Optimal control of double delayed HIV-1 infection model of fighting a virus with another virus

Document Type : Research Paper


Department of Mathematics, University of Malakand, Chakdara Dir (Lower) Khyber Pakhtunkhawa, Pakistan.


A double delayed- HIV-1 infection model with optimal controls is taken into account. The proposed model consists of double-time delays and the following five compartments: uninfected cells CD4+ T cells, infected CD4+ T cells, double infected CD4+ T cells, human immunodeficiency virus, and recombinant virus. Further, the optimal controls functions are introduced into the model. Objective functional is constituted which aims to (i) minimize the infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood Then, the existence and uniqueness results for the optimal control pair are established. The optimality system is derived and then solved numerically using an iterative method with Runge-Kutta fourth-order scheme.


[1] N. Ali, G. Zaman, and M. I. Chohan, Mathematical analysis of delayed HIV-1 infection model for the competition of two viruses, Coge Math Stat, 4(1) (2017), 1332821.
[2] N. Ali, G. Zaman, and M. I. Chohan, Global Stability of a Delayed HIV-1 Model with Saturations Response, Appl. Math, 11(1) (2017), 1–6.
[3] N. Ali, G. Zaman, and M. Ikhlaq Chohan, Dynamical behavior of HIV-1 epidemic model with time dependent delay, J. Math. Comput. Scie, 6(3) (2016), 377–389.
[4] S. Bonhoeffer, J. M. Coffin, and M. A. Nowak, Human immunodeficiency virus drug therapy and virus load, J. Virol, 71(4) (1997), 3275–3278.
[5] R. Culshaw, S. Ruan, and R. Spiteri, Optimal HIV treatment by maximising immune response, J. Math. Bio. 48(5) (2004), 545–562.
[6] J. H. David, H. Tran,and Banks, HIV model analysis and estimation implementation under optimal control based treatment strategies, Int. J .Pure Appl. Math, 57(3) (2009), 357–392.
[7] E. Eisele and R. F. Siliciano, Redefining the Viral Reservoirs That Prevent HIV-1 Eradication, Immunity, 3(37) (2012), 377-388.
[8] K. R. Fister, S. Lenhart, and J. S. Mc Nally, Optimizing Chemotherapy in an HIV Model, Electron. J. Diff. Eqns, 32(32) (1998), 1–12.
[9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer Verlag, New York, 1975.
[10] L. Gollmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Special Issue on Computational Methods for Optimization and Control J. Ind. Manag. Optim, 10(2) (2014), 413–441.
[11] T. Khan, G. Zaman, and M. I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, J. Biol Dyn, 11 (1) (2019), 172–189.
[12] T. Khan, G. Zaman, and M. I. Chohan, The transmission dynamic of different hepatitis Binfected individuals with the effect of hospitalization , J. Biol Dyn, 12 (1) (2018), 611–631.
[13] T. Khan, Z. Ullah, N. Ali, and G. Zaman, Modeling and control of the hepatitis B virus spreading using an epidemic model, Chao. Soli. Frac, 124 (2019), 1–9.
[14] C. Michie, A. McLean, C. Alcock, and P. Beverly, Lifespan of human lymphocyte subsets defined by cd45 isoforms, Nature, 360(6401) (1992), 264–265.
[15] G. Nolan, Harnessing viral devices as pharmaceuticals: fighting HIV-1s fire with fire, Cell, 90 (1997), 821–824.
[16] M. Nowak and C. R. Bangham, Population Dynamics of Immune Respons Persistent Viruses, Science, 272 (5258) (1996), 74–79.
[17] N. Nowak and R. May, Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford, 2000.
[18] A. Prelson and P. W. Nelson, Mathematical models of HIV dynamics in vivo, Siam Review, 4(1) (1992), 3-44.
[19] J. A. Sharp, A. P. Browning, A.P, T. Mapder, C. M. Baker, K. Burrage, and M. J. Simpson, Designing combination therapies using multiple optimal controls, J. Theo. Biol, 497 (2020), 110277.
[20] D. Tully, Optimizing Chemotherapy in an HIV Model, College of the Redwoods, Springer, 1999.
[21] W. Wodarz and M. A. Nowak, Specific therapy regimes could lead to long-term immunological control of HIV, Proc. Natl. Acad. Sci.96, 95(25) (1999), 14464–14469.
[22] X. Zhou, X. Song, and X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate, J. Math. Anal. Appl, 342(2) (2008), 1342–1355.