September  2021, 11(3): 363-375. doi: 10.3934/naco.2020031

Optimal control of viral infection model with saturated infection rate

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

Received  November 2019 Revised  February 2020 Published  September 2021 Early access  May 2020

This paper deals with an optimal control problem for a viral infection model with cytotoxic T-lymphocytes (CTL) immune response. The model under consideration describes the interaction between the uninfected cells, the 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 maximum 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. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031
References:
[1]

B. M. AdamsH. T. BanksH. 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 (2076-3417), 7 (2017).

[3]

K. AllaliS. Harroudi and D. F. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Mathematics in Computer Science, 12 (2018), 111-127.  doi: 10.1007/s11786-018-0333-9.

[4]

R. CulshawS. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562.  doi: 10.1007/s00285-003-0245-3.

[5]

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. 

[6]

J. DananeA. Meskaf and K. Allali, Optimal control of a delayed hepatitis B viral infection model with HBV DNA containing capsids and CTL immune response, Optimal Control Applications and Methods, 39 (2018), 1262-1272.  doi: 10.1002/oca.2407.

[7]

J. Danane and K. Allali, Mathematical analysis and treatment for a delayed hepatitis B viral infection model with the adaptive immune response and DNA-containing capsids, High-throughput, 7 (2018), 35.

[8]

J. Danane and K. Allali, Optimal control of an HIV model with CTL cells and latently infected cells, Numerical Algebra, Control and Optimization, 10 (2020), 207-225.  doi: 10.3934/naco.2019048.

[9]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: a comparison, J. Theor. Biol., 190 (1998), 201-214. 

[10]

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

[11]

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

[12]

T. J. Liang, Hepatitis B: the virus and disease, Hepatology, 49 (2009), S13–S21.

[13]

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.

[14]

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 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.

[15]

A. MeskafY. Tabit and K. Allali, Global analysis of a HCV model with CTL, antibody responses and therapy, Applied Mathematical Sciences, 9 (2015), 3997-4008. 

[16]

A. MeskafK. Allali and Y. Tabit, Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses, International Journal of Dynamics and Control, 5 (2017), 893-902.  doi: 10.1007/s40435-016-0231-4.

[17]

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

[18]

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.

[19]

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

[20]

D. RochaC. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., 41 (2018), 2251-2260.  doi: 10.1002/mma.4207.

[21]

L. B. Seeff, Natural history of chronic hepatitis C, Hepatology, 36 (2002), S35–S46. doi: 10.1002/hep.1840360706.

[22]

M. A. StaffordL. CoreyY. CaoE. S. DaarD. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. 

[23]

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. 

[24]

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.

[25]

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

[26]

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

[27]

World Health Organization HIV/AIDS Key facts, November 2017, http://www.who.int/mediacentre/factsheets/fs360/en/index.html.,

[28]

H. ZhuY. Luo and M. Chen, Stability and Hopfbifurcation 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. AdamsH. T. BanksH. 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 (2076-3417), 7 (2017).

[3]

K. AllaliS. Harroudi and D. F. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Mathematics in Computer Science, 12 (2018), 111-127.  doi: 10.1007/s11786-018-0333-9.

[4]

R. CulshawS. Ruan and R. J. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Biol., 48 (2004), 545-562.  doi: 10.1007/s00285-003-0245-3.

[5]

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. 

[6]

J. DananeA. Meskaf and K. Allali, Optimal control of a delayed hepatitis B viral infection model with HBV DNA containing capsids and CTL immune response, Optimal Control Applications and Methods, 39 (2018), 1262-1272.  doi: 10.1002/oca.2407.

[7]

J. Danane and K. Allali, Mathematical analysis and treatment for a delayed hepatitis B viral infection model with the adaptive immune response and DNA-containing capsids, High-throughput, 7 (2018), 35.

[8]

J. Danane and K. Allali, Optimal control of an HIV model with CTL cells and latently infected cells, Numerical Algebra, Control and Optimization, 10 (2020), 207-225.  doi: 10.3934/naco.2019048.

[9]

R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: a comparison, J. Theor. Biol., 190 (1998), 201-214. 

[10]

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

[11]

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

[12]

T. J. Liang, Hepatitis B: the virus and disease, Hepatology, 49 (2009), S13–S21.

[13]

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.

[14]

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 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.

[15]

A. MeskafY. Tabit and K. Allali, Global analysis of a HCV model with CTL, antibody responses and therapy, Applied Mathematical Sciences, 9 (2015), 3997-4008. 

[16]

A. MeskafK. Allali and Y. Tabit, Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses, International Journal of Dynamics and Control, 5 (2017), 893-902.  doi: 10.1007/s40435-016-0231-4.

[17]

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

[18]

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.

[19]

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

[20]

D. RochaC. J. Silva and D. F. M. Torres, Stability and optimal control of a delayed HIV model, Math. Methods Appl. Sci., 41 (2018), 2251-2260.  doi: 10.1002/mma.4207.

[21]

L. B. Seeff, Natural history of chronic hepatitis C, Hepatology, 36 (2002), S35–S46. doi: 10.1002/hep.1840360706.

[22]

M. A. StaffordL. CoreyY. CaoE. S. DaarD. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301. 

[23]

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. 

[24]

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.

[25]

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

[26]

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

[27]

World Health Organization HIV/AIDS Key facts, November 2017, http://www.who.int/mediacentre/factsheets/fs360/en/index.html.,

[28]

H. ZhuY. Luo and M. Chen, Stability and Hopfbifurcation 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.

Figure 1.  The uninfected cells as function of time
Figure 2.  The infected cells as function of time
Figure 3.  The viral load as function of time
Figure 4.  The CTL cells as function of time
Figure 5.  The optimal control $ u_1 $ (left) and the optimal control $ u_2 $ (right) versus time
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