    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:
  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. 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. 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.  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.  F. Brauer, P. Van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6.  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.  F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerospace and Electronic Systems Magazine, 20 (2005), 57-69. doi: 10.1109/MAES.2005.1499276. G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5.  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.  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. 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. 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.  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. 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. 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.  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.  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.  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. 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.  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.  K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer-Verlag, Berlin, Heidelberg, 2005. Hepatitis B Foudation, http://www.hepb.org.

show all references

References:
  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. 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. 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.  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.  F. Brauer, P. Van den Driessche and J. Wu, Mathematical Epidemiology, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78911-6.  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.  F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerospace and Electronic Systems Magazine, 20 (2005), 57-69. doi: 10.1109/MAES.2005.1499276. G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-642-03711-5.  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.  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. 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. 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.  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. 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. 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.  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.  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.  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. 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.  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.  K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, Springer-Verlag, Berlin, Heidelberg, 2005. Hepatitis B Foudation, http://www.hepb.org. 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. .
 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 $\frac{1}{day}$
  Michael Basin, Mark A. Pinsky. Control of Kalman-like filters using impulse and continuous feedback design. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 169-184. doi: 10.3934/dcdsb.2002.2.169  Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks & Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359  Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-24. doi: 10.3934/dcdsb.2018336  João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks & Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303  Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917  Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893  Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653  Judy Day, Jonathan Rubin, Gilles Clermont. Using nonlinear model predictive control to find optimal therapeutic strategies to modulate inflammation. Mathematical Biosciences & Engineering, 2010, 7 (4) : 739-763. doi: 10.3934/mbe.2010.7.739  Gregory Zitelli, Seddik M. Djouadi, Judy D. Day. Combining robust state estimation with nonlinear model predictive control to regulate the acute inflammatory response to pathogen. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1127-1139. doi: 10.3934/mbe.2015.12.1127  Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098  Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028  Alexander Bibov, Heikki Haario, Antti Solonen. Stabilized BFGS approximate Kalman filter. Inverse Problems & Imaging, 2015, 9 (4) : 1003-1024. doi: 10.3934/ipi.2015.9.1003  Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177  Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139  Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2019052  Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301  Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607  H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119  Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631  Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2019066

2017 Impact Factor: 1.23