# American Institute of Mathematical Sciences

January  2018, 23(1): 79-99. doi: 10.3934/dcdsb.2018006

## Optimal control of normalized SIMR models with vaccination and treatment

 1 SYSTEC, DEEC, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal 2 Universität Münster, Institut für Numerische und Angewandte Mathematik, Einsteinstr. 62,49143 Münster, Germany 3 ENSTA-Paris Tech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France

* Corresponding author: MdR de Pinho, mrpinho@fe.up.pt

Received  December 2016 Published  January 2018

We study a model based on the so called SIR model to control the spreading of a disease in a varying population via vaccination and treatment. Since we assume that medical treatment is not immediate we add a new compartment, $M$, to the SIR model. We work with the normalized version of the proposed model. For such model we consider the problem of steering the system to a specified target. We consider both a fixed time optimal control problem with $L^1$ cost and the minimum time problem to drive the system to the target. In contrast to the literature, we apply different techniques of optimal control to our problems of interest.Using the direct method, we first solve the fixed time problem and then proceed to validate the computed solutions using both necessary conditions and second order sufficient conditions. Noteworthy, we perform a sensitivity analysis of the solutions with respect to some parameters in the model. We also use the Hamiltonian Jacobi approach to study how the minimum time function varies with respect to perturbations of the initial conditions. Additionally, we consider a multi-objective approach to study the trade off between the minimum time and the social costs of the control of diseases. Finally, we propose the application of Model Predictive Control to deal with uncertainties of the model.

Citation: Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006
##### References:
 [1] M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic orderconditions for bang-singular extremals, Numerical Algebra, Control and Optimization, 2 (2012), 511-546.  doi: 10.3934/naco.2012.2.511. [2] M. Assellaou, O. Bokanowski, A. Desilles and H. Zidani, A Hamilton-Jacobi-Bellman Approach for the Optimal Control of an Abort Landing Problem 55th IEEE Conference on Decision and Control, 2016. doi: 10.1109/CDC.2016.7798815. [3] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [4] M. H. A. Biswas, L. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761. [5] O. Bokanowski, A. Briani and H. Zidani, Minumum time control problems for non autonomous differential equations, Systems and Control Letters, 58 (2009), 742-746.  doi: 10.1016/j.sysconle.2009.08.003. [6] O. Bokanowski, A. Désilles, H. Zidani and J. Zhao, ROC-HJ software "Reachability, Optimal Control, and Hamilton-Jacobi equations, http://uma.ensta-paristech.fr/soft/ROC-HJ(2016). [7] O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM Journal on Control and Optimization, 48 (2010), 4292-4316.  doi: 10.1137/090762075. [8] H. Bonnel and Y. C. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), 93-112.  doi: 10.1007/s10957-010-9709-y. [9] C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. [10] 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, Journal of Computational and Applied Mathematics, 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8. [11] C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-Verlag, Berlin, (2001), 3-16. [12] C. Büskens and H. Maurer, Sensitivity analysis and real--time control of parametric optimal control problems using nonlinear programming methods, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-erlag, Berlin, (2001), 57-68. [13] M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: Pros and cons, Mathematical Biosciences and Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008. [14] G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization Springer, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-79159-1. [15] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations SIAM, Philadelphia, PA, 2014. [16] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis Academic Press, New York, London, 1983. [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Brooks-Cole Publishing Company, Duxbury Press, 1993. [18] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms Springer, 2011. doi: 10.1007/978-0-85729-501-9. [19] M. R. Hestenes, Calculus of Variations and Optimal Control Theory John Wiley & Sons, Inc., New York-London-Sydney, 1966. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [21] H. W. Hethcote, The basic epidemiology models: Models, expressions for $R_0$, parameter estimation, and applications, In:Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), 16 (2009), 1-61.  doi: 10.1142/9789812834836_0001. [22] Y. C. Kaya and H. Maurer, A numerical method for nonconvex multi-objective optimal control problems, Computational Optimization and Applications, 57 (2014), 685-702.  doi: 10.1007/s10589-013-9603-2. [23] A. J. Krener, The high order Maximal Principle and its application to singular extremals, SIAM J.on Control and Optimization, 15 (1977), 256-293.  doi: 10.1137/0315019. [24] U. Ledzewicz and H. Schättler, On optimal singular control for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, 2 (2011), 981-990. [25] H. Maurer and M. d. R. de Pinho, Optimal control of epdimiological SEIR models with $L^1$ objective and control-state constraints, Pacific Journal of Optimization, 12 (2016), 415-436. [26] H. Maurer, C. Büskens, J.-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. [27] R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. [28] 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 Advances in Design and Control, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368. [29] L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes (in Russian), Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964. [30] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439-471.  doi: 10.1016/0021-9991(88)90177-5. [31] A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.

