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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

W. BlattnerR. C. Gallo and H. M. Temin, HIV causes AIDS, Science, 241 (1988), 515-516.   Google Scholar

[4]

E. S. DaarT. MoudgilR. 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.   Google Scholar

[5]

K. R. FisterS. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12.   Google Scholar

[6]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.  Google Scholar

[7]

J. O. Kahn and B. D. Walker, Acute human immunodeficiency virus type 1 infection, New Engl. J. Med., 339 (1998), 33-39.   Google Scholar

[8]

G. R. KaufmannP. CunninghamA. D. KelleherJ. ZaudersA. CarrJ. VizzardM. Law and D. A. Cooper, Patterns of viral dynamics during primary human immunodeficiency virus type 1 infection, J. Infec. Dis., 178 (1998), 1812-1815.   Google Scholar

[9]

C. LiuR. 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.  Google Scholar

[10]

C. LiuZ. GongH. 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.  Google Scholar

[11]

J. M. Orellana, Optimal drug scheduling for HIV therapy effciency improvement, Biomed. Signal Process, 6 (2011), 379-386.   Google Scholar

[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.  Google Scholar

[13]

L. Pontryagin, V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.  Google Scholar

[14]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   Google Scholar

[15]

T. SchackerA. CollierJ. HughesT. Shea and L. Corey, Clinical and epidemiologic features of primary HIV infection, Ann. Int. Med., 125 (1996), 257-264.   Google Scholar

[16]

Q. SunL. 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.   Google Scholar

[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.  Google Scholar

[18]

G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.   Google Scholar

[19]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991.  Google Scholar

[20]

X. WangA. 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.  Google Scholar

[21]

X. WangY. 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.  Google Scholar

[22]

World Health Organization, HIV/AIDS key facts, Available from http://www.who.int/mediacentre/factsheets/fs360/en/index.html. Google Scholar

[23]

H. ZhuY. 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.  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, Math. Biosci. Eng., 1 (2004), 223–241. doi: 10.3934/mbe.2004.1.223.  Google Scholar

[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.  Google Scholar

[3]

W. BlattnerR. C. Gallo and H. M. Temin, HIV causes AIDS, Science, 241 (1988), 515-516.   Google Scholar

[4]

E. S. DaarT. MoudgilR. 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.   Google Scholar

[5]

K. R. FisterS. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12.   Google Scholar

[6]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.  Google Scholar

[7]

J. O. Kahn and B. D. Walker, Acute human immunodeficiency virus type 1 infection, New Engl. J. Med., 339 (1998), 33-39.   Google Scholar

[8]

G. R. KaufmannP. CunninghamA. D. KelleherJ. ZaudersA. CarrJ. VizzardM. Law and D. A. Cooper, Patterns of viral dynamics during primary human immunodeficiency virus type 1 infection, J. Infec. Dis., 178 (1998), 1812-1815.   Google Scholar

[9]

C. LiuR. 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.  Google Scholar

[10]

C. LiuZ. GongH. 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.  Google Scholar

[11]

J. M. Orellana, Optimal drug scheduling for HIV therapy effciency improvement, Biomed. Signal Process, 6 (2011), 379-386.   Google Scholar

[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.  Google Scholar

[13]

L. Pontryagin, V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.  Google Scholar

[14]

M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   Google Scholar

[15]

T. SchackerA. CollierJ. HughesT. Shea and L. Corey, Clinical and epidemiologic features of primary HIV infection, Ann. Int. Med., 125 (1996), 257-264.   Google Scholar

[16]

Q. SunL. 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.   Google Scholar

[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.  Google Scholar

[18]

G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.   Google Scholar

[19]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Longman Scientific and Technical, 1991.  Google Scholar

[20]

X. WangA. 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.  Google Scholar

[21]

X. WangY. 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.  Google Scholar

[22]

World Health Organization, HIV/AIDS key facts, Available from http://www.who.int/mediacentre/factsheets/fs360/en/index.html. Google Scholar

[23]

H. ZhuY. 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.  Google Scholar

Figure 1.  Surface plot of $ R_0 $ (left) and contour plot of $ R_0 $ (right)
Figure 2.  The uninfected cells as function of time
Figure 3.  The latently infected cells as function of time
Figure 4.  The infected cells as function of time
Figure 5.  The HIV virus as function of time
Figure 6.  The CTL response as function of time
Figure 7.  The optimal control $ u_1 $ (left) and the optimal control $ u_2 $ (right) versus time
Figure 8.  The optimal control $ u_1 $ (left) and the optimal control $ u_2 $ (right) versus time
Figure 9.  The behavior of the infection dynamics
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