Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics
A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of m...
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| Format: | Article |
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MDPI AG
2015-10-01
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| Series: | Mathematics |
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| Online Access: | http://www.mdpi.com/2227-7390/3/4/961 |
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| author | Ellina V. Grigorieva Evgenii N. Khailov |
| author_facet | Ellina V. Grigorieva Evgenii N. Khailov |
| author_sort | Ellina V. Grigorieva |
| collection | DOAJ |
| description | A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided. |
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| id | doaj.art-1d9449f1fa0e491e937b6bb68b71f05a |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-13T04:58:49Z |
| publishDate | 2015-10-01 |
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| spelling | doaj.art-1d9449f1fa0e491e937b6bb68b71f05a2022-12-21T23:58:50ZengMDPI AGMathematics2227-73902015-10-013496198310.3390/math3040961math3040961Optimal Intervention Strategies for a SEIR Control Model of Ebola EpidemicsEllina V. Grigorieva0Evgenii N. Khailov1Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX 76204, USAFaculty of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow 119992, RussiaA SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided.http://www.mdpi.com/2227-7390/3/4/961SEIR modelcontrol the spread of Ebola epidemicnonlinear control systemPontryagin maximum principlenon-autonomous quadratic differential systemgeneralized Rolle’s theorem |
| spellingShingle | Ellina V. Grigorieva Evgenii N. Khailov Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics Mathematics SEIR model control the spread of Ebola epidemic nonlinear control system Pontryagin maximum principle non-autonomous quadratic differential system generalized Rolle’s theorem |
| title | Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics |
| title_full | Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics |
| title_fullStr | Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics |
| title_full_unstemmed | Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics |
| title_short | Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics |
| title_sort | optimal intervention strategies for a seir control model of ebola epidemics |
| topic | SEIR model control the spread of Ebola epidemic nonlinear control system Pontryagin maximum principle non-autonomous quadratic differential system generalized Rolle’s theorem |
| url | http://www.mdpi.com/2227-7390/3/4/961 |
| work_keys_str_mv | AT ellinavgrigorieva optimalinterventionstrategiesforaseircontrolmodelofebolaepidemics AT evgeniinkhailov optimalinterventionstrategiesforaseircontrolmodelofebolaepidemics |