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

Junyoung Jang; Kihoon Jang; Hee-Dae Kwon; Jeehyun Lee

doi:10.3934/mbe.2018030

Article Contents

2018, Volume 15, Issue 3: 667-691. Doi: 10.3934/mbe.2018030

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

1.
Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Republic of Korea
2.
Department of Mathematics, Inha University, 100 Inharo, Nam-gu, Incheon 22212, Republic of Korea
3.
Department of Mathematics, and Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Republic of Korea

* Corresponding author: Hee-Dae Kwon
* Corresponding author: Hee-Dae Kwon

Received: March 13, 2017

Revised: September 14, 2017

Published: June 2018

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Abstract

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.

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

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Figure 1. The bifurcation diagram of the model system

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Figure 2. Feedback control algorithm

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

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

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

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

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

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

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

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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%.

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

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

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

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

	Description	value	units
$S$	production rate of target cells	$5\times10^5$	$ \frac{cells}{mL \cdot day} $
$d_T$	death rate of target cells	0.003	$ \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 cells	0.1	$ \cdot $
$m$	mitotic production rate of infected cells	0.003	$\frac{1}{day} $
$d_I$	death rate of infected cells	0.043	$ \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 cells	6.24	$ \frac{virions}{cells \cdot day} $
$c$	clearance rate of free virions	0.7	$ \frac{1}{day} $
$S_E$	production rate of immune effectors	9.33	$ \frac{cells}{mL \cdot day} $
$B_E$	maximum birth rate for immune effectors	0.5	$ \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 effectors	0.52	$ \frac{1}{day} $

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