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 & Continuous Dynamical Systems - S, 2022, 15 (3) : 501-518. doi: 10.3934/dcdss.2021148
References:
[1]

K. AllaliS. 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.  Google Scholar

[2]

K. AllaliY. 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.  Google Scholar

[3]

M. S. CiupeB. L. BivortD. 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.  Google Scholar

[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.  Google Scholar

[5]

M. P. DavenportR. 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.  Google Scholar

[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.  Google Scholar

[7]

R. DenysiukC. 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.  Google Scholar

[8]

J. DjordjevicC. 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.  Google Scholar

[9]

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

[10]

I. S. Gradshteǐn and I. M. Ryzhik, Table of integrals, series, and products, Math. Comp., 39 (1982), 747-757.   Google Scholar

[11]

O. KostylenkoH. 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.  Google Scholar

[12]

D. L. Lukes, Differential Equations, Mathematics in Science and Engineering, 162, Academic Press, Inc. 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000.   Google Scholar
[16]

K. A. PawelekS. LiuF. 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.  Google Scholar

[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.  Google Scholar

[18]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

F. RodriguesC. J. SilvaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. WangY. ZhouF. 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.  Google Scholar

[30]

D. Wodarz, Killer Cell Dynamics, Interdisciplinary Applied Mathematics, 32, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68733-9.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

K. AllaliS. 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.  Google Scholar

[2]

K. AllaliY. 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.  Google Scholar

[3]

M. S. CiupeB. L. BivortD. 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.  Google Scholar

[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.  Google Scholar

[5]

M. P. DavenportR. 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.  Google Scholar

[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.  Google Scholar

[7]

R. DenysiukC. 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.  Google Scholar

[8]

J. DjordjevicC. 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.  Google Scholar

[9]

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

[10]

I. S. Gradshteǐn and I. M. Ryzhik, Table of integrals, series, and products, Math. Comp., 39 (1982), 747-757.   Google Scholar

[11]

O. KostylenkoH. 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.  Google Scholar

[12]

D. L. Lukes, Differential Equations, Mathematics in Science and Engineering, 162, Academic Press, Inc. 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] M. A. Nowak and R. M. May, Virus Dynamics, Oxford University Press, Oxford, 2000.   Google Scholar
[16]

K. A. PawelekS. LiuF. 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.  Google Scholar

[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.  Google Scholar

[18]

A. S. PerelsonA. U. NeumannM. MarkowitzJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

F. RodriguesC. J. SilvaD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

Y. WangY. ZhouF. 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.  Google Scholar

[30]

D. Wodarz, Killer Cell Dynamics, Interdisciplinary Applied Mathematics, 32, Springer-Verlag, New York, 2007. doi: 10.1007/978-0-387-68733-9.  Google Scholar

[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.  Google Scholar

Figure 1.  The evolution of the uninfected cells during time
Figure 2.  The evolution of the infected cells during time
Figure 3.  The evolution of the HIV virus during time
Figure 4.  The evolution of the CTL cells during time
Figure 5.  The evolution of the antibodies during time
Figure 6.  The behaviour of the two optimal controls
Table 1.  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
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