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2014, 11(4): 761-784. doi: 10.3934/mbe.2014.11.761

## A SEIR model for control of infectious diseases with constraints

 1 Faculdade de Engenharia da Universidade do Porto, DEEC and ISR-Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal, Portugal, Portugal

Received  April 2013 Revised  December 2013 Published  March 2014

Optimal control can be of help to test and compare different vaccination strategies of a certain disease. In this paper we propose the introduction of constraints involving state variables on an optimal control problem applied to a compartmental SEIR (Susceptible. Exposed, Infectious and Recovered) model. We study the solution of such problems when mixed state control constraints are used to impose upper bounds on the available vaccines at each instant of time. We also explore the possibility of imposing upper bounds on the number of susceptible individuals with and without limitations on the number of vaccines available. In the case of mere mixed constraints a numerical and analytical study is conducted while in the other two situations only numerical results are presented.
Citation: 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
##### References:
 [1] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag. New York, 2001. [2] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983. [3] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer-Verlag, London, 2013. doi: 10.1007/978-1-4471-4820-3. [4] F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48, (2010), 4500-4524. doi: 10.1137/090757642. [5] M. d. R. de Pinho, M. M. Ferreira, U. Ledzewicz and H. Schaettler, A model for cancer chemotherapy with state-space constraints, Nonlinear Analysis, 63 (2005), e2591-e2602. [6] M. d. R. de Pinho, P. Loewen and G. N. Silva, A weak maximum principle for optimal control problems with nonsmooth mixed constraints, Set-Valued and Variational Analysis, 17 (2009), 203-2219. doi: 10.1007/s11228-009-0108-1. [7] E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate, MdR de Pinho,Hacet. J. Math. Stat., 40 (2011), 287-295. [8] P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS), Department of Electrical and Electronic Engineering, Imperial College London, London, England, UK, 2010. [9] R. F. Hartl, S. 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. [10] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, $2^{nd}$ Edition (405 pages), John Wiley, New York, 1980. [11] 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.), Vol. 16. Chap. 1, pp. 1-61, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008. doi: 10.1142/9789812834836_0001. [12] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Bulletin of Mathematical Biology, 53 (1991), 35-55. [13] H. Maurer and S. Pickenhain, Second order sufficient conditions for optimal control problems with mixed control-state constraints, J. Optim. Theory Appl., 86 (1995), 649-667. doi: 10.1007/BF02192163. [14] Helmut Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach, SIAM J. Control Optm., 41 (2002), 380-403. doi: 10.1137/S0363012900377419. [15] 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, 2012. doi: 10.1137/1.9781611972368. [16] D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes, Optim. Control Appl. Meth., 32 (2011), 181-184. doi: 10.1002/oca.990. [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] L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems, Project Report, 2013, http://paginas.fe.up.pt/~faf/ProjectFCT2009/report.pdf. [19] O. Prosper, O. Saucedo, D. Thompson, G. T. Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, Mathematical Biosciences and Engineering, 8 (2011), 141-170. doi: 10.3934/mbe.2011.8.141. [20] P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations, Journal of Applied Mathematics, 2012 (2012), 1-20. doi: 10.1155/2012/824192. [21] H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [22] C. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination, Applied Mathematical Modelling, 34 (2010), 2685-2697. doi: 10.1016/j.apm.2009.12.005. [23] R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. [24] 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.

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##### References:
 [1] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer-Verlag. New York, 2001. [2] F. Clarke, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983. [3] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Springer-Verlag, London, 2013. doi: 10.1007/978-1-4471-4820-3. [4] F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints, SIAM J. Control Optim., 48, (2010), 4500-4524. doi: 10.1137/090757642. [5] M. d. R. de Pinho, M. M. Ferreira, U. Ledzewicz and H. Schaettler, A model for cancer chemotherapy with state-space constraints, Nonlinear Analysis, 63 (2005), e2591-e2602. [6] M. d. R. de Pinho, P. Loewen and G. N. Silva, A weak maximum principle for optimal control problems with nonsmooth mixed constraints, Set-Valued and Variational Analysis, 17 (2009), 203-2219. doi: 10.1007/s11228-009-0108-1. [7] E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate, MdR de Pinho,Hacet. J. Math. Stat., 40 (2011), 287-295. [8] P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS), Department of Electrical and Electronic Engineering, Imperial College London, London, England, UK, 2010. [9] R. F. Hartl, S. 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. [10] M. R. Hestenes, Calculus of Variations and Optimal Control Theory, $2^{nd}$ Edition (405 pages), John Wiley, New York, 1980. [11] 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.), Vol. 16. Chap. 1, pp. 1-61, World Scientific Publishing Co. Pte. Ltd., Singapore, 2008. doi: 10.1142/9789812834836_0001. [12] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Bulletin of Mathematical Biology, 53 (1991), 35-55. [13] H. Maurer and S. Pickenhain, Second order sufficient conditions for optimal control problems with mixed control-state constraints, J. Optim. Theory Appl., 86 (1995), 649-667. doi: 10.1007/BF02192163. [14] Helmut Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach, SIAM J. Control Optm., 41 (2002), 380-403. doi: 10.1137/S0363012900377419. [15] 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, 2012. doi: 10.1137/1.9781611972368. [16] D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes, Optim. Control Appl. Meth., 32 (2011), 181-184. doi: 10.1002/oca.990. [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] L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems, Project Report, 2013, http://paginas.fe.up.pt/~faf/ProjectFCT2009/report.pdf. [19] O. Prosper, O. Saucedo, D. Thompson, G. T. Garcia, X. Wang and C. Castillo-Chavez, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza, Mathematical Biosciences and Engineering, 8 (2011), 141-170. doi: 10.3934/mbe.2011.8.141. [20] P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations, Journal of Applied Mathematics, 2012 (2012), 1-20. doi: 10.1155/2012/824192. [21] H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples, Springer, New York, 2012. doi: 10.1007/978-1-4614-3834-2. [22] C. Sun and Y. H. Hsieh, Global analysis of an SEIR model with varying population size and vaccination, Applied Mathematical Modelling, 34 (2010), 2685-2697. doi: 10.1016/j.apm.2009.12.005. [23] R. Vinter, Optimal Control, Birkhäuser, Boston, 2000. [24] 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.
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