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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).   Google Scholar [2] F. Clarke, Optimization and Nonsmooth Analysis,, John Wiley, (1983).   Google Scholar [3] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control,, Springer-Verlag, (2013).  doi: 10.1007/978-1-4471-4820-3.  Google Scholar [4] F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., 48 (2010), 4500.  doi: 10.1137/090757642.  Google Scholar [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).   Google Scholar [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.  doi: 10.1007/s11228-009-0108-1.  Google Scholar [7] E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate,, MdR de Pinho, 40 (2011), 287.   Google Scholar [8] P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS),, Department of Electrical and Electronic Engineering, (2010).   Google Scholar [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.  doi: 10.1137/1037043.  Google Scholar [10] M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, $2^{nd}$ Edition (405 pages), (1980).   Google Scholar [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, (2008), 1.  doi: 10.1142/9789812834836_0001.  Google Scholar [12] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53 (1991), 35.   Google Scholar [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.  doi: 10.1007/BF02192163.  Google Scholar [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.  doi: 10.1137/S0363012900377419.  Google Scholar [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.  Google Scholar [16] D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes,, Optim. Control Appl. Meth., 32 (2011), 181.  doi: 10.1002/oca.990.  Google Scholar [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.   Google Scholar [18] L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems,, Project Report, (2013).   Google Scholar [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.  doi: 10.3934/mbe.2011.8.141.  Google Scholar [20] P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations,, Journal of Applied Mathematics, 2012 (2012), 1.  doi: 10.1155/2012/824192.  Google Scholar [21] H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples,, Springer, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar [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.  doi: 10.1016/j.apm.2009.12.005.  Google Scholar [23] R. Vinter, Optimal Control,, Birkhäuser, (2000).   Google Scholar [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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar

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##### References:
 [1] F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, Springer-Verlag. New York, (2001).   Google Scholar [2] F. Clarke, Optimization and Nonsmooth Analysis,, John Wiley, (1983).   Google Scholar [3] F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control,, Springer-Verlag, (2013).  doi: 10.1007/978-1-4471-4820-3.  Google Scholar [4] F. Clarke and MdR de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., 48 (2010), 4500.  doi: 10.1137/090757642.  Google Scholar [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).   Google Scholar [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.  doi: 10.1007/s11228-009-0108-1.  Google Scholar [7] E. Demirci, A. Unal and N. Ozalp, A fractional order seir model with density dependent death rate,, MdR de Pinho, 40 (2011), 287.   Google Scholar [8] P. Falugi, E. Kerrigan and E. van Wyk, Imperial College London Optimal Control Software User Guide (ICLOCS),, Department of Electrical and Electronic Engineering, (2010).   Google Scholar [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.  doi: 10.1137/1037043.  Google Scholar [10] M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, $2^{nd}$ Edition (405 pages), (1980).   Google Scholar [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, (2008), 1.  doi: 10.1142/9789812834836_0001.  Google Scholar [12] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53 (1991), 35.   Google Scholar [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.  doi: 10.1007/BF02192163.  Google Scholar [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.  doi: 10.1137/S0363012900377419.  Google Scholar [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.  Google Scholar [16] D. S. Naidu, T. Fernando and K. R. Fister, Optimal control in diabetes,, Optim. Control Appl. Meth., 32 (2011), 181.  doi: 10.1002/oca.990.  Google Scholar [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.   Google Scholar [18] L.T. Paiva, Optimal Control in Constrained and Hybrid Nonlinear Systems,, Project Report, (2013).   Google Scholar [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.  doi: 10.3934/mbe.2011.8.141.  Google Scholar [20] P. Shi and L. Dong, Dynamical models for infectious diseases with varying population size and vaccinations,, Journal of Applied Mathematics, 2012 (2012), 1.  doi: 10.1155/2012/824192.  Google Scholar [21] H. Schäettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples,, Springer, (2012).  doi: 10.1007/978-1-4614-3834-2.  Google Scholar [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.  doi: 10.1016/j.apm.2009.12.005.  Google Scholar [23] R. Vinter, Optimal Control,, Birkhäuser, (2000).   Google Scholar [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.  doi: 10.1007/s10107-004-0559-y.  Google Scholar
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