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# Optimal control of a delayed HIV model

• * Corresponding author: delfim@ua.pt
• 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.

Mathematics Subject Classification: Primary: 34C60, 49K15; Secondary: 92D30.

 Citation: • • Figure 1.  Endemic equilibrium $E_2$ for the parameter values of Table 1 and time delay $\tau=0.5$

Figure 2.  State variables with time delay $\tau = 0.5$ (dashed curves) versus without delay (continuous curves)

Figure 3.  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

Figure 4.  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 $[0, 10]$

Figure 5.  State variables in the case of an intracellular delay only ($\tau = 0.5$ and $\xi=0$): controlled (dashed lines) versus uncontrolled situations (continuous lines)

Table 1.  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
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