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December 2018, 11(6): 1071-1101. doi: 10.3934/dcdss.2018062

Determination of the optimal controls for an Ebola epidemic model

1. 

Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204, USA

2. 

Faculty of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia

* Corresponding author: Ellina Grigorieva

Received  February 2017 Revised  April 2017 Published  June 2018

A control SEIR type model describing the spread of an Ebola epidemic in a population of a constant size is considered on the given time interval. This model contains four bounded control functions, three of which are distancing controls in the community, at the hospital, and during burial; the fourth is burial control. We consider the optimal control problem of minimizing the fraction of infectious individuals in the population at the given terminal time and analyze the corresponding optimal controls with the Pontryagin maximum principle. We use values of the model parameters and control constraints for which the optimal controls are bang-bang. To estimate the number of zeros of the switching functions that determine the behavior of these controls, a linear non-autonomous homogenous system of differential equations for these switching functions and corresponding to them auxiliary functions are obtained. Subsequent study of the properties of solutions of this system allows us to find analytically the estimates of the number of switchings and the type of the optimal controls for the model parameters and control constraints related to all Ebola epidemics from 1995 until 2014. Corresponding numerical calculations confirming the results are presented.

Citation: Ellina Grigorieva, Evgenii Khailov. Determination of the optimal controls for an Ebola epidemic model. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1071-1101. doi: 10.3934/dcdss.2018062
References:
[1]

M. D. Ahmad, M. Usman, A. Khan and M. Imran, Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination, Infectious Diseases of Poverty, 5 (2016), p72. doi: 10.1186/s40249-016-0161-6.

[2]

P. S. Aleksandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977.

[3]

A. I. Astrovskii and I. V. Gaishun, Quasidifferentiability and observability of linear nonstationary systems, Diff. Equat., 45 (2009), 1602-1611. doi: 10.1134/S0012266109110061.

[4]

A. I. Astrovskii and I. V. Gaishun, Controllability of linear nonstationary systems with scalar input and quasidifferentiable coefficients, Diff. Equat., 49 (2013), 1018-1026. doi: 10.1134/S0012266113080107.

[5]

B. P. Demidovich, Lectures on Stability Theory, Nauka, Moscow, 1967.

[6]

P. Diaz, P. Constantine, K. Kalmbach, E. Jones and S. Pankavich, A modified SEIR model for the spread of Ebola in Western Africa and metrics for resource allocation, Appl. Math. Comput., 324 (2018), 141–155, arXiv: 1603.04955. doi: 10.1016/j.amc.2017.11.039.

[7]

A. V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM J. Control Optim., 30 (1992), 1087-1091. doi: 10.1137/0330057.

[8]

B. EbenezerK. Badu and A.-A. S. Kwesi, Optimal control application to an Ebola model, Health Science Journal, 10 (2016), 1-7.

[9]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469.

[10]

M. F. C. Gomes, A. P. y Pointti, L. Rossi, D. Chao, I. Longini, M. E. Halloran and A. Vespignani, Assessing the international spreading risk associated with the 2014 West African Ebola outbreak, PLOC Current Outbreaks, 2014 Sep 2, Edition 1. doi: 10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5.

[11]

E. Grigorieva and E. Khailov, Analytic study of optimal control intervention strategies for Ebola epidemic model, in Proceedings of the SIAM Conference on Control and its Applications (CT15), Paris, France, July 8–10, (2015), 392–399.

[12]

E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983.

[13]

E. V. Grigorieva and E. N. Khailov, Estimating the number of switchings of the optimal intervention strategies for SEIR control model of Ebola epidemics, Pure and Applied Functional Analysis, 1 (2016), 541-572.

[14]

E. Grigorieva and E. Khailov, Optimal priventive strategies for SEIR type model of 2014 Ebola epidemics, Dynam. Cont. Dis. Ser. B, 24 (2017), 155-182.

[15]

E. GrigorievaE. Khailov and A. Korobeinikov, Optimal control for an epidemic in populations of varying size, Discret. Contin. Dyn. Syst., supplement (2015), 549-561. doi: 10.3934/proc.2015.0549.

