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

Maria do Rosário de Pinho; Filipa Nunes Nogueira

doi:10.3934/mbe.2017008

Article Contents

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

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

Maria do Rosário de Pinho^1, and
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

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

Received: November 22, 2015

Accepted: June 02, 2016

Published: September 2017

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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.

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Mathematics Subject Classification: Primary: 49K15, 49N90; Secondary: 49M37.

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Figure 1. Fluid Analogy of the Normalized SEIR compartmental model.

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Figure 2. 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 $R_0\times 100$.

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Figure 3. 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.

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Figure 4. 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 $\phi(t)<0$.

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Figure 5. Case 1: optimal trajectories (including $r$).

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Figure 6. Case 2: Computed optimal control $u^*$ plotted together with the scaled switching function $\phi$. During the last seventeen years $\phi(t)<0$.

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Figure 7. Case 2: optimal trajectories.

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Figure 8. 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 $\phi(t)<0$.

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Figure 9. Case 3: Optimal trajectories.

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Figure 10. Optimal vaccinated rate, $u^*$, in red. Approximate control $u_{apr}$, in blue dash. $\rho=500$ and $u \in [0,1]$.

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Table 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

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Table 2. 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

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Table 3. 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

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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.