# American Institute of Mathematical Sciences

June  2020, 25(6): 2293-2306. doi: 10.3934/dcdsb.2020035

## Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays

 a. University of Cheikh Anta Diop, Department of Mathematics and Computer SciencesFaculty of Science and Technic, P.O. Box 5005 Dakar, Senegal b. University of Yaounde I, Department of MathematicsFaculty of Science, Yaounde, Cameroon c. University of Yaounde I, Department of Mathematics and PhysicsNational Advanced School of Engineering, Yaounde, Cameroon

* Corresponding author: Mboya Ba

Received  January 2019 Revised  August 2019 Published  June 2020 Early access  February 2020

Fund Project: The first author is supported by NSF grant

HIV infection is divided into stages of infection which are determined by the CD4 cells count progression. Through each stage, the time delay for the progression is important because the duration of HIV infection varies according to the infectious. Retarded optimal control theory is applied to a system of delays ordinary differential equations modeling the evolution of HIV with differential infectivity. Seeking to reduce the population of the infective individuals with low CD$4$ cells, we use the ARV drug to control the fraction of infective individuals that is identified and will be put under treatment. We use optimal control theory to study our proposed system. Numerical simulations are provided to illustrate the effect of the Antiretroviral treatment (ART) taking into account the delays.

Citation: Mboya Ba, P. Tchinda Mouofo, Mountaga Lam, Jean-Jules Tewa. Optimal intervention strategies of a SI-HIV models with differential infectivity and time delays. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2293-2306. doi: 10.3934/dcdsb.2020035
##### References:
 [1] AIDS INFO - U.S. Department of Health and Human Services, Offering Information On HIV/AIDS Treatment, Prevention, and Research. Understanding HIV/AIDS, Side Effects of HIV Medicines, 2017. Available from: https://aidsinfo.nih.gov. [2] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.  doi: 10.1006/bulm.2001.0266. [3] Canada's Source for HIV and Hepatitis C Information (CATIE), HIV Treatment and an Undetectable Viral Load to Prevent HIV Transmission, 2018. Available from: www.catie.ca./reports/reports.webpage. [4] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [5] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^{+}$ T-cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7. [6] D. C. Douek, M. Roederer and R. A. Koup, Emerging concepts in the immunopathogenesis of AIDS, Annual Rev. Medicine, 60 (2009), 471-484.  doi: 10.1146/annurev.med.60.041807.123549. [7] P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Modell., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002. [8] J. W. Eaton and T. B. Hallett, Why the proportion of transmission during early-stage HIV infection does not predict the long-term impact of treatment on HIV incidence, Proceedings of the National Academy of Sciences, 111, 2014, 16202â€"16207. doi: 10.1073/pnas.1323007111. [9] H. R. Erfanian and M. H. Noori Skandari, Optimal control of an HIV model, Internat. J. Appl. Math. Comput. Sci., 2 (2011), 650-658.  doi: 10.22436/jmcs.02.04.09. [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, 1, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.  doi: 10.1038/373123a0. [13] J. F. Hutchinson, The biology and evolution of HIV, Annual Rev. Anthropology, 30 (2001), 85-108.  doi: 10.1146/annurev.anthro.30.1.85. [14] H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Appl. Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710. [15] S. E. Langford, J. Ananworanich and D. A. Cooper, Predictors of disease progression in HIV infection: A review, AIDS Res. Therapy, 4 (2007), 4-11.  doi: 10.1186/1742-6405-4-11. [16] C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  doi: 10.1016/S0025-5564(02)00149-9. [17] P. T. Mouofo, J. J. Tewa, B. Mewoli and S. Bowong, Optimal control of a delayed system subject to mixed control-state constraints with application to a within-host model of hepatitis virus B, Annual Rev. Control, 37 (2013), 246-259.  doi: 10.1016/j.arcontrol.2013.09.004. [18] J. M. Murray, G. Kaufmann, A. D. Kelleher and D. A. Cooper, A model of primary HIV-1 infection, Math. Biosci., 154 (1998), 57-85.  doi: 10.1016/S0025-5564(98)10046-9. [19] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.  doi: 10.1016/S0025-5564(99)00055-3. [20] P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8. [21] 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. [22] D. Powell, J. Fair, R. J. Leclaire and L. M. Moore, Sensitivity analysis of an infectious disease model, International System Dynamics Conference, Boston, MA, 2005. [23] F. Rodrigues, C. J. Silva and D. F. M. Torres, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030. [24] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Sensitivity analysis in a dengue epidemiological model, Conference Papers in Mathematics, 2013 (2013), 7pp. doi: 10.1155/2013/721406. [25] M. Shirazian and M. H. Farahi, Optimal control strategy for a fully determined HIV model, Intelligent Control Automation, 1 (2010), 15-19.  doi: 10.4236/ica.2010.11002. [26] R. A. Weiss, How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571. [27] C. G. Wermuth, The Practice of Medicinal Chemistry, Elsevier Science, 2003.

