# American Institute of Mathematical Sciences

March  2022, 15(3): 501-518. doi: 10.3934/dcdss.2021148

## Optimal control of an HIV model with a trilinear antibody growth function

 1 Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, P. O. Box 146, Mohammedia, Morocco 2 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

* Corresponding author: Delfim F. M. Torres

Received  June 2019 Revised  May 2021 Published  March 2022 Early access  November 2021

We propose and study a new mathematical model of the human immunodeficiency virus (HIV). The main novelty is to consider that the antibody growth depends not only on the virus and on the antibodies concentration but also on the uninfected cells concentration. The model consists of five nonlinear differential equations describing the evolution of the uninfected cells, the infected ones, the free viruses, and the adaptive immunity. The adaptive immune response is represented by the cytotoxic T-lymphocytes (CTL) cells and the antibodies with the growth function supposed to be trilinear. The model includes two kinds of treatments. The objective of the first one is to reduce the number of infected cells, while the aim of the second is to block free viruses. Firstly, the positivity and the boundedness of solutions are established. After that, the local stability of the disease free steady state and the infection steady states are characterized. Next, an optimal control problem is posed and investigated. Finally, numerical simulations are performed in order to show the behavior of solutions and the effectiveness of the two incorporated treatments via an efficient optimal control strategy.

Citation: Karam Allali, Sanaa Harroudi, Delfim F. M. Torres. Optimal control of an HIV model with a trilinear antibody growth function. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 501-518. doi: 10.3934/dcdss.2021148
##### References:
 [1] K. Allali, S. Harroudi and D. F. M. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci., 12 (2018), 111-127.  doi: 10.1007/s11786-018-0333-9. [2] K. Allali, Y. Tabit and S. Harroudi, On HIV model with adaptive immune response, two saturated rates and therapy, Math. Model. Nat. Phenom., 12 (2017), 1-14.  doi: 10.1051/mmnp/201712501. [3] M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27.  doi: 10.1016/j.mbs.2005.12.006. [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. [5] M. P. Davenport, R. M. Ribeiro and A. S. Perelson, Kinetics of virus-specific CD8+ T cells and the control of human immunodeficiency virus infection, J. Vir., 78 (2004), 10096-10103.  doi: 10.1128/JVI.78.18.10096-10103.2004. [6] R. J. De Boer and A. S. Perelson, Quantifying T lymphocyte turnover, J. Theoret. Biol., 327 (2013), 45-87.  doi: 10.1016/j.jtbi.2012.12.025. [7] R. Denysiuk, C. J. Silva and D. F. M. Torres, Multiobjective optimization to a TB-HIV/AIDS coinfection optimal control problem, Comput. Appl. Math., 37 (2018), 2112-2128.  doi: 10.1007/s40314-017-0438-9. [8] J. Djordjevic, C. J. Silva and D. F. M. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168-175.  doi: 10.1016/j.aml.2018.05.005. [9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, 1975. [10] I. S. Gradshteǐn and I. M. Ryzhik, Table of integrals, series, and products, Math. Comp., 39 (1982), 747-757. [11] O. Kostylenko, H. S. Rodrigues and D. F. M. Torres, The spread of a financial virus through Europe and beyond, AIMS Math., 4 (2019), 86-98.  doi: 10.3934/Math.2019.1.86. [12] D. L. Lukes, Differential Equations, Mathematics in Science and Engineering, 162, Academic Press, Inc. 1982. [13] C. C. McCluskey and M. Santoprete, A bare-bones mathematical model of radicalization, J. Dyn. Games, 5 (2018), 243-264.  doi: 10.3934/jdg.2018016. [14] M. A. Nowak and R. M. May, Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Math. Biosci., 106 (1991), 1-21.  doi: 10.1016/0025-5564(91)90037-J. [15] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000. [16] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.  doi: 10.1016/j.mbs.2011.11.002. [17] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107. [18] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.  doi: 10.1126/science.271.5255.1582. [19] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York, 1962. [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. [21] F. Rodrigues, C. J. Silva, D. F. M. Torres and H. Maurer, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030. [22] S. Saha and G. P. Samanta, Modelling and optimal control of HIV/AIDS prevention through PrEP and limited treatment, Phys. A, 516 (2019), 280-307.  doi: 10.1016/j.physa.2018.10.033. [23] C. J. Silva and D. F. M. Torres, Modeling TB-HIV syndemic and treatment, J. Appl. Math., 2014 (2014), 248407, 14 pp. doi: 10.1155/2014/248407. [24] C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.  doi: 10.3934/dcds.2015.35.4639. [25] C. J. Silva and D. F. M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecological Complexity, 30 (2017), 70-75.  doi: 10.1016/j.ecocom.2016.12.001. [26] C. J. Silva and D. F. M. Torres, Global stability for a HIV/AIDS model, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 67 (2018), 93-101.  doi: 10.1501/Commua1_0000000833. [27] C. J. Silva and D. F. M. Torres, Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 119-141.  doi: 10.3934/dcdss.2018008. [28] P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002. [29] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3. [30] D. Wodarz, Killer Cell Dynamics, Interdisciplinary Applied Mathematics, 32, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68733-9. [31] H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.

