# American Institute of Mathematical Sciences

June 2018, 15(3): 667-691. doi: 10.3934/mbe.2018030

## 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

Received  March 13, 2017 Revised  September 14, 2017 Published  December 2017

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.

Citation: Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030
##### References:
 [1] Z. Abbas and A. R. Siddiqui, Management of hepatitis B in developing countries, World Journal of Hepatology, 3 (2011), 292-299. doi: 10.4254/wjh.v3.i12.292. [2] D. Lavanchy, Hepatitis B virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures, Journal of Viral Hepatitis, 11 (2004), 97-107. doi: 10.1046/j.1365-2893.2003.00487.x. [3] B. M. Adams, H. T. Banks, M. Davidian, Hee-Dae Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, Journal of Computational and Applied Mathematics, 184 (2005), 10-49. doi: 10.1016/j.cam.2005.02.004. [4] K. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems-series B, 11 (2009), 587-611. doi: 10.3934/dcdsb.2009.11.587. [5] F. Brauer, P. Van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6. [6] C. Castillo-Chavez, Blower, P. van den Driessche, D. Kirschner and A. -A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0065-6. [7] F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerospace and Electronic Systems Magazine, 20 (2005), 57-69. doi: 10.1109/MAES.2005.1499276. [8] G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5. [9] T. Fujimoto and R. R. Ranade, Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle, Electronic Journal of Linear Algebra, 11 (2004), 59-65. doi: 10.13001/1081-3810.1122. [10] J. Guedj, Y. Rotman, S. J. Cotler, C. Koh and P. Schmid, Understanding early serum hepatitis D virus and hepatitis B surface antigen kinetics during pegylated interferon-alpha therapy via mathematical modeling, Hepatology, 60 (2014), 1902-1910. doi: 10.1002/hep.27357. [11] L. G. Guidotti, R. Rochford, J. Chung, M. Shapiro and R. Purcell, Viral clearance without destruction of infected cells during acute HBV infection, Science, 284 (1999), 825-829. doi: 10.1126/science.284.5415.825. [12] K. Ito and K. Kunisch, Asymptotic properties of receding horizon optimal control problems, SIAM Journal on Control and Optimization, 40 (2002), 1585-1610. doi: 10.1137/S0363012900369423. [13] H. Y. Kim, H. -D. Kwon, T. S. Jang, J. Lim and H. Lee, Mathematical modeling of triphasic viral dynamics in patients with HBeAg-positive chronic hepatitis B showing response to 24-week clevudine therapy, PloS One, 7 (2012), e50377. doi: 10.1371/journal.pone.0050377. [14] S. B. Kim, M. Yoon, N. S. Ku, M. H. Kim and J. E. Song, et, al., Mathematical modeling of HIV prevention measures including pre-exposure prophylaxis on hiv incidence in south korea, PloS One, 9 (2014), e90080. doi: 10.1371/journal.pone.0090080. [15] J. Lee, J. Kim and H.-D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, Journal of Theoretical Biology, 317 (2013), 310-320. doi: 10.1016/j.jtbi.2012.10.032. [16] S. Lee, M. Golinski and G. Chowell, Modeling optimal age-specific vaccination strategies against pandemic influenza, Bulletin of Mathematical Biology, 74 (2012), 958-980. doi: 10.1007/s11538-011-9704-y. [17] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems -Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [18] N. K. Martin, P. Vickerman, G. R. Foster, S. J. Hutchinson, D. J. Goldberg and M. Hickman, Can antiviral therapy for hepatitis C reduce the prevalence of HCV among injecting drug user populations? A modeling analysis of its prevention utility, Journal of Hepatology, 54 (2011), 1137-1144. doi: 10.1016/j.jhep.2010.08.029. [19] R. Storn and K. Price, Differential evolution -a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359. doi: 10.1023/A:1008202821328. [20] R. Thimme, S. Wieland, C. Steiger, J. Ghrayeb and K. A. Reimann, CD8(+) T cells mediate viral clearance and disease pathogenesis during acute hepatitis B virus infection, J. Virol, 77 (2003), 68-76. [21] K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer-Verlag, Berlin, Heidelberg, 2005. [22] Hepatitis B Foudation, http://www.hepb.org.