show all references

##### References:
 [1] M. S. Aronna, J. F. Bonnans, A. V. Dmitruk and P. A. Lotito, Quadratic orderconditions for bang-singular extremals, Numerical Algebra, Control and Optimization, 2 (2012), 511-546.  doi: 10.3934/naco.2012.2.511. [2] M. Assellaou, O. Bokanowski, A. Desilles and H. Zidani, A Hamilton-Jacobi-Bellman Approach for the Optimal Control of an Abort Landing Problem 55th IEEE Conference on Decision and Control, 2016. doi: 10.1109/CDC.2016.7798815. [3] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Systems and Control: Foundations and Applications. Birkhäuser, Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [4] M. H. A. Biswas, L. T. Paiva and M. d. R. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014), 761-784.  doi: 10.3934/mbe.2014.11.761. [5] O. Bokanowski, A. Briani and H. Zidani, Minumum time control problems for non autonomous differential equations, Systems and Control Letters, 58 (2009), 742-746.  doi: 10.1016/j.sysconle.2009.08.003. [6] O. Bokanowski, A. Désilles, H. Zidani and J. Zhao, ROC-HJ software "Reachability, Optimal Control, and Hamilton-Jacobi equations, http://uma.ensta-paristech.fr/soft/ROC-HJ(2016). [7] O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption, SIAM Journal on Control and Optimization, 48 (2010), 4292-4316.  doi: 10.1137/090762075. [8] H. Bonnel and Y. C. Kaya, Optimization over the efficient set of multi-objective convex optimal control problems, J. Optim. Theory Appl., 147 (2010), 93-112.  doi: 10.1007/s10957-010-9709-y. [9] C. Büskens, Optimierungsmethoden und Sensitivitätsanalyse Für Optimale Steuerprozesse mit Steuer-und Zustands-Beschränkungen Dissertation, Institut für Numerische Mathematik, Universität Münster, Germany, 1998. [10] 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, Journal of Computational and Applied Mathematics, 120 (2000), 85-108.  doi: 10.1016/S0377-0427(00)00305-8. [11] C. Büskens and H. Maurer, Sensitivity analysis and real-time optimization of parametric nonlinear programming problems, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-Verlag, Berlin, (2001), 3-16. [12] C. Büskens and H. Maurer, Sensitivity analysis and real--time control of parametric optimal control problems using nonlinear programming methods, in: Online Optimization of Large Scale Systems (M. Grötschel, S. O. Krumke, J. Rambau, eds.), Springer-erlag, Berlin, (2001), 57-68. [13] M. d. R. de Pinho and F. N. Nogueira, On application of optimal control to SEIR normalized models: Pros and cons, Mathematical Biosciences and Engineering, 14 (2017), 111-126.  doi: 10.3934/mbe.2017008. [14] G. Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization Springer, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-79159-1. [15] M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations SIAM, Philadelphia, PA, 2014. [16] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis Academic Press, New York, London, 1983. [17] R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Brooks-Cole Publishing Company, Duxbury Press, 1993. [18] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms Springer, 2011. doi: 10.1007/978-0-85729-501-9. [19] M. R. Hestenes, Calculus of Variations and Optimal Control Theory John Wiley & Sons, Inc., New York-London-Sydney, 1966. