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October  2018, 15(5): 1137-1154. doi: 10.3934/mbe.2018051

Optimal control problems with time delays: Two case studies in biomedicine

1. 

Münster University of Applied Sciences, Department of Mechanical Engineering, Stegerwaldstr. 39, 48565 Steinfurt, Germany

2. 

University of Münster, Applied Mathematics: Institute of Analysis and Numerics, Einsteinstr. 62, 49143 Münster, Germany

* Corresponding author: H. Maurer, maurer@math.uni-muenster.de

Received  April 30, 2017 Accepted  March 18, 2018 Published  May 2018

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming $ L^1$-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For $ L^1$-type objectives we obtain extremal controls of bang-bang type.

Citation: Laurenz Göllmann, Helmut Maurer. Optimal control problems with time delays: Two case studies in biomedicine. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1137-1154. doi: 10.3934/mbe.2018051
References:
[1]

B. Buonomo and M. Cerasuolo, The effect of time delay in plant-pathogen interactions with host demography, Math. Biosciences and Engineering, 12 (2015), 473-490.  doi: 10.3934/mbe.2015.12.473.  Google Scholar

[2]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Westfälische Wilhelms-Universität Münster, Germany, 1998. Google Scholar

[3]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math., 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[4]

C. Büskens and M. Gerdts, WORHP: Large-Scale Sparse Nonlinear Optimization Solver, http://www.worhp.de. Google Scholar

[5]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A class of optimal state-delay control Problems, Nonlinear Analysis: Real World Applications, 14 (2013), 1536-1550.  doi: 10.1016/j.nonrwa.2012.10.017.  Google Scholar

[6]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Mathematical Biosciences, 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[7]

S. EikenberryS. HewsJ. D. Nagy and Y. Kuang, The dynamics of a delay model of Hepatitis B virus infection with logistic hepatocyte growth, Mathematical Biosciences, 6 (2009), 283-299.  doi: 10.3934/mbe.2009.6.283.  Google Scholar

[8]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for MathematicalProgramming, The Scientific Press, South San Francisco, California, 1993. Google Scholar

[9]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control and mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar

[10]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.   Google Scholar

[11]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems, Journal of Optimization Theory and Applications, 18 (1976), 371-377.  doi: 10.1007/BF00933818.  Google Scholar

[12]

R. F. HartlS. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218.  doi: 10.1137/1037043.  Google Scholar

[13]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.  Google Scholar

[14]

S. C. Huang, Optimal Control problems with retardations and restricted phase coordinates, Journal of Optimization Theory and Applications, 3 (1969), 316-360.  doi: 10.1007/BF00931371.  Google Scholar

[15]

J. KlamkaH. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Mathematical Biosciences and Engineering, 14 (2017), 195-216.  doi: 10.3934/mbe.2017013.  Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.  Google Scholar

[17]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications, 26 (2005), 129-156.  doi: 10.1002/oca.756.  Google Scholar

[18]

R. M. May, Time-delay versus stability in population models with two and three tropic levels, Ecology, 54 (1973), 315-325.   Google Scholar

[19]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[20]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translation by K. N. Trirogoff, Wiley, New York, 1962.  Google Scholar

[21]

F. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim and M. A. Abdeen, Delay differential model for tumour-immune-response with chemoimmunotherapy and optimal control. Computational and Mathematical Methods in Medicine, Hindawi Publishing Corporation, Vol. 2014, Article ID 982978, (2014). doi: 10.1155/2014/982978.  Google Scholar

[22]

H. SchättlerU. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with L2-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.  Google Scholar

[23]

C. SilvaH. Maurer and D.F.M. Torres, Optimal control of a tuberculosis model with state and control delays, Mathematical Biosciences and Engineering, 14 (2017), 321-337.  doi: 10.3934/mbe.2017021.  Google Scholar

[24]

C. T. SreeramareddyK. V. PanduruJ. Menten and J. V. den Ende, Time delays in diagnosis of pulmonary tuberculosis: A systematic review of literature, BMC Infectious Diseases, 9 (2009), 91-100.  doi: 10.1186/1471-2334-9-91.  Google Scholar

[25]

J. Stoer and R. Bulirsch, Introduction ot Numerical Analysis, Third Edition, Texts in Applied Mathematics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-22250-8.  Google Scholar

[26]

D. G. StorlaS. Yimer and G. A. Bjune, A systematic review in delay in the diagnosis and treatment of tuberculosis, BMC Public Health, 8 (2008), p15.  doi: 10.1186/1471-2458-8-15.  Google Scholar

[27]

P. van den Driessche, Some Epidemiological Models with Delays, Report DMS-679-IR, University of Victoria, Department of Mathematics, 1994. Google Scholar

[28]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[29]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.  Google Scholar

show all references

References:
[1]

B. Buonomo and M. Cerasuolo, The effect of time delay in plant-pathogen interactions with host demography, Math. Biosciences and Engineering, 12 (2015), 473-490.  doi: 10.3934/mbe.2015.12.473.  Google Scholar

[2]