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##### References:
 [1] K. Allali, S. Harroudi and D. F. M. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response, Math. Comput. Sci., 12 (2018), 111-127.  doi: 10.1007/s11786-018-0333-9. [2] K. Allali, Y. Tabit and S. Harroudi, On HIV model with adaptive immune response, two saturated rates and therapy, Math. Model. Nat. Phenom., 12 (2017), 1-14.  doi: 10.1051/mmnp/201712501. [3] M. S. Ciupe, B. L. Bivort, D. M. Bortz and P. W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models, Math. Biosci., 200 (2006), 1-27.  doi: 10.1016/j.mbs.2005.12.006. [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. [5] M. P. Davenport, R. M. Ribeiro and A. S. Perelson, Kinetics of virus-specific CD8+ T cells and the control of human immunodeficiency virus infection, J. Vir., 78 (2004), 10096-10103.  doi: 10.1128/JVI.78.18.10096-10103.2004. [6] R. J. De Boer and A. S. Perelson, Quantifying T lymphocyte turnover, J. Theoret. Biol., 327 (2013), 45-87.  doi: 10.1016/j.jtbi.2012.12.025. [7] R. Denysiuk, C. J. Silva and D. F. M. Torres, Multiobjective optimization to a TB-HIV/AIDS coinfection optimal control problem, Comput. Appl. Math., 37 (2018), 2112-2128.  doi: 10.1007/s40314-017-0438-9. [8] J. Djordjevic, C. J. Silva and D. F. M. Torres, A stochastic SICA epidemic model for HIV transmission, Appl. Math. Lett., 84 (2018), 168-175.  doi: 10.1016/j.aml.2018.05.005. [9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, Berlin, 1975. [10] I. S. Gradshteǐn and I. M. Ryzhik, Table of integrals, series, and products, Math. Comp., 39 (1982), 747-757. [11] O. Kostylenko, H. S. Rodrigues and D. F. M. Torres, The spread of a financial virus through Europe and beyond, AIMS Math., 4 (2019), 86-98.  doi: 10.3934/Math.2019.1.86. [12] D. L. Lukes, Differential Equations, Mathematics in Science and Engineering, 162, Academic Press, Inc. 1982. [13] C. C. McCluskey and M. Santoprete, A bare-bones mathematical model of radicalization, J. Dyn. Games, 5 (2018), 243-264.  doi: 10.3934/jdg.2018016. [14] M. A. Nowak and R. M. May, Mathematical biology of HIV infections: Antigenic variation and diversity threshold, Math. Biosci., 106 (1991), 1-21.  doi: 10.1016/0025-5564(91)90037-J. [15] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000. [16] K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Math. Biosci., 235 (2012), 98-109.  doi: 10.1016/j.mbs.2011.11.002. [17] A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Rev., 41 (1999), 3-44.  doi: 10.1137/S0036144598335107. [18] A. S. Perelson, A. U. Neumann, M. Markowitz, J. M. Leonard and D. D. Ho, HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time, Science, 271 (1996), 1582-1586.  doi: 10.1126/science.271.5255.1582. [19] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley & Sons, Inc., New York, 1962. [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. [21] F. Rodrigues, C. J. Silva, D. F. M. Torres and H. Maurer, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030. [22] S. Saha and G. P. Samanta, Modelling and optimal control of HIV/AIDS prevention through PrEP and limited treatment, Phys. A, 516 (2019), 280-307.  doi: 10.1016/j.physa.2018.10.033. [23] C. J. Silva and D. F. M. Torres, Modeling TB-HIV syndemic and treatment, J. Appl. Math., 2014 (2014), 248407, 14 pp. doi: 10.1155/2014/248407. [24] C. J. Silva and D. F. M. Torres, A TB-HIV/AIDS coinfection model and optimal control treatment, Discrete Contin. Dyn. Syst., 35 (2015), 4639-4663.  doi: 10.3934/dcds.2015.35.4639. [25] C. J. Silva and D. F. M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, Ecological Complexity, 30 (2017), 70-75.  doi: 10.1016/j.ecocom.2016.12.001. [26] C. J. Silva and D. F. M. Torres, Global stability for a HIV/AIDS model, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., 67 (2018), 93-101.  doi: 10.1501/Commua1_0000000833. [27] C. J. Silva and D. F. M. Torres, Modeling and optimal control of HIV/AIDS prevention through PrEP, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 119-141.  doi: 10.3934/dcdss.2018008. [28] P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Model., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002. [29] Y. Wang, Y. Zhou, F. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3. [30] D. Wodarz, Killer Cell Dynamics, Interdisciplinary Applied Mathematics, 32, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68733-9. [31] H. Zhu and X. Zou, Dynamics of a HIV-1 infection model with cell-mediated immune response and intracellular delay, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 511-524.  doi: 10.3934/dcdsb.2009.12.511.
The evolution of the uninfected cells during time
The evolution of the infected cells during time
The evolution of the HIV virus during time
The evolution of the CTL cells during time
The evolution of the antibodies during time
The behaviour of the two optimal controls
Parameters, their symbols and meaning, and default values used in HIV literature
 Parameters Meaning Value References $\lambda$ source rate of CD4+ T cells $1$-$10$ cells $\mu l^{-1}$ days $^{-1}$ [4] $d$ decay rate of healthy cells $0.007$-$0.1$ days $^{-1}$ [4] $\beta$ rate at which CD4+ T cells become infected $0.00025$-$0.5$ $\mu l$ virion $^{-1}$ days $^{-1}$ [4] $a$ death rate of infected CD4+ T cells, not by CTL $0.2$-$0.3$ days $^{-1}$ [4] $\mu$ clearance rate of virus $2.06$-$3.81$ days $^{-1}$ [18] $N$ number of virions produced by infected CD4+ T-cells $6.25$-$23599.9$ virion $^{-1}$ [3,29] $p$ clearance rate of infection $1$-$4.048 \times 10^{-4}$ ml virion days $^{-1}$ [3,16] $c$ activation rate of CTL cells $0.0051$-$3.912$ days $^{-1}$ [3] $h$ death rate of CTL cells $0.004$-$8.087$ days $^{-1}$ [3] $q$ Neutralization rate of virions $0.12$ days $^{-1}$ Assumed $g$ activation rate of antibodies $0.00013$ days $^{-1}$ Assumed $\alpha$ death rate of antibodies $0.12$ days $^{-1}$ Assumed
 Parameters Meaning Value References $\lambda$ source rate of CD4+ T cells $1$-$10$ cells $\mu l^{-1}$ days $^{-1}$ [4] $d$ decay rate of healthy cells $0.007$-$0.1$ days $^{-1}$ [4] $\beta$ rate at which CD4+ T cells become infected $0.00025$-$0.5$ $\mu l$ virion $^{-1}$ days $^{-1}$ [4] $a$ death rate of infected CD4+ T cells, not by CTL $0.2$-$0.3$ days $^{-1}$ [4] $\mu$ clearance rate of virus $2.06$-$3.81$ days $^{-1}$ [18] $N$ number of virions produced by infected CD4+ T-cells $6.25$-$23599.9$ virion $^{-1}$ [3,29] $p$ clearance rate of infection $1$-$4.048 \times 10^{-4}$ ml virion days $^{-1}$ [3,16] $c$ activation rate of CTL cells $0.0051$-$3.912$ days $^{-1}$ [3] $h$ death rate of CTL cells $0.004$-$8.087$ days $^{-1}$ [3] $q$ Neutralization rate of virions $0.12$ days $^{-1}$ Assumed $g$ activation rate of antibodies $0.00013$ days $^{-1}$ Assumed $\alpha$ death rate of antibodies $0.12$ days $^{-1}$ Assumed
 [1] Hee-Dae Kwon, Jeehyun Lee, Myoungho Yoon. An age-structured model with immune response of HIV infection: Modeling and optimal control approach. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 153-172. doi: 10.3934/dcdsb.2014.19.153 [2] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311 [3] Urszula Ledzewicz, Mohammad Naghnaeian, Heinz Schättler. Dynamics of tumor-immune interaction under treatment as an optimal control problem. Conference Publications, 2011, 2011 (Special) : 971-980. doi: 10.3934/proc.2011.2011.971 [4] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 [5] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [6] Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008 [7] Mudassar Imran, Hal L. Smith. The dynamics of bacterial infection, innate immune response, and antibiotic treatment. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 127-143. doi: 10.3934/dcdsb.2007.8.127 [8] Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009 [9] Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621 [10] Aiping Wang, Michael Y. Li. Viral dynamics of HIV-1 with CTL immune response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2257-2272. doi: 10.3934/dcdsb.2020212 [11] Cristiana J. Silva. Stability and optimal control of a delayed HIV/AIDS-PrEP model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 639-654. doi: 10.3934/dcdss.2021156 [12] Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120 [13] Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030 [14] Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086 [15] Jinhu Xu, Yicang Zhou. Bifurcation analysis of HIV-1 infection model with cell-to-cell transmission and immune response delay. Mathematical Biosciences & Engineering, 2016, 13 (2) : 343-367. doi: 10.3934/mbe.2015006 [16] Huiyan Zhu, Xingfu Zou. Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay. Discrete and Continuous Dynamical Systems - B, 2009, 12 (2) : 511-524. doi: 10.3934/dcdsb.2009.12.511 [17] Zhijun Liu, Lianwen Wang, Ronghua Tan. Spatiotemporal dynamics for a diffusive HIV-1 infection model with distributed delays and CTL immune response. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2767-2790. doi: 10.3934/dcdsb.2021159 [18] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [19] M. Teresa T. Monteiro, Isabel Espírito Santo, Helena Sofia Rodrigues. An optimal control problem applied to a wastewater treatment plant. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153 [20] Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129

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