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

Maria do Rosário de Pinho; Filipa Nunes Nogueira; Maria do Rosário de Pinho; Filipa Nunes Nogueira

doi:10.3934/mbe.2017008

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 1: 111-126. doi: 10.3934/mbe.2017008

Previous Article Next Article

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

Maria do Rosário de Pinho ¹,
Filipa Nunes Nogueira ^{2
,
,}

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

Received: 23 November 2015 Accepted: 03 June 2016 Published: 01 February 2017
MSC : Primary: 49K15, 49N90; Secondary: 49M37

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.
- Optimal control,
- necessary confitions,
- normalized SEIR model,
- public health,
- epidemiology
Citation: Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 111-126. doi: 10.3934/mbe.2017008

Related Papers:

Abstract

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.

References

[1]	[ M. Biswas,L. Paiva,M. de Pinho, A SEIR model for control of infectious diseases with constraints, Mathematical Biosciences and Engineering, 11 (2014): 761-784.
[2]	[ F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology Springer-Verlag, New York, 2001.
[3]	[ C. Buskens,D. Wassel, The ESA NLP solver WORHP, Modeling and optimization in space engineering, Springer Optim. Appl., Springer, New York, 73 (2013): 85-110.
[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,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,B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks–Cole Publishing Company, USA, null (1993).
[8]	[ A. Friedman and C. Kao, Mathematical Modeling of Biological Processes Springer, Switzerland, 2014.
[9]	[ H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000): 599-653.
[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.
[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,E. N. Khailov, Optimal control for an epidemic in a population of varying size, Discrete Contin. Dyn. Syst., null (2015): 549-561.
[13]	[ U. Ledzewicz,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,J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 160 (1999): 191-213.
[15]	[ H. Maurer,C. Buskens,J.-H. R. Kim,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.
[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,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,N. Marcus, Optimal control strategies and cost-effectiveness analysis of a malaria model, Biosystems, 111 (2013): 83-101.
[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.
[20]	[ H. Schaettler and U. Ledzewicz, Geometric Optimal Control. Theory, Methods and Examples Springer, New York, 2012.
[21]	[ A. Wächter,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.

Reader Comments

Your name:*

Email:*
© 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)