C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustands-Beschränkungen, PhD thesis, Institut für Numerische Mathematik, Westfälische Wilhelms-Universität Münster, Germany, 1998. Google Scholar

[3]

C. Büskens and H. Maurer, SQP methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. Comput. Appl. Math., 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8.  Google Scholar

[4]

C. Büskens and M. Gerdts, WORHP: Large-Scale Sparse Nonlinear Optimization Solver, http://www.worhp.de. Google Scholar

[5]

Q. ChaiR. LoxtonK. L. Teo and C. Yang, A class of optimal state-delay control Problems, Nonlinear Analysis: Real World Applications, 14 (2013), 1536-1550.  doi: 10.1016/j.nonrwa.2012.10.017.  Google Scholar

[6]

R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4+ T-cells, Mathematical Biosciences, 165 (2000), 27-39.  doi: 10.1016/S0025-5564(00)00006-7.  Google Scholar

[7]

S. EikenberryS. HewsJ. D. Nagy and Y. Kuang, The dynamics of a delay model of Hepatitis B virus infection with logistic hepatocyte growth, Mathematical Biosciences, 6 (2009), 283-299.  doi: 10.3934/mbe.2009.6.283.  Google Scholar

[8]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for MathematicalProgramming, The Scientific Press, South San Francisco, California, 1993. Google Scholar

[9]

L. GöllmannD. Kern and H. Maurer, Optimal control problems with delays in state and control and mixed control-state constraints, Optimal Control Applications and Methods, 30 (2009), 341-365.  doi: 10.1002/oca.843.  Google Scholar

[10]

L. Göllmann and H. Maurer, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and Management Optimization, 10 (2014), 413-441.   Google Scholar

[11]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems, Journal of Optimization Theory and Applications, 18 (1976), 371-377.  doi: 10.1007/BF00933818.  Google Scholar

[12]

R. F. HartlS. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints, SIAM Review, 37 (1995), 181-218.  doi: 10.1137/1037043.  Google Scholar

[13]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory, John Wiley, New York, 1966.  Google Scholar

[14]

S. C. Huang, Optimal Control problems with retardations and restricted phase coordinates, Journal of Optimization Theory and Applications, 3 (1969), 316-360.  doi: 10.1007/BF00931371.  Google Scholar

[15]

J. KlamkaH. Maurer and A. Swierniak, Local controllability and optimal control for a model of combined anticancer therapy with control delays, Mathematical Biosciences and Engineering, 14 (2017), 195-216.  doi: 10.3934/mbe.2017013.  Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, San Diego, 1993.  Google Scholar

[17]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optimal Control Methods and Applications, 26 (2005), 129-156.  doi: 10.1002/oca.756.  Google Scholar

[18]

R. M. May, Time-delay versus stability in population models with two and three tropic levels, Ecology, 54 (1973), 315-325.   Google Scholar

[19]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, Vol. DC 24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.  Google Scholar

[20]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translation by K. N. Trirogoff, Wiley, New York, 1962.  Google Scholar

[21]

F. Rihan, D. H. Abdelrahman, F. Al-Maskari, F. Ibrahim and M. A. Abdeen, Delay differential model for tumour-immune-response with chemoimmunotherapy and optimal control. Computational and Mathematical Methods in Medicine, Hindawi Publishing Corporation, Vol. 2014, Article ID 982978, (2014). doi: 10.1155/2014/982978.  Google Scholar

[22]

H. SchättlerU. Ledzewicz and H. Maurer, Sufficient conditions for strong local optimality in optimal control problems with L2-type objectives and control constraints, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2657-2679.  doi: 10.3934/dcdsb.2014.19.2657.  Google Scholar

[23]

C. SilvaH. Maurer and D.F.M. Torres, Optimal control of a tuberculosis model with state and control delays, Mathematical Biosciences and Engineering, 14 (2017), 321-337.  doi: 10.3934/mbe.2017021.  Google Scholar

[24]

C. T. SreeramareddyK. V. PanduruJ. Menten and J. V. den Ende, Time delays in diagnosis of pulmonary tuberculosis: A systematic review of literature, BMC Infectious Diseases, 9 (2009), 91-100.  doi: 10.1186/1471-2334-9-91.  Google Scholar

[25]

J. Stoer and R. Bulirsch, Introduction ot Numerical Analysis, Third Edition, Texts in Applied Mathematics, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-22250-8.  Google Scholar

[26]

D. G. StorlaS. Yimer and G. A. Bjune, A systematic review in delay in the diagnosis and treatment of tuberculosis, BMC Public Health, 8 (2008), p15.  doi: 10.1186/1471-2458-8-15.  Google Scholar

[27]

P. van den Driessche, Some Epidemiological Models with Delays, Report DMS-679-IR, University of Victoria, Department of Mathematics, 1994. Google Scholar

[28]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

[29]

H. Yang and J. Wei, Global behaviour of a delayed viral kinetic model with general incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1573-1582.  doi: 10.3934/dcdsb.2015.20.1573.  Google Scholar

