-830.  doi: 10.1007/s11071-011-9954-0. [22] World Health Organization, HIV/AIDS key facts, Available from http://www.who.int/mediacentre/factsheets/fs360/en/index.html. [23] H. Zhu, Y. Luo and M. Chen, Stability and Hopf bifurcation of a HIV infection model with CTL-response delay, Computers and Mathematics with Applications, 62 (2011), 3091-3102.  doi: 10.1016/j.camwa.2011.08.022.
Surface plot of $R_0$ (left) and contour plot of $R_0$ (right)
The uninfected cells as function of time
The latently infected cells as function of time
The infected cells as function of time
The HIV virus as function of time
The CTL response as function of time
The optimal control $u_1$ (left) and the optimal control $u_2$ (right) versus time
The optimal control $u_1$ (left) and the optimal control $u_2$ (right) versus time
The behavior of the infection dynamics
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