[16]

E. V. GrigorievaE. N. Khailov and A. Korobeinikov, Optimal control for a SIR epidemic model with nonlinear incidence rate, Math. Model. Nat. Phenom., 11 (2016), 89-104. doi: 10.1051/mmnp/201611407.

[17]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964.

[18]

D. Hincapié-PalacioJ. Ospina and D. F. M. Torres, Approximated analytical solution to an Ebola optimal control problem, International Journal for Computational Methods in Engineering Science and Mechanics, 17 (2016), 382-390. doi: 10.1080/15502287.2016.1231236.

[19]

E. N. Khailov and E. V. Grigorieva, On the extensibility of solutions of nonautonomous quadratic differential systems, Trudy Inst. Mat. i Mekh. UrO RAN, 19 (2013), 279-288.

[20]

E. N. Khailov and E. V. Grigorieva, On splitting quadratic system of differential equations, in Systems Analysis: Modeling and Control, Abstracts of the Intrenational Conference in memory of Academician Arkady Kryazhimskiy, Ekaterinburg, Russia, October 3–8, (2016), 64–66.

[21]

U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discret. Contin. Dyn. Syst., supplement (2011), 981-990.

[22]

E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967.

[23]

J. LegrandR. F. GraisP. Y. BoelleA. J. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.

[24]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optim. Contr. Appl. Met., 26 (2005), 129-156. doi: 10.1002/oca.756.

[25]

F. T. Oduro, G. Apaaboah and J. Baafi, Optimal control of Ebola transmission dynamics with interventions, British Journal of Mathematics & Computer Sciences, 19 (2016), Article BJMCS. 29372, 1–19.

[26]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: SecondOrder Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, vol. DC24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.

[27]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962.

[28]

A. Rachah and D. F. M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa, Discrete Dynamics in Nature and Society, (2015), Art. ID 842792, 9 pp. doi: 10.1155/2015/842792.

[29]

C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis, Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia, PLOC Current Outbreaks, 2014 Oct 16, Edition 1. doi: 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c.

[30]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. doi: 10.1007/978-1-4614-3834-2.

[31]

A. N. Tikhonov, A. B. vasileva and A. G. Sveshnikov, Differential Equations, SpringerVerlag, Berlin-Heidelberg-New York, 1985. doi: 10.1007/978-3-642-82175-2.

[32]

F. P. vasilev, Optimization Methods, Factorial Press, Moscow, 2002.

show all references

References:
[1]

M. D. Ahmad, M. Usman, A. Khan and M. Imran, Optimal control analysis of Ebola disease with control strategies of quarantine and vaccination, Infectious Diseases of Poverty, 5 (2016), p72. doi: 10.1186/s40249-016-0161-6.

[2]

P. S. Aleksandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977.

[3]

A. I. Astrovskii and I. V. Gaishun, Quasidifferentiability and observability of linear nonstationary systems, Diff. Equat., 45 (2009), 1602-1611. doi: 10.1134/S0012266109110061.

[4]

A. I. Astrovskii and I. V. Gaishun, Controllability of linear nonstationary systems with scalar input and quasidifferentiable coefficients, Diff. Equat., 49 (2013), 1018-1026. doi: 10.1134/S0012266113080107.

[5]

B. P. Demidovich, Lectures on Stability Theory, Nauka, Moscow, 1967.

[6]

P. Diaz, P. Constantine, K. Kalmbach, E. Jones and S. Pankavich, A modified SEIR model for the spread of Ebola in Western Africa and metrics for resource allocation, Appl. Math. Comput., 324 (2018), 141–155, arXiv: 1603.04955. doi: 10.1016/j.amc.2017.11.039.

[7]

A. V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations, SIAM J. Control Optim., 30 (1992), 1087-1091. doi: 10.1137/0330057.

[8]

B. EbenezerK. Badu and A.-A. S. Kwesi, Optimal control application to an Ebola model, Health Science Journal, 10 (2016), 1-7.

[9]

H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469.

[10]

M. F. C. Gomes, A. P. y Pointti, L. Rossi, D. Chao, I. Longini, M. E. Halloran and A. Vespignani, Assessing the international spreading risk associated with the 2014 West African Ebola outbreak, PLOC Current Outbreaks, 2014 Sep 2, Edition 1. doi: 10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5.