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##### References:
 [1] Z. Abbas and A. R. Siddiqui, Management of hepatitis B in developing countries, World Journal of Hepatology, 3 (2011), 292-299. doi: 10.4254/wjh.v3.i12.292. [2] D. Lavanchy, Hepatitis B virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures, Journal of Viral Hepatitis, 11 (2004), 97-107. doi: 10.1046/j.1365-2893.2003.00487.x. [3] B. M. Adams, H. T. Banks, M. Davidian, Hee-Dae Kwon, H. T. Tran, S. N. Wynne and E. S. Rosenberg, HIV dynamics: Modeling, data analysis, and optimal treatment protocols, Journal of Computational and Applied Mathematics, 184 (2005), 10-49. doi: 10.1016/j.cam.2005.02.004. [4] K. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems-series B, 11 (2009), 587-611. doi: 10.3934/dcdsb.2009.11.587. [5] F. Brauer, P. Van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6. [6] C. Castillo-Chavez, Blower, P. van den Driessche, D. Kirschner and A. -A. Yakubu, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0065-6. [7] F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerospace and Electronic Systems Magazine, 20 (2005), 57-69. doi: 10.1109/MAES.2005.1499276. [8] G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5. [9] T. Fujimoto and R. R. Ranade, Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle, Electronic Journal of Linear Algebra, 11 (2004), 59-65. doi: 10.13001/1081-3810.1122. [10] J. Guedj, Y. Rotman, S. J. Cotler, C. Koh and P. Schmid, Understanding early serum hepatitis D virus and hepatitis B surface antigen kinetics during pegylated interferon-alpha therapy via mathematical modeling, Hepatology, 60 (2014), 1902-1910. doi: 10.1002/hep.27357. [11] L. G. Guidotti, R. Rochford, J. Chung, M. Shapiro and R. Purcell, Viral clearance without destruction of infected cells during acute HBV infection, Science, 284 (1999), 825-829. doi: 10.1126/science.284.5415.825. [12] K. Ito and K. Kunisch, Asymptotic properties of receding horizon optimal control problems, SIAM Journal on Control and Optimization, 40 (2002), 1585-1610. doi: 10.1137/S0363012900369423. [13] H. Y. Kim, H. -D. Kwon, T. S. Jang, J. Lim and H. Lee, Mathematical modeling of triphasic viral dynamics in patients with HBeAg-positive chronic hepatitis B showing response to 24-week clevudine therapy, PloS One, 7 (2012), e50377. doi: 10.1371/journal.pone.0050377. [14] S. B. Kim, M. Yoon, N. S. Ku, M. H. Kim and J. E. Song, et, al., Mathematical modeling of HIV prevention measures including pre-exposure prophylaxis on hiv incidence in south korea, PloS One, 9 (2014), e90080. doi: 10.1371/journal.pone.0090080. [15] J. Lee, J. Kim and H.-D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, Journal of Theoretical Biology, 317 (2013), 310-320. doi: 10.1016/j.jtbi.2012.10.032. [16] S. Lee, M. Golinski and G. Chowell, Modeling optimal age-specific vaccination strategies against pandemic influenza, Bulletin of Mathematical Biology, 74 (2012), 958-980. doi: 10.1007/s11538-011-9704-y. [17] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete and Continuous Dynamical Systems -Series B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [18] N. K. Martin, P. Vickerman, G. R. Foster, S. J. Hutchinson, D. J. Goldberg and M. Hickman, Can antiviral therapy for hepatitis C reduce the prevalence of HCV among injecting drug user populations? A modeling analysis of its prevention utility, Journal of Hepatology, 54 (2011), 1137-1144. doi: 10.1016/j.jhep.2010.08.029. [19] R. Storn and K. Price, Differential evolution -a simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359. doi: 10.1023/A:1008202821328. [20] R. Thimme, S. Wieland, C. Steiger, J. Ghrayeb and K. A. Reimann, CD8(+) T cells mediate viral clearance and disease pathogenesis during acute hepatitis B virus infection, J. Virol, 77 (2003), 68-76. [21] K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer-Verlag, Berlin, Heidelberg, 2005. [22] Hepatitis B Foudation, http://www.hepb.org.
The bifurcation diagram of the model system
Feedback control algorithm
Mutation step to create a mutant vector $v_i$
Crossover step to yield one of the vectors $v_i$, $u'_i$, $u''_i$ and $x_i$ as a new candidate
Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-5}$, $w_3=10^{-5}$.
Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-3}$, $w_3=10^{-3}$.
Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-1}$, $w_3=10^{-1}$.
Feedback controls $\mu_1$, $\mu_2$ and the corresponding states $T, V, I, E$ with $w_2=10^{-3}$, $w_3=10^{-2}$.
The difference between the total amount of $\mu_2$ and $\mu_1$ using same treatment efficacy ($\eta = \epsilon = 0.9$).
The difference between the total amount of $\mu_2$ and $\mu_1$ using various combinations of treatment efficacy assuming the total efficacy of 99%.
Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-5}$, $w_3=10^{-5}$.
Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-3}$, $w_3=10^{-3}$.
Piecewise constant controls with monthly, biweekly, and weekly measurement and piecewise continuous controls with monthly measurement using $w_2=10^{-1}$, $w_3=10^{-1}$.
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 $\frac{1}{day}$
 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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