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907. [21] H. W. Hethcote, The basic epidemiology models: Models, expressions for $R_0$, parameter estimation, and applications, In:Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), 16 (2009), 1-61.  doi: 10.1142/9789812834836_0001. [22] Y. C. Kaya and H. Maurer, A numerical method for nonconvex multi-objective optimal control problems, Computational Optimization and Applications, 57 (2014), 685-702.  doi: 10.1007/s10589-013-9603-2. [23] A. J. Krener, The high order Maximal Principle and its application to singular extremals, SIAM J.on Control and Optimization, 15 (1977), 256-293.  doi: 10.1137/0315019. [24] U. Ledzewicz and H. Schättler, On optimal singular control for a general SIR-model with vaccination and treatment, Discrete and Continuous Dynamical Systems, 2 (2011), 981-990. [25] H. Maurer and M. d. R. de Pinho, Optimal control of epdimiological SEIR models with $L^1$ objective and control-state constraints, Pacific Journal of Optimization, 12 (2016), 415-436. [26] H. Maurer, C. Büskens, J.-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. [27] R. M. Neilan and S. Lenhart, An introduction to optimal control with an application in disease modeling, DIMACS Series in Discrete Mathematics, 75 (2010), 67-81. [28] 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 Advances in Design and Control, 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2012. doi: 10.1137/1.9781611972368. [29] L. S. Pontryagin, V. G. Boltyanski, R. V. Gramkrelidze and E. F. Miscenko, The Mathematical Theory of Optimal Processes (in Russian), Fitzmatgiz, Moscow; English translation: Pergamon Press, New York, 1964. [30] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439-471.  doi: 10.1016/0021-9991(88)90177-5. [31] A. Wächter and L. T. Biegler, On the implementation of a primal-dual interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.
SIMR compartmental model with treatment and vaccination.
Case 1: state variables for $(P_1)$ with parameters as in Table 1.
Case 1 with parameters as in Table 1. (left) comparison between the computed values $u_*(t)$ and the values of the singular control $u_{sing}(x_*(t),p_*(t))$ (19), (right) control $v_*(t)$ and switching function $\phi_2$ showing that the control law (14) for $v_*$ is precisely satisfied.
Optimal controls for $(P_1)$. Left: Control $u_*(t)$ for Case 1 in red and, for Case 2, control $u_*(t)$ in blue and its switching function $\phi_1(t)$ in dashed blue. Right: Control $v_*(t)$ for Case 1 in red and, for Case 2, control $v_*(t)$ in blue and its switching function $\phi_2(t)$ in dashed blue,
Case 1: controls for $(P_1)$ with parameters as in Table 1.
Value of the minimum time function for different values of $(s_0,i_0)$. The vertical axis corresponds to the values of the minimum time.
State variables for optimal trajectories corresponding to different values of the initial position $(s_0,i_0,m_0)$.
Case 1 :$A=10, B=1, C=3$. Minimize the scalarized objective $J^{(w)}$ (23). Top row: (left) Pareto front $(J_1,T)$ for $w \in [0,1]$, (right) objective $J_1$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.
Case 2 : $A=10, B=3, C=1$. Minimize the scalarized objective $J^{(w)}$ (23) for $w \in [0,1]$. Top row: (left) Pareto front $(J_1,J_2)$ for $w \in [0,1]$, (right) objective $J_1(u,v)$ as function of $w \in [0,1]$. Bottom row: (left) Distance of Pareto curve to the origin, (right) objective $J_2=T$ as function of $w \in [0,1]$.
Perturbed incidence rate $c_p$.