[11]

E. Grigorieva and E. Khailov, Analytic study of optimal control intervention strategies for Ebola epidemic model, in Proceedings of the SIAM Conference on Control and its Applications (CT15), Paris, France, July 8–10, (2015), 392–399.

[12]

E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983.

[13]

E. V. Grigorieva and E. N. Khailov, Estimating the number of switchings of the optimal intervention strategies for SEIR control model of Ebola epidemics, Pure and Applied Functional Analysis, 1 (2016), 541-572.

[14]

E. Grigorieva and E. Khailov, Optimal priventive strategies for SEIR type model of 2014 Ebola epidemics, Dynam. Cont. Dis. Ser. B, 24 (2017), 155-182.

[15]

E. GrigorievaE. Khailov and A. Korobeinikov, Optimal control for an epidemic in populations of varying size, Discret. Contin. Dyn. Syst., supplement (2015), 549-561. doi: 10.3934/proc.2015.0549.

[16]

E. V. GrigorievaE. N. Khailov and A. Korobeinikov, Optimal control for a SIR epidemic model with nonlinear incidence rate, Math. Model. Nat. Phenom., 11 (2016), 89-104. doi: 10.1051/mmnp/201611407.

[17]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964.

[18]

D. Hincapié-PalacioJ. Ospina and D. F. M. Torres, Approximated analytical solution to an Ebola optimal control problem, International Journal for Computational Methods in Engineering Science and Mechanics, 17 (2016), 382-390. doi: 10.1080/15502287.2016.1231236.

[19]

E. N. Khailov and E. V. Grigorieva, On the extensibility of solutions of nonautonomous quadratic differential systems, Trudy Inst. Mat. i Mekh. UrO RAN, 19 (2013), 279-288.

[20]

E. N. Khailov and E. V. Grigorieva, On splitting quadratic system of differential equations, in Systems Analysis: Modeling and Control, Abstracts of the Intrenational Conference in memory of Academician Arkady Kryazhimskiy, Ekaterinburg, Russia, October 3–8, (2016), 64–66.

[21]

U. Ledzewicz and H. Schättler, On optimal singular controls for a general SIR-model with vaccination and treatment, Discret. Contin. Dyn. Syst., supplement (2011), 981-990.

[22]

E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967.

[23]

J. LegrandR. F. GraisP. Y. BoelleA. J. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.

[24]

H. MaurerC. BüskensJ.-H. R. Kim and Y. Kaya, Optimization methods for the verification of second-order sufficient conditions for bang-bang controls, Optim. Contr. Appl. Met., 26 (2005), 129-156. doi: 10.1002/oca.756.

[25]

F. T. Oduro, G. Apaaboah and J. Baafi, Optimal control of Ebola transmission dynamics with interventions, British Journal of Mathematics & Computer Sciences, 19 (2016), Article BJMCS. 29372, 1–19.

[26]

N. P. Osmolovskii and H. Maurer, Applications to Regular and Bang-Bang Control: SecondOrder Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control, SIAM Advances in Design and Control, vol. DC24, SIAM Publications, Philadelphia, 2012. doi: 10.1137/1.9781611972368.

[27]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962.

[28]

A. Rachah and D. F. M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa, Discrete Dynamics in Nature and Society, (2015), Art. ID 842792, 9 pp. doi: 10.1155/2015/842792.

[29]

C. M. Rivers, E. T. Lofgren, M. Marathe, S. Eubank and B. L. Lewis, Modeling the impact of interventions on an epidemic of Ebola in Sierra Leone and Liberia, PLOC Current Outbreaks, 2014 Oct 16, Edition 1. doi: 10.1371/currents.outbreaks.4d41fe5d6c05e9df30ddce33c66d084c.

[30]

H. Schättler and U. Ledzewicz, Geometric Optimal Control: Theory, Methods and Examples, Springer, New York-Heidelberg-Dordrecht-London, 2012. doi: 10.1007/978-1-4614-3834-2.

[31]

A. N. Tikhonov, A. B. vasileva and A. G. Sveshnikov, Differential Equations, SpringerVerlag, Berlin-Heidelberg-New York, 1985. doi: 10.1007/978-3-642-82175-2.

