American Institute of Mathematical Sciences

Advanced
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

Junyoung Jang 1, , Kihoon Jang 2, , Hee-Dae Kwon 2,, and  Jeehyun Lee 3,

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.

Keywords: Feedback control, HBV, model predictive control, ensemble Kalman filter, differential evolution.
Mathematics Subject Classification: Primary: 92B05; Secondary: 49K15.
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.

show all references

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.

Figure 1.  The bifurcation diagram of the model system
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Feedback control algorithm
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download as excel

Descriptionvalueunits
$S$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} $
[1]

Wawan Hafid Syaifudin, Endah R. M. Putri. The application of model predictive control on stock portfolio optimization with prediction based on Geometric Brownian Motion-Kalman Filter. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021119
[2]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial and Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099
[3]

Andreas Bock, Colin J. Cotter. Learning landmark geodesics using the ensemble Kalman filter. Foundations of Data Science, 2021, 3 (4) : 701-727. doi: 10.3934/fods.2021020
[4]

Lars Grüne, Marleen Stieler. Multiobjective model predictive control for stabilizing cost criteria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3905-3928. doi: 10.3934/dcdsb.2018336
[5]

Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks and Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359
[6]

Torsten Trimborn, Lorenzo Pareschi, Martin Frank. Portfolio optimization and model predictive control: A kinetic approach. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6209-6238. doi: 10.3934/dcdsb.2019136
[7]

Michael Basin, Mark A. Pinsky. Control of Kalman-like filters using impulse and continuous feedback design. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 169-184. doi: 10.3934/dcdsb.2002.2.169
[8]

João M. Lemos, Fernando Machado, Nuno Nogueira, Luís Rato, Manuel Rijo. Adaptive and non-adaptive model predictive control of an irrigation channel. Networks and Heterogeneous Media, 2009, 4 (2) : 303-324. doi: 10.3934/nhm.2009.4.303
[9]

Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917
[10]

Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893
[11]

Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653
[12]

Luís Tiago Paiva, Fernando A. C. C. Fontes. Sampled–data model predictive control: Adaptive time–mesh refinement algorithms and guarantees of stability. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2335-2364. doi: 10.3934/dcdsb.2019098
[13]

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
[14]

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
[15]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control and Related Fields, 2021, 11 (3) : 579-599. doi: 10.3934/mcrf.2021013
[16]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations and Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[17]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[18]

Alexander Bibov, Heikki Haario, Antti Solonen. Stabilized BFGS approximate Kalman filter. Inverse Problems and Imaging, 2015, 9 (4) : 1003-1024. doi: 10.3934/ipi.2015.9.1003
[19]

Russell Johnson, Carmen Núñez. The Kalman-Bucy filter revisited. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4139-4153. doi: 10.3934/dcds.2014.34.4139
[20]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (350)
  • HTML views (626)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy