# American Institute of Mathematical Sciences

January  2018, 23(1): 443-458. doi: 10.3934/dcdsb.2018030

## Optimal control of a delayed HIV model

 1 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal 2 Institute of Computational and Applied Mathematics, University of Münster, D-48149 Münster, Germany

* Corresponding author: delfim@ua.pt

Received  July 2016 Published  January 2018

We propose a model for the human immunodeficiency virus type 1 (HIV-1) infection with intracellular delay and prove the local asymptotical stability of the equilibrium points. Then we introduce a control function representing the efficiency of reverse transcriptase inhibitors and consider the pharmacological delay associated to the control. Finally, we propose and analyze an optimal control problem with state and control delays. Through numerical simulations, extremal solutions are proposed for minimization of the virus concentration and treatment costs.

Citation: Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
##### References:

show all references

##### References:
Endemic equilibrium $E_2$ for the parameter values of Table 1 and time delay $\tau=0.5$
State variables with time delay $\tau = 0.5$ (dashed curves) versus without delay (continuous curves)
Bang-bang control $c(t)$ (20) (continuous curve) and switching function $\phi$ (18) matching the control law (19). Zoom into a neighborhood of the switching time $t_s$: (left) Case 1, (middle) Case 2, (right) Case 3
A comparison of state trajectories in Case 1 (no delays, dashed line) and Case 3 (delays $\tau = 0.5$ and $\xi = 0.2$, continuous line). (left) zoom of infected cells $I(t)$ into $[0, 5]$, (middle) zoom of free virus particles $V(t)$ into $[0, 10]$, (right) zoom of CTL cells $T(t)$ into $[0, 10]$
State variables in the case of an intracellular delay only ($\tau = 0.5$ and $\xi=0$): controlled (dashed lines) versus uncontrolled situations (continuous lines)
Parameter values
 Parameter Value Units λ 5 day-1mm-3 m 0.03 day-1 r 0.0014 mm3virion-1day-1 u 0.32 day-1 s 0.05 mm3day-1 k 153.6 day-1 v 1 day-1 a 0.2 mm3day-1 n 0.3 day-1 tf 50 day τ 0.5 day ξ 0.2 day
 Parameter Value Units λ 5 day-1mm-3 m 0.03 day-1 r 0.0014 mm3virion-1day-1 u 0.32 day-1 s 0.05 mm3day-1 k 153.6 day-1 v 1 day-1 a 0.2 mm3day-1 n 0.3 day-1 tf 50 day τ 0.5 day ξ 0.2 day
 [1] Shengqiang Liu, Lin Wang. Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy. Mathematical Biosciences & Engineering, 2010, 7 (3) : 675-685. doi: 10.3934/mbe.2010.7.675 [2] Rachid Ouifki, Gareth Witten. A model of HIV-1 infection with HAART therapy and intracellular delays. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 229-240. doi: 10.3934/dcdsb.2007.8.229 [3] Laurenz Göllmann, Helmut Maurer. Theory and applications of optimal control problems with multiple time-delays. Journal of Industrial & Management Optimization, 2014, 10 (2) : 413-441. doi: 10.3934/jimo.2014.10.413 [4] Laurenz Göllmann, Helmut Maurer. Optimal control problems with time delays: Two case studies in biomedicine. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1137-1154. doi: 10.3934/mbe.2018051 [5] 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 [6] Jinliang Wang, Lijuan Guan. Global stability for a HIV-1 infection model with cell-mediated immune response and intracellular delay. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 297-302. doi: 10.3934/dcdsb.2012.17.297 [7] Zhaohua Gong, Chongyang Liu, Yujing Wang. Optimal control of switched systems with multiple time-delays and a cost on changing control. Journal of Industrial & Management Optimization, 2018, 14 (1) : 183-198. doi: 10.3934/jimo.2017042 [8] Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343 [9] Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021 [10] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [11] Haitao Song, Weihua Jiang, Shengqiang Liu. Virus dynamics model with intracellular delays and immune response. Mathematical Biosciences & Engineering, 2015, 12 (1) : 185-208. doi: 10.3934/mbe.2015.12.185 [12] Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007 [13] Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101 [14] Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639 [15] Cristiana J. Silva, Delfim F. M. Torres. Modeling and optimal control of HIV/AIDS prevention through PrEP. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 119-141. doi: 10.3934/dcdss.2018008 [16] B. M. Adams, H. T. Banks, Hee-Dae Kwon, Hien T. Tran. Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches. Mathematical Biosciences & Engineering, 2004, 1 (2) : 223-241. doi: 10.3934/mbe.2004.1.223 [17] Jaouad Danane, Karam Allali. Optimal control of an HIV model with CTL cells and latently infected cells. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019048 [18] M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743 [19] Xia Wang, Shengqiang Liu, Libin Rong. Permanence and extinction of a non-autonomous HIV-1 model with time delays. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1783-1800. doi: 10.3934/dcdsb.2014.19.1783 [20] Wei Feng. Dynamics in 30species preadtor-prey models with time delays. Conference Publications, 2007, 2007 (Special) : 364-372. doi: 10.3934/proc.2007.2007.364

2018 Impact Factor: 1.008