# American Institute of Mathematical Sciences

February  2017, 14(1): 111-126. doi: 10.3934/mbe.2017008

## On application of optimal control to SEIR normalized models: Pros and cons

 1 SYSTEC, DEEC, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200–465 Porto, Portugal 2 CTAC, Departamento de Engenharia Civil, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal

* Corresponding author: Filipa N. Nogueira, dma09030@fe.up.pt

Received  November 23, 2015 Accepted  June 03, 2016 Published  October 2016

In this work we normalize a SEIR model that incorporates exponential natural birth and death, as well as disease-caused death. We use optimal control to control by vaccination the spread of a generic infectious disease described by a normalized model with $L^1$ cost. We discuss the pros and cons of SEIR normalized models when compared with classical models when optimal control with $L^1$ costs are considered. Our discussion highlights the role of the cost. Additionally, we partially validate our numerical solutions for our optimal control problem with normalized models using the Maximum Principle.

Citation: Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences & Engineering, 2017, 14 (1) : 111-126. doi: 10.3934/mbe.2017008
##### References:
 [1] M. Biswas, L. Paiva and M. 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. [2] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [3] C. Buskens and D. Wassel, The ESA NLP solver WORHP, Modeling and optimization in space engineering, Springer Optim. Appl., Springer, New York, 73 (2013), 85-110. doi: 10.1007/978-1-4614-4469-5_4. [4] V. Capasso, Mathematical Structures of Epidemic Systems Springer-Verlag, Berlin Heidelberg, 2008. [5] F. Clarke, Optimization and Nonsmooth Analysis John Wiley & Sons, New York, 1983. [6] M. de Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and $L^1$ cost, Springer International Publishing, Switzerland, Lecture Notes in Electrical Engineering, 312 (2015), 135-145. [7] R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks–Cole Publishing Company, USA, (1993). [8] A. Friedman and C. Kao, Mathematical Modeling of Biological Processes Springer, Switzerland, 2014. doi: 10.1007/978-3-319-08314-8. [9] H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [10] H. Hethcote, The basic epidemiology models: Models, expressions for $r_0$, parameter estimation, and applications, Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), World Scientific Publishing Co. Pte. Ltd., Singapore, 16 (2009), 1-61. doi: 10.1142/9789812834836_0001. [11] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A Containing papers of a Mathematical and Physical Nature, 115. [12] A. Korobeinikov, E. V. Grigorieva and E. N. Khailov, Optimal control for an epidemic in a population of varying size, Discrete Contin. Dyn. Syst., (2015), 549-561. doi: 10.3934/proc.2015.0549. [13] U. Ledzewicz and H. Schaettler, On optimal singular controls for a general sir-model with vaccination and treatment supplement, Discrete and Continuous Dynamical Systems, Series B, 2 (2011), 981-990. [14] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [15] H. Maurer, C. Buskens, J.-H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Applications and Methods, 26 (2015), 129-156. doi: 10.1002/oca.756. [16] H. Maurer and M. R. de Pinho, Optimal control of seir models in epidemiology with l$^1$ objectives, https://hal.inria.fr/hal-01101291/file/Maurer-dePinho-SEIR.pdf. (Accepted) [17] 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. [18] K. Okosun, O. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013), 83-101. doi: 10.1016/j.biosystems.2012.09.008. [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, 24 SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. [20] H. Schaettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [21] 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.

