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February  2017, 14(1): 111-126. doi: 10.3934/mbe.2017008

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

Received  November 23, 2015 Accepted  June 03, 2016 Published  October 2016

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.

Keywords: Optimal control, necessary confitions, normalized SEIR model, public health, epidemiology.
Mathematics Subject Classification: Primary: 49K15, 49N90; Secondary: 49M37.
Citation: Maria do Rosário de Pinho, Filipa Nunes Nogueira. On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences & Engineering, 2017, 14 (1) : 111-126. doi: 10.3934/mbe.2017008
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

V. Capasso, Mathematical Structures of Epidemic Systems Springer-Verlag, Berlin Heidelberg, 2008.  Google Scholar

[5]

F. Clarke, Optimization and Nonsmooth Analysis John Wiley & Sons, New York, 1983.  Google Scholar

[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.   Google Scholar

[7]

R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks–Cole Publishing Company, USA, (1993).   Google Scholar

[8]

A. Friedman and C. Kao, Mathematical Modeling of Biological Processes Springer, Switzerland, 2014. doi: 10.1007/978-3-319-08314-8.  Google Scholar

[9]

H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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) 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), 67-81.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

V. Capasso, Mathematical Structures of Epidemic Systems Springer-Verlag, Berlin Heidelberg, 2008.  Google Scholar

[5]

F. Clarke, Optimization and Nonsmooth Analysis John Wiley & Sons, New York, 1983.  Google Scholar

[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.   Google Scholar

[7]

R. Fourer, D. Gay and B. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Duxbury Press, Brooks–Cole Publishing Company, USA, (1993).   Google Scholar

[8]

A. Friedman and C. Kao, Mathematical Modeling of Biological Processes Springer, Switzerland, 2014. doi: 10.1007/978-3-319-08314-8.  Google Scholar

[9]

H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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) 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), 67-81.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Fluid Analogy of the Normalized SEIR compartmental model.
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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 $R_0\times 100$.">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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tables 1, 3, and 2.">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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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
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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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
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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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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