A SEIR model for control of infectious diseases with constraints. (English) Zbl 1327.92055

Summary: 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.


92D30 Epidemiology
49K15 Optimality conditions for problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems


Full Text: DOI


[1] F. Brauer, <em>Mathematical Models in Population Biology and Epidemiology</em>,, Springer-Verlag. New York, (2001) · Zbl 0967.92015
[2] F. Clarke, <em>Optimization and Nonsmooth Analysis</em>,, John Wiley, (1983) · Zbl 0582.49001
[3] F. Clarke, <em>Functional Analysis, Calculus of Variations and Optimal Control</em>,, Springer-Verlag, (2013) · Zbl 1277.49001
[4] F. Clarke, Optimal control problems with mixed constraints,, SIAM J. Control Optim., 48, 4500, (2010) · Zbl 1208.49018
[5] M. d. R. de Pinho, A model for cancer chemotherapy with state-space constraints,, Nonlinear Analysis, 63, (2005) · Zbl 1222.92050
[6] M. d. R. de Pinho, A weak maximum principle for optimal control problems with nonsmooth mixed constraints,, Set-Valued and Variational Analysis, 17, 203, (2009) · Zbl 1168.49023
[7] E. Demirci, A fractional order seir model with density dependent death rate,, MdR de Pinho, 40, 287, (2011) · Zbl 1262.92032
[8] P. Falugi, <em>Imperial College London Optimal Control Software User Guide (ICLOCS)</em>,, Department of Electrical and Electronic Engineering, (2010)
[9] R. F. Hartl, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Review, 37, 181, (1995) · Zbl 0832.49013
[10] M. R. Hestenes, <em>Calculus of Variations and Optimal Control Theory</em>,, \(2^{nd}\) Edition (405 pages), (1980) · Zbl 0481.49001
[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, 1, (2008)
[12] W. O. Kermack, Contributions to the mathematical theory of epidemics,, Bulletin of Mathematical Biology, 53, 35, (1991)
[13] H. Maurer, Second order sufficient conditions for optimal control problems with mixed control-state constraints,, J. Optim. Theory Appl., 86, 649, (1995) · Zbl 0874.49020
[14] Helmut Maurer, Second order sufficient conditions for optimal control problems with free final time: The Riccati approach,, SIAM J. Control Optm., 41, 380, (2002) · Zbl 1012.49018
[15] N. P. Osmolovskii, <em>Applications to Regular and Bang-Bang Control: Second-Order Necessary And Sufficient Optimality Conditions In Calculus Of Variations And Optimal Control</em>,, SIAM Advances in Design and Control, 24, (2012) · Zbl 1263.49002
[16] D. S. Naidu, Optimal control in diabetes,, Optim. Control Appl. Meth., 32, 181, (2011)
[17] R. M. Neilan, An introduction to optimal control with an application in disease modeling,, DIMACS Series in Discrete Mathematics, 75, 67, (2010) · Zbl 1352.92164
[18] L. T. Paiva, <em>Optimal Control in Constrained and Hybrid Nonlinear Systems</em>,, Project Report, (2013)
[19] O. Prosper, Modeling control strategies for concurrent epidemics of seasonal and pandemic H1N1 influenza,, Mathematical Biosciences and Engineering, 8, 141, (2011) · Zbl 1260.92070
[20] P. Shi, Dynamical models for infectious diseases with varying population size and vaccinations,, Journal of Applied Mathematics, 2012, 1, (2012) · Zbl 1235.37036
[21] H. Schäettler, <em>Geometric Optimal Control. Theory, Methods and Examples</em>,, Springer, (2012) · Zbl 1276.49002
[22] C. Sun, Global analysis of an SEIR model with varying population size and vaccination,, Applied Mathematical Modelling, 34, 2685, (2010) · Zbl 1201.34076
[23] R. Vinter, <em>Optimal Control</em>,, Birkhäuser, (2000) · Zbl 1050.49022
[24] A. Wächter, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106, 25, (2006) · Zbl 1134.90542
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.