show all references

##### References:
 [1] M. Biswas, L. Paiva and M. 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. [2] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-3516-1. [3] C. Buskens and D. Wassel, The ESA NLP solver WORHP, Modeling and optimization in space engineering, Springer Optim. Appl., Springer, New York, 73 (2013), 85-110. doi: 10.1007/978-1-4614-4469-5_4. [4] V. Capasso, Mathematical Structures of Epidemic Systems Springer-Verlag, Berlin Heidelberg, 2008. [5] F. Clarke, Optimization and Nonsmooth Analysis John Wiley & Sons, New York, 1983. [6] M. de Pinho, I. Kornienko and H. Maurer, Optimal control of a SEIR model with mixed constraints and $L^1$ cost, Springer International Publishing, Switzerland, Lecture Notes in Electrical Engineering, 312 (2015), 135-145. [7] R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks–Cole Publishing Company, USA, (1993). [8] A. Friedman and C. Kao, Mathematical Modeling of Biological Processes Springer, Switzerland, 2014. doi: 10.1007/978-3-319-08314-8. [9] H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [10] H. Hethcote, The basic epidemiology models: Models, expressions for $r_0$, parameter estimation, and applications, Mathematical Understanding of Infectious Disease Dynamics (S. Ma and Y. Xia, Eds.), World Scientific Publishing Co. Pte. Ltd., Singapore, 16 (2009), 1-61. doi: 10.1142/9789812834836_0001. [11] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A Containing papers of a Mathematical and Physical Nature, 115. [12] A. Korobeinikov, E. V. Grigorieva and E. N. Khailov, Optimal control for an epidemic in a population of varying size, Discrete Contin. Dyn. Syst., (2015), 549-561. doi: 10.3934/proc.2015.0549. [13] U. Ledzewicz and H. Schaettler, On optimal singular controls for a general sir-model with vaccination and treatment supplement, Discrete and Continuous Dynamical Systems, Series B, 2 (2011), 981-990. [14] M. Y. Li, J. R. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9. [15] H. Maurer, C. Buskens, J.-H. R. Kim and C. Y. Kaya, Optimization methods for the verification of second order sufficient conditions for bang-bang controls, Optimal Control Applications and Methods, 26 (2015), 129-156. doi: 10.1002/oca.756. [16] H. Maurer and M. R. de Pinho, Optimal control of seir models in epidemiology with l$^1$ objectives, https://hal.inria.fr/hal-01101291/file/Maurer-dePinho-SEIR.pdf. (Accepted) [17] 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. [18] K. Okosun, O. Rachid and N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013), 83-101. doi: 10.1016/j.biosystems.2012.09.008. [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, 24 SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368. [20] H. Schaettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [21] 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.
Fluid Analogy of the Normalized SEIR compartmental model.
Optimal control for $(P)$: parameters in tables 1 and 2. Left: Optimal control different dead rates: in red for $d=0.0099$ and in blue for $d=0.0005$. Right: Optimal control with different initial values. In red for $S_0$, $E_0$, $I_0$ and $R_0$ as in the table 2 in blue for initial conditions $S_0\times 100$, $E_0\times 100$, $I_0\times 100$ and $R_0\times 100$.
Optimal control for $(P_n)$ with $\rho=500$ in blue. Optimal control calculated for $(P)$ in blue. The parameters are described in tables 1, 3, and 2.
Case 1: Computed optimal control $u^*$ plotted together with the singular control computed according to (22) and with the switching function $\phi$. During the first five years $\phi(t)>1$ and during the last eight years $\phi(t)<0$.
Case 1: optimal trajectories (including $r$).
Case 2: Computed optimal control $u^*$ plotted together with the scaled switching function $\phi$. During the last seventeen years $\phi(t)<0$.
Case 2: optimal trajectories.
Case 3: Computed optimal control $u^*$ plotted together with the scaled switching function $\phi$. During the first sixteen years $\phi(t)>0$ and during the last three years $\phi(t)<0$.
Case 3: Optimal trajectories.
Optimal vaccinated rate, $u^*$, in red. Approximate control $u_{apr}$, in blue dash. $\rho=500$ and $u \in [0,1]$.
Parameters for SEIR models
 Parameter Description Value b Natural birth rate 0.01 d Death rate 0.0099 c Incidence coefficient 1.1 f Exposed to infectious rate 0.5 g Recovery rate 0.1 a Disease induced death rate 0.2 T Number of years 20
 Parameter Description Value b Natural birth rate 0.01 d Death rate 0.0099 c Incidence coefficient 1.1 f Exposed to infectious rate 0.5 g Recovery rate 0.1 a Disease induced death rate 0.2 T Number of years 20
Initial Conditions and cost parameters for problems with classical SEIR model
 Parameter Description Value A weight parameter 1 B weight parameter 2 S0 Initial susceptible population 1000 E0 Initial exposed population 100 I0 Initial infected population 50 R0 Initial recovered population 15 N0 Initial population 1165
 Parameter Description Value A weight parameter 1 B weight parameter 2 S0 Initial susceptible population 1000 E0 Initial exposed population 100 I0 Initial infected population 50 R0 Initial recovered population 15 N0 Initial population 1165
Initial Conditions and cost parameters for problems with classical SEIR model normalized model.
 Parameter Description Value s0 Percentage of initial susceptible population 0.858 e0 Percentage of initial exposed population 0.086 i0 Percentage of initial infected population 0.043
 Parameter Description Value s0 Percentage of initial susceptible population 0.858 e0 Percentage of initial exposed population 0.086 i0 Percentage of initial infected population 0.043
 [1] Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 [2] Vincenzo Capasso, Sebastian AniȚa. The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally). Mathematical Biosciences & Engineering, 2018, 15 (1) : 1-20. doi: 10.3934/mbe.2018001 [3] Bruno Buonomo, Giuseppe Carbone, Alberto d'Onofrio. Effect of seasonality on the dynamics of an imitation-based vaccination model with public health intervention. Mathematical Biosciences & Engineering, 2018, 15 (1) : 299-321. doi: 10.3934/mbe.2018013 [4] Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 519-537. doi: 10.3934/dcdss.2020029 [5] Julien Arino, Chris Bauch, Fred Brauer, S. Michelle Driedger, Amy L. Greer, S.M. Moghadas, Nick J. Pizzi, Beate Sander, Ashleigh Tuite, P. van den Driessche, James Watmough, Jianhong Wu, Ping Yan. Pandemic influenza: Modelling and public health perspectives. Mathematical Biosciences & Engineering, 2011, 8 (1) : 1-20. doi: 10.3934/mbe.2011.8.1 [6] M. H. A. Biswas, L. T. Paiva, MdR de Pinho. A SEIR model for control of infectious diseases with constraints. Mathematical Biosciences & Engineering, 2014, 11 (4) : 761-784. doi: 10.3934/mbe.2014.11.761 [7] Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485 [8] Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 [9] Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559 [10] Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022 [11] Markus Thäter, Kurt Chudej, Hans Josef Pesch. Optimal vaccination strategies for an SEIR model of infectious diseases with logistic growth. Mathematical Biosciences & Engineering, 2018, 15 (2) : 485-505. doi: 10.3934/mbe.2018022 [12] Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101 [13] Christine Burggraf, Wilfried Grecksch, Thomas Glauben. Stochastic control of individual's health investments. Conference Publications, 2015, 2015 (special) : 159-168. doi: 10.3934/proc.2015.0159 [14] Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323 [15] Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003 [16] Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445 [17] Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019 [18] Avner Friedman, Najat Ziyadi, Khalid Boushaba. A model of drug resistance with infection by health care workers. Mathematical Biosciences & Engineering, 2010, 7 (4) : 779-792. doi: 10.3934/mbe.2010.7.779 [19] Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030 [20] Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021

2018 Impact Factor: 1.313