[32]

F. P. vasilev, Optimization Methods, Factorial Press, Moscow, 2002.

Figure 1.  Graphs of the optimal solutions for the Ebola epidemic in Liberia: top row: S*(t), E*(t); middle row: I*(t), H*(t); bottom row: F*(t), R*(t).
Figure 2.  Graphs of the optimal solutions for the Ebola epidemic in Sierra Leone: top row: S*(t), E*(t); middle row: I*(t), H*(t); bottom row: F*(t), R*(t).
Table 1.  Values of parameters for system (7) and control constraints (3) for 1995 Ebola epidemic in Congo, 2000 Ebola epidemic in Uganda and 2014 Ebola epidemics in Liberia, Guinea and Sierra Leone.
parameter or constraint Liberia
(2014)
Sierra Leone
(2014)
Guinea
(2014)
Uganda
(2000)
Congo
(1995)
$ \alpha $ $ 0.083333 $ $ 0.100000 $ $ 0.111111 $ $ 0.083333 $ $ 0.142857 $
$ \gamma $ $ 0.060802 $ $ 0.047815 $ $ 0.500000 $ $ 0.154762 $ $ 0.134000 $
$ \delta $ $ 0.026767 $ $ 0.010038 $ $ 0.005900 $ $ 0.018550 $ $ 0.006600 $
$ \sigma $ $ 0.030165 $ $ 0.058020 $ $ 0.289100 $ $ 0.020562 $ $ 0.027500 $
$ \rho $ $ 0.049652 $ $ 0.119808 $ $ 0.014880 $ $ 0.041186 $ $ 0.053692 $
$ \chi $ $ 0.031486 $ $ 0.015743 $ $ 0.001120 $ $ 0.036524 $ $ 0.012594 $
$ \mu $ $ 0.497512 $ $ 0.222222 $ $ 0.300000 $ $ 0.500000 $ $ 0.500000 $
$ u_{\max} = \beta_{I} $ $ 0.160000 $ $ 0.128000 $ $ 0.315000 $ $ 3.532000 $ $ 0.588000 $
$ u_{\min} $ $ 0.123077 $ $ 0.098462 $ $ 0.242308 $ $ 2.716923 $ $ 0.452308 $
$ v_{\max} = \beta_{H} $ $ 0.062000 $ $ 0.080000 $ $ 0.016500 $ $ 0.012000 $ $ 0.794000 $
$ v_{\min} $ $ 0.047692 $ $ 0.061538 $ $ 0.012692 $ $ 0.009231 $ $ 0.610769 $
$ w_{\max} = \beta_{F} $ $ 0.489000 $ $ 0.111000 $ $ 0.160000 $ $ 0.462000 $ $ 7.653000 $
$ w_{\min} $ $ 0.376154 $ $ 0.085385 $ $ 0.123077 $ $ 0.355385 $ $ 5.886923 $
$ \eta_{\max} $ $ 0.797512 $ $ 0.522222 $ $ 0.600000 $ $ 0.800000 $ $ 0.800000 $
$ \eta_{\min} = \mu $ $ 0.497512 $ $ 0.222222 $ $ 0.300000 $ $ 0.500000 $ $ 0.500000 $
$ \lambda = \rho + \chi $ $ 0.081138 $ $ 0.135551 $ $ 0.016000 $ $ 0.077710 $ $ 0.066286 $
$ \nu = \gamma + \delta + \sigma $ $ 0.117734 $ $ 0.115873 $ $ 0.795000 $ $ 0.193874 $ $ 0.168100 $
parameter or constraint Liberia
(2014)
Sierra Leone
(2014)
Guinea
(2014)
Uganda
(2000)
Congo
(1995)
$ \alpha $ $ 0.083333 $ $ 0.100000 $ $ 0.111111 $ $ 0.083333 $ $ 0.142857 $
$ \gamma $ $ 0.060802 $ $ 0.047815 $ $ 0.500000 $ $ 0.154762 $ $ 0.134000 $
$ \delta $ $ 0.026767 $ $ 0.010038 $ $ 0.005900 $ $ 0.018550 $ $ 0.006600 $
