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Feedback control of an HBV model based on ensemble kalman filter and differential evolution

  • * Corresponding author: Hee-Dae Kwon

    * Corresponding author: Hee-Dae Kwon 
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  • In this paper, we derive efficient drug treatment strategies for hepatitis B virus (HBV) infection by formulating a feedback control problem. We introduce and analyze a dynamic mathematical model that describes the HBV infection during antiviral therapy. We determine the reproduction number and then conduct a qualitative analysis of the model using the number. A control problem is considered to minimize the viral load with consideration for the treatment costs. In order to reflect the status of patients at both the initial time and the follow-up visits, we consider the feedback control problem based on the ensemble Kalman filter (EnKF) and differential evolution (DE). EnKF is employed to estimate full information of the state from incomplete observation data. We derive a piecewise constant drug schedule by applying DE algorithm. Numerical simulations are performed using various weights in the objective functional to suggest optimal treatment strategies in different situations.

    Mathematics Subject Classification: Primary: 92B05; Secondary: 49K15.


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  • Figure 1.  The bifurcation diagram of the model system

    Figure 2.  Feedback control algorithm

    Figure 3.  Mutation step to create a mutant vector $v_i$

    Figure 4.  Crossover step to yield one of the vectors $v_i$, $u'_i$, $u''_i$ and $x_i$ as a new candidate

    Figure 5.  Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-5}$, $w_3=10^{-5}$.

    Figure 6.  Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-3}$, $w_3=10^{-3}$.

    Figure 7.  Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-1}$, $w_3=10^{-1}$.

    Figure 8.  Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-3}$, $w_3=10^{-2}$.

    Figure 9.  The difference between the total amount of $\mu_2$ and $\mu_1$ using same treatment efficacy ($\eta = \epsilon = 0.9$).

    Figure 10.  The difference between the total amount of $\mu_2$ and $\mu_1$ using various combinations of treatment efficacy assuming the total efficacy of 99%.

    Figure 11.  Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-5}$, $w_3=10^{-5}$.

    Figure 12.  Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-3}$, $w_3=10^{-3}$.

    Figure 13.  Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-1}$, $w_3=10^{-1}$.

    Table 1.  Parameters used in the model (1). They are principally extracted from Kim et al. [13].

    $S$production rate of target cells $5\times10^5$ $ \frac{cells}{mL \cdot day} $
    $d_T$death rate of target cells0.003 $ \frac{1}{day} $
    $\eta$treatment efficacy of inhibiting de novo infection $\in [0, 1]$ $ \cdot $
    $b$de novo infection rate of target cells $4\times10^{-10}$$ \frac{mL}{virions \cdot day} $
    $f $calibration coefficient of $\alpha$ for target cells0.1 $ \cdot $
    $m$mitotic production rate of infected cells0.003 $\frac{1}{day} $
    $d_I$death rate of infected cells0.043 $ \frac{1}{day} $
    $\alpha$immune effector-induced clearance rate of infected cells $7\times10^{-4}$$ \frac{mL}{cells \cdot day} $
    $\epsilon$treatment efficacy of inhibiting viral production $\in [0, 1]$ $ \cdot $
    $p$viral production rate by infected cells6.24$ \frac{virions}{cells \cdot day} $
    $c$clearance rate of free virions0.7$ \frac{1}{day} $
    $S_E$production rate of immune effectors9.33 $ \frac{cells}{mL \cdot day} $
    $B_E$maximum birth rate for immune effectors0.5 $ \frac{1}{day} $
    $K_E$Michaelis-Menten type coefficient for immune effectors $4.07\times10^5$ $ \frac{cells}{mL} $
    $D_E$death rate of immune effectors0.52 $ \frac{1}{day} $
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