Determination of the optimal controls for an Ebola epidemic model

Ellina Grigorieva; Evgenii Khailov

doi:10.3934/dcdss.2018062

Article Contents

2018, Volume 11, Issue 6: 1071-1101. Doi: 10.3934/dcdss.2018062

This issue Previous Article On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations Next Article Linear openness and feedback stabilization of nonlinear control systems

Determination of the optimal controls for an Ebola epidemic model

Ellina Grigorieva^1, , and
Evgenii Khailov^2,

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
* Corresponding author: Ellina Grigorieva

Received: January 31, 2017

Revised: March 31, 2017

Published: December 2018

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Abstract

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.

Keywords:

SEIR model,
control the spread of Ebola epidemic,
nonlinear control system,
Pontryagin maximum principle,
non-autonomous quadratic differential system,
generalized Rolle's theorem.

Mathematics Subject Classification: Primary: 49N90, 90C90; Secondary: 93C95.

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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).

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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).

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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 $

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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 $

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