Six computed values of the state variables for the Fixed Horizon MPC described above assuming that the "real" system differs from the model of $(P_1)$ solely with respect to the incidence rate $c$. From left to right on the first row, we show the $s$ and $i$ state variables; on the second row we show the $m$ state variable. All the three graphs have different scales.
Parameters for the normalized SIMR model (7)-(10)
 Parameter Description Value $b$ Natural birth rate 0.01 $c$ Incidence coefficient 1.1 $a_1$ Infection induced death rate 0.08 $a_2$ Treatment induced death rate 0.005 $g_1$ Recovery rate of those infected 0.02 $g_2$ Recovery rate of those under treatment 0.5 $\eta$ Efficiency of vaccine 0.8 $u_{max}$ Maximum rate of vaccination 0.8 $v_{max}$ Maximum rate of treatment 0.8 $s_0$ Initial percentage of susceptible population 0.95 $i_0$ Initial percentage of infected population 0.05 $m_0$ Initial percentage of infected population under treatment 0
 Parameter Description Value $b$ Natural birth rate 0.01 $c$ Incidence coefficient 1.1 $a_1$ Infection induced death rate 0.08 $a_2$ Treatment induced death rate 0.005 $g_1$ Recovery rate of those infected 0.02 $g_2$ Recovery rate of those under treatment 0.5 $\eta$ Efficiency of vaccine 0.8 $u_{max}$ Maximum rate of vaccination 0.8 $v_{max}$ Maximum rate of treatment 0.8 $s_0$ Initial percentage of susceptible population 0.95 $i_0$ Initial percentage of infected population 0.05 $m_0$ Initial percentage of infected population under treatment 0
 Numerical results for $(P_1)$ with $A=10, B=1, C=3 : {\text{Cost}} \ J_1= 22.02571 ,$ $s(T)= 1.201593 e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 8.932 084 e-06$ $p_s(0)= -6.3486 ,$ $p_i(0) = -7.6544e+01,$ $p_m(0) = -6.2821e-02,$ $p_s(T)= -3.2237e-03,$ $p_i(T) = -5.8613e+03,$ $p_m(T) = -1.4648e-05 .$
 Numerical results for $(P_1)$ with $A=10, B=1, C=3 : {\text{Cost}} \ J_1= 22.02571 ,$ $s(T)= 1.201593 e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 8.932 084 e-06$ $p_s(0)= -6.3486 ,$ $p_i(0) = -7.6544e+01,$ $p_m(0) = -6.2821e-02,$ $p_s(T)= -3.2237e-03,$ $p_i(T) = -5.8613e+03,$ $p_m(T) = -1.4648e-05 .$
 Numerical results fors $(P_1)$ with $A=10, B=3, C=1 : {\text{Cost}} J_1= 15.56816 ,$ $s(T)= 3.337 73e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 1.370 10 e-05$ $p_s(0)= -5.614 47 ,$ $p_i(0) = -3.432 49e+01,$ $p_m(0) = -5.360 07 e-02,$ $p_s(T)= -5.4088e-04 ,$ $p_i(T) = -1.9668 e+03 ,$ $p_m(T) = -2.4584e-06.$
 Numerical results fors $(P_1)$ with $A=10, B=3, C=1 : {\text{Cost}} J_1= 15.56816 ,$ $s(T)= 3.337 73e-01 ,$ $i( T)= 5e-04 ,$ $m(T)= 1.370 10 e-05$ $p_s(0)= -5.614 47 ,$ $p_i(0) = -3.432 49e+01,$ $p_m(0) = -5.360 07 e-02,$ $p_s(T)= -5.4088e-04 ,$ $p_i(T) = -1.9668 e+03 ,$ $p_m(T) = -2.4584e-06.$
 parameter $dt_1/dq$ $dt_2/dq$ $dt_3/dq$ $du_c/dq$ $q = B$ -0.5892, 0.7584, 0.7584, -0.1868, $q = C$ 0.1520, -0.6556, -0.6556, 0.07068, $q = a_1$ -1.960, -23.52, -23.52, 0.4620.
 parameter $dt_1/dq$ $dt_2/dq$ $dt_3/dq$ $du_c/dq$ $q = B$ -0.5892, 0.7584, 0.7584, -0.1868, $q = C$ 0.1520, -0.6556, -0.6556, 0.07068, $q = a_1$ -1.960, -23.52, -23.52, 0.4620.
 parameter $dt_1/dq$ $dt_2/dq$ $q = B$ -0.49896, 1.45249 $q = C$ 1.4525, -5.4816 $q = a_1$ 0.27758, -24.112
 parameter $dt_1/dq$ $dt_2/dq$ $q = B$ -0.49896, 1.45249 $q = C$ 1.4525, -5.4816 $q = a_1$ 0.27758, -24.112
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