# American Institute of Mathematical Sciences

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  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 & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031
##### 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 (2076-3417), 7 (2017).  Google Scholar [3] K. Allali, S. 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.  Google Scholar [4] R. Culshaw, S. 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.  Google Scholar [5] 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.   Google Scholar [6] J. Danane, A. 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12.   Google Scholar [11] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.  Google Scholar [12] T. J. Liang, Hepatitis B: the virus and disease, Hepatology, 49 (2009), S13–S21. Google Scholar [13] 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.  Google Scholar [14] 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 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.  Google Scholar [15] A. Meskaf, Y. Tabit and K. Allali, Global analysis of a HCV model with CTL, antibody responses and therapy, Applied Mathematical Sciences, 9 (2015), 3997-4008.   Google Scholar [16] A. Meskaf, K. 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.  Google Scholar [17] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   Google Scholar [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.  Google Scholar [19] L. Pontryagin and V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.  Google Scholar [20] D. Rocha, C. 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.  Google Scholar [21] L. B. Seeff, Natural history of chronic hepatitis C, Hepatology, 36 (2002), S35–S46. doi: 10.1002/hep.1840360706.  Google Scholar [22] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301.   Google Scholar [23] 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.   Google Scholar [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.  Google Scholar [25] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.   Google Scholar [26] 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 [27] World Health Organization HIV/AIDS Key facts, November 2017, http://www.who.int/mediacentre/factsheets/fs360/en/index.html., Google Scholar [28] H. Zhu, Y. 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.  Google Scholar

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##### 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 (2076-3417), 7 (2017).  Google Scholar [3] K. Allali, S. 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.  Google Scholar [4] R. Culshaw, S. 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.  Google Scholar [5] 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.   Google Scholar [6] J. Danane, A. 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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.   Google Scholar [10] K. R. Fister, S. Lenhart and J. S. McNally, Optimizing chemotherapy in an HIV model, Electron J. Differ. Equ., 32 (1998), 1-12.   Google Scholar [11] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, Berlin, 1975.  Google Scholar [12] T. J. Liang, Hepatitis B: the virus and disease, Hepatology, 49 (2009), S13–S21. Google Scholar [13] 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.  Google Scholar [14] 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 (2019), 170-182.  doi: 10.1016/j.jprocont.2018.10.001.  Google Scholar [15] A. Meskaf, Y. Tabit and K. Allali, Global analysis of a HCV model with CTL, antibody responses and therapy, Applied Mathematical Sciences, 9 (2015), 3997-4008.   Google Scholar [16] A. Meskaf, K. 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.  Google Scholar [17] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses, Science, 272 (1996), 74-79.   Google Scholar [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.  Google Scholar [19] L. Pontryagin and V. Boltyanskii, et al., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.  Google Scholar [20] D. Rocha, C. 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.  Google Scholar [21] L. B. Seeff, Natural history of chronic hepatitis C, Hepatology, 36 (2002), S35–S46. doi: 10.1002/hep.1840360706.  Google Scholar [22] M. A. Stafford, L. Corey, Y. Cao, E. S. Daar, D. D. Ho and A. S. Perelson, Modeling plasma virus concentration during primary HIV infection, J. Theor. Biol., 203 (2000), 285-301.   Google Scholar [23] 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.   Google Scholar [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.  Google Scholar [25] G. W. Swan, Role of optimal control theory in cancer chemotherapy, Math. Biosci., 101 (1990), 237-284.   Google Scholar [26] 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 [27] World Health Organization HIV/AIDS Key facts, November 2017, http://www.who.int/mediacentre/factsheets/fs360/en/index.html., Google Scholar [28] H. Zhu, Y. 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.  Google Scholar
The uninfected cells as function of time
The infected cells as function of time
The viral load as function of time
The CTL cells as function of time
The optimal control $u_1$ (left) and the optimal control $u_2$ (right) versus time
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