Figure 1.  Optimal solution of the non-delayed control problem with $\tau_1 = \tau_2 = 0$ and weights $B_1 = 5, B_2 = 10$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) effector cells $E(t)$, (c) tumour cells $T(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) healthy cells $N(t)$, (c) cytostatic agent $U(t)$.
Figure 2.  Optimal controls $u_k(t)$ and switching functions $\phi_k(t), \,(k = 1,2)$ in a neighborhood of the switching times $t_k$ illustrating the control-law (35) and the strict bang-bang property (37).
Figure 3.  Optimal controls $u_1(t)$ and $u_2(t)$ for functionals $J_p(p,u), p = 1,2,$ with weights $B_1 = 5, B_2 = 10$.
Figure 4.  Optimal solution of the delayed control problem with state delay $\tau_1 = 1.5$, control delay $\tau_2 = 3.0$ and weights $B_1 = 5, B_2 = 10$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) effector cells $E(t)$, (c) tumour cells $T(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) healthy cells $N(t)$, (c) cytostatic agent $U(t)$.
Figure 5.  Delayed solution with $\tau_1 = 1.5$ and $\tau_2 = 3.0$: controls $u_k(t)$ and switching functions $\phi_k(t), \,(k = 1,2)$ in a neighborhood of the switching times $t_k$ illustrating the control-law (35) and the strict bang-bang property (37).
Figure 6.  Optimal controls $u_1(t)$ and $u_2(t)$ for functionals $J_1(x,u)$ and $J_2(x,u)$ with delays $\tau_1 = 1.5, \tau_2 = 3.0$ and weights $B_1 = 5, B_2 = 10$.
Figure 7.  Optimal solution of the delayed control problem with state delay $\tau_1 = 1.5$, control delay $\tau_2 = 3.0$ and mixed control-state constraint $U(t) + u_2(t) \leq 3$. Top row: (a) dose control $u_1(t)$ of chemotherapy, (b) function $U(t)+u_2(t)$, (c) effector cells $E(t)$. Bottom row: (a) dose control $u_2(t)$ of immune therapy, (b) multiplier $\mu(t)$ for mixed constraint, (c) tumour cells $T(t)$.
Figure 8.  Controls and switching functions (51) for delays $\tau = 0$, $\tau = 10$ and $\tau = 15$. For all delays the control law (52) is satisfied and the strict bang-bang property holds.
Figure 9.  Comparison of state variables for delays $\tau = 0, 10, 15$. Top row: (a) healthy cells $x$, (b) exposed cells $p$. Bottom row: (a) infected cells $y$, (b) free virions $v$.
Table 1.  Parameters in the control problem of chemo-immunotherapy [21].
Parameter Description Value
$t_f$ final time $30$ d (days)
$\tau_1$ state delay $1.5$ d
$\tau_2$ control delay $3.0$ d
$(u_{k,\min},u_{k,\max})$ control bounds $(0, 1)$ for $\,k=1,2$
$(a_1,\,a_2,\,a_3)$ cell kill rate response $(0.2,\,0.4,\,0.1)$
$(\beta,\, \beta_2)$ reciprocal carrying capacities of tumour and host cells $(0.002,\,1.0)$
$(c_1,\, c_2)$ scaling parameters $(3\times 10^{-5},\,3\times 10^{-8})$
$d_1$ drug decay rate $0.01$
$\delta$ immune cell death rate $0.2$
$\eta$ steepness of immune response $0.3$
$\mu_e$ uninfected effector cell decrease rate $0.003611$
$(\sigma,\,\rho)$ immune cell influx and decay rate resp. $(0.2,\,0.2)$
$(s_1,\, r_2,\, r_3)$ cell growth rates $(0.3,\,1.03,\,1.0)$
$n_T$ immune effector cell decrease rate $1.0$
$(B_1,B_2) $ weights $ (5,\,10)$
Parameter Description Value
$t_f$ final time $30$ d (days)
$\tau_1$ state delay $1.5$ d
$\tau_2$ control delay $3.0$ d
$(u_{k,\min},u_{k,\max})$ control bounds $(0, 1)$ for $\,k=1,2$
$(a_1,\,a_2,\,a_3)$ cell kill rate response $(0.2,\,0.4,\,0.1)$
$(\beta,\, \beta_2)$ reciprocal carrying capacities of tumour and host cells $(0.002,\,1.0)$
$(c_1,\, c_2)$ scaling parameters $(3\times 10^{-5},\,3\times 10^{-8})$
$d_1$ drug decay rate $0.01$
$\delta$ immune cell death rate $0.2$
$\eta$ steepness of immune response $0.3$
$\mu_e$ uninfected effector cell decrease rate $0.003611$
$(\sigma,\,\rho)$ immune cell influx and decay rate resp. $(0.2,\,0.2)$
$(s_1,\, r_2,\, r_3)$ cell growth rates $(0.3,\,1.03,\,1.0)$
$n_T$ immune effector cell decrease rate $1.0$
$(B_1,B_2) $ weights $ (5,\,10)$
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