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##### References:
 [1] AIDS INFO - U.S. Department of Health and Human Services, Offering Information On HIV/AIDS Treatment, Prevention, and Research. Understanding HIV/AIDS, Side Effects of HIV Medicines, 2017. Available from: https://aidsinfo.nih.gov. [2] D. S. Callaway and A. S. Perelson, HIV-1 infection and low steady state viral loads, Bull. Math. Biol., 64 (2002), 29-64.  doi: 10.1006/bulm.2001.0266. [3] Canada's Source for HIV and Hepatitis C Information (CATIE), HIV Treatment and an Undetectable Viral Load to Prevent HIV Transmission, 2018. Available from: www.catie.ca./reports/reports.webpage. [4] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [5] R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of $CD4^{+}$ T-cells, Math. Biosci., 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7. [6] D. C. Douek, M. Roederer and R. A. Koup, Emerging concepts in the immunopathogenesis of AIDS, Annual Rev. Medicine, 60 (2009), 471-484.  doi: 10.1146/annurev.med.60.041807.123549. [7] P. van den Driessche, Reproduction numbers of infectious disease models, Infect. Dis. Modell., 2 (2017), 288-303.  doi: 10.1016/j.idm.2017.06.002. [8] J. W. Eaton and T. B. Hallett, Why the proportion of transmission during early-stage HIV infection does not predict the long-term impact of treatment on HIV incidence, Proceedings of the National Academy of Sciences, 111, 2014, 16202â€"16207. doi: 10.1073/pnas.1323007111. [9] H. R. Erfanian and M. H. Noori Skandari, Optimal control of an HIV model, Internat. J. Appl. Math. Comput. Sci., 2 (2011), 650-658.  doi: 10.22436/jmcs.02.04.09. [10] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Applications of Mathematics, 1, Springer-Verlag, Berlin-New York, 1975. doi: 10.1007/978-1-4612-6380-7. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] D. D. Ho, A. U. Neumann, A. S. Perelson, W. Chen, J. M. Leonard and M. Markowitz, Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection, Nature, 373 (1995), 123-126.  doi: 10.1038/373123a0. [13] J. F. Hutchinson, The biology and evolution of HIV, Annual Rev. Anthropology, 30 (2001), 85-108.  doi: 10.1146/annurev.anthro.30.1.85. [14] H. R. Joshi, Optimal control of an HIV immunology model, Optimal Control Appl. Methods, 23 (2002), 199-213.  doi: 10.1002/oca.710. [15] S. E. Langford, J. Ananworanich and D. A. Cooper, Predictors of disease progression in HIV infection: A review, AIDS Res. Therapy, 4 (2007), 4-11.  doi: 10.1186/1742-6405-4-11. [16] C. C. McCluskey, A model of HIV/AIDS with staged progression and amelioration, Math. Biosci., 181 (2003), 1-16.  doi: 10.1016/S0025-5564(02)00149-9. [17] P. T. Mouofo, J. J. Tewa, B. Mewoli and S. Bowong, Optimal control of a delayed system subject to mixed control-state constraints with application to a within-host model of hepatitis virus B, Annual Rev. Control, 37 (2013), 246-259.  doi: 10.1016/j.arcontrol.2013.09.004. [18] J. M. Murray, G. Kaufmann, A. D. Kelleher and D. A. Cooper, A model of primary HIV-1 infection, Math. Biosci., 154 (1998), 57-85.  doi: 10.1016/S0025-5564(98)10046-9. [19] P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includes an intracellular delay, Math. Biosci., 163 (2000), 201-215.  doi: 10.1016/S0025-5564(99)00055-3. [20] P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179 (2002), 73-94.  doi: 10.1016/S0025-5564(02)00099-8. [21] 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. [22] D. Powell, J. Fair, R. J. Leclaire and L. M. Moore, Sensitivity analysis of an infectious disease model, International System Dynamics Conference, Boston, MA, 2005. [23] F. Rodrigues, C. J. Silva and D. F. M. Torres, Optimal control of a delayed HIV model, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 443-458.  doi: 10.3934/dcdsb.2018030. [24] H. S. Rodrigues, M. T. T. Monteiro and D. F. M. Torres, Sensitivity analysis in a dengue epidemiological model, Conference Papers in Mathematics, 2013 (2013), 7pp. doi: 10.1155/2013/721406. [25] M. Shirazian and M. H. Farahi, Optimal control strategy for a fully determined HIV model, Intelligent Control Automation, 1 (2010), 15-19.  doi: 10.4236/ica.2010.11002. [26] R. A. Weiss, How does HIV cause AIDS?, Science, 260 (1993), 1273-1279.  doi: 10.1126/science.8493571. [27] C. G. Wermuth, The Practice of Medicinal Chemistry, Elsevier Science, 2003.
Impact of time delay on dynamics of model system (1)
Simulation results of the delayed HIV model (5) with control (red line) and without control (blue line)
State variable with control and time delay (continuous curves) versus with control and without delay (dashed curves)
Sensitivity indices of $\mathcal R_{0}$
 Parameter Sensitivity index Value $\Lambda$ 0 $10^4$ $c$ $1$ $+0.3$ $\beta_1$ $+0.7557$ 0.03 $\beta_2$ $+0.2033$ 0.0384 $\beta_3$ $+0.0388$ 0.03 $\beta_4$ +0.0021 0.02 $\beta_5$ $+3.095\times10^{-8}$ 0.01 $k_1$ $+0.033$ $3\times 10^{-5}$ $k_2$ $-0.010$ $4\times 10^{-5}$ $k_3$ $-0.0012$ $10^{-5}$ $k_4$ $-1.7060\times 10^{-4}$ $10^{-5}$ $d_5$ $-2.99\times 10^{-5}$ $3.3\times 10^{-3}$
 Parameter Sensitivity index Value $\Lambda$ 0 $10^4$ $c$ $1$ $+0.3$ $\beta_1$ $+0.7557$ 0.03 $\beta_2$ $+0.2033$ 0.0384 $\beta_3$ $+0.0388$ 0.03 $\beta_4$ +0.0021 0.02 $\beta_5$ $+3.095\times10^{-8}$ 0.01 $k_1$ $+0.033$ $3\times 10^{-5}$ $k_2$ $-0.010$ $4\times 10^{-5}$ $k_3$ $-0.0012$ $10^{-5}$ $k_4$ $-1.7060\times 10^{-4}$ $10^{-5}$ $d_5$ $-2.99\times 10^{-5}$ $3.3\times 10^{-3}$
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