$ \sigma $ $ 0.030165 $ $ 0.058020 $ $ 0.289100 $ $ 0.020562 $ $ 0.027500 $
$ \rho $ $ 0.049652 $ $ 0.119808 $ $ 0.014880 $ $ 0.041186 $ $ 0.053692 $
$ \chi $ $ 0.031486 $ $ 0.015743 $ $ 0.001120 $ $ 0.036524 $ $ 0.012594 $
$ \mu $ $ 0.497512 $ $ 0.222222 $ $ 0.300000 $ $ 0.500000 $ $ 0.500000 $
$ u_{\max} = \beta_{I} $ $ 0.160000 $ $ 0.128000 $ $ 0.315000 $ $ 3.532000 $ $ 0.588000 $
$ u_{\min} $ $ 0.123077 $ $ 0.098462 $ $ 0.242308 $ $ 2.716923 $ $ 0.452308 $
$ v_{\max} = \beta_{H} $ $ 0.062000 $ $ 0.080000 $ $ 0.016500 $ $ 0.012000 $ $ 0.794000 $
$ v_{\min} $ $ 0.047692 $ $ 0.061538 $ $ 0.012692 $ $ 0.009231 $ $ 0.610769 $
$ w_{\max} = \beta_{F} $ $ 0.489000 $ $ 0.111000 $ $ 0.160000 $ $ 0.462000 $ $ 7.653000 $
$ w_{\min} $ $ 0.376154 $ $ 0.085385 $ $ 0.123077 $ $ 0.355385 $ $ 5.886923 $
$ \eta_{\max} $ $ 0.797512 $ $ 0.522222 $ $ 0.600000 $ $ 0.800000 $ $ 0.800000 $
$ \eta_{\min} = \mu $ $ 0.497512 $ $ 0.222222 $ $ 0.300000 $ $ 0.500000 $ $ 0.500000 $
$ \lambda = \rho + \chi $ $ 0.081138 $ $ 0.135551 $ $ 0.016000 $ $ 0.077710 $ $ 0.066286 $
$ \nu = \gamma + \delta + \sigma $ $ 0.117734 $ $ 0.115873 $ $ 0.795000 $ $ 0.193874 $ $ 0.168100 $
Table 2.  Values of expressions $ B_{j}^{2} - 4A_{j}C_{j} > 0 $, $ j = \overline{1,4} $ for 1995 Ebola epidemic in Congo, 2000 Ebola epidemic in Uganda and 2014 Ebola epidemics in Liberia, Guinea and Sierra Leone.
value Liberia
(2014)
Sierra Leone
(2014)
Guinea
(2014)
Uganda
(2000)
Congo
(1995)
$ B_{1}^{2} - 4A_{1}C_{1} $ $ 3.521008 $ $ 2.109150 $ $ 3.623030 $ $ 32.461802 $ $ 083.275796 $
$ B_{2}^{2} - 4A_{2}C_{2} $ $ 5.670857 $ $ 3.829651 $ $ 5.286568 $ $ 62.011891 $ $ 165.813664 $
$ B_{3}^{2} - 4A_{3}C_{3} $ $ 1.528994 $ $ 0.923566 $ $ 3.221486 $ $ 86.006805 $ $ 220.629650 $
$ B_{4}^{2} - 4A_{4}C_{4} $ $ 3.609691 $ $ 1.779715 $ $ 3.068885 $ $ 64.706358 $ $ 323.796477 $
value Liberia
(2014)
Sierra Leone
(2014)
Guinea
(2014)
Uganda
(2000)
Congo
(1995)
$ B_{1}^{2} - 4A_{1}C_{1} $ $ 3.521008 $ $ 2.109150 $ $ 3.623030 $ $ 32.461802 $ $ 083.275796 $
$ B_{2}^{2} - 4A_{2}C_{2} $ $ 5.670857 $ $ 3.829651 $ $ 5.286568 $ $ 62.011891 $ $ 165.813664 $
$ B_{3}^{2} - 4A_{3}C_{3} $ $ 1.528994 $ $ 0.923566 $ $ 3.221486 $ $ 86.006805 $ $ 220.629650 $
$ B_{4}^{2} - 4A_{4}C_{4} $ $ 3.609691 $ $ 1.779715 $ $ 3.068885 $ $ 64.706358 $ $ 323.796477 $
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