    • Previous Article
Linear openness and feedback stabilization of nonlinear control systems
• DCDS-S Home
• This Issue
• Next Article
On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations
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:
  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. Google Scholar  P. S. Aleksandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977. Google Scholar  A. I. Astrovskii and I. V. Gaishun, Quasidifferentiability and observability of linear nonstationary systems, Diff. Equat., 45 (2009), 1602-1611.  doi: 10.1134/S0012266109110061.  Google Scholar  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.  Google Scholar  B. P. Demidovich, Lectures on Stability Theory, Nauka, Moscow, 1967. Google Scholar  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.  Google Scholar  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.  Google Scholar  B. Ebenezer, K. Badu and A.-A. S. Kwesi, Optimal control application to an Ebola model, Health Science Journal, 10 (2016), 1-7.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983.   Google Scholar  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. Google Scholar  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. Google Scholar  E. Grigorieva, E. 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.  Google Scholar  E. V. Grigorieva, E. 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.  Google Scholar  P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. Google Scholar  D. Hincapié-Palacio, J. 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.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. Google Scholar  J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.   Google Scholar  H. Maurer, C. Büskens, J.-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.  Google Scholar  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. Google Scholar  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.  Google Scholar  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  F. P. vasilev, Optimization Methods, Factorial Press, Moscow, 2002. Google Scholar

show all references

References:
  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. Google Scholar  P. S. Aleksandrov, Introduction to Set Theory and General Topology, Nauka, Moscow, 1977. Google Scholar  A. I. Astrovskii and I. V. Gaishun, Quasidifferentiability and observability of linear nonstationary systems, Diff. Equat., 45 (2009), 1602-1611.  doi: 10.1134/S0012266109110061.  Google Scholar  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.  Google Scholar  B. P. Demidovich, Lectures on Stability Theory, Nauka, Moscow, 1967. Google Scholar  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.  Google Scholar  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.  Google Scholar  B. Ebenezer, K. Badu and A.-A. S. Kwesi, Optimal control application to an Ebola model, Health Science Journal, 10 (2016), 1-7.   Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  E. V. Grigorieva and E. N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics, Mathematics, 3 (2015), 961-983.   Google Scholar  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. Google Scholar  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. Google Scholar  E. Grigorieva, E. 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.  Google Scholar  E. V. Grigorieva, E. 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.  Google Scholar  P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, 1964. Google Scholar  D. Hincapié-Palacio, J. 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.  Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  E. B. Lee and L. Marcus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. Google Scholar  J. Legrand, R. F. Grais, P. Y. Boelle, A. J. Valleron and A. Flahault, Understanding the dynamics of Ebola epidemics, Epidemiol. Infect., 135 (2007), 610-621.   Google Scholar  H. Maurer, C. Büskens, J.-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.  Google Scholar  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. Google Scholar  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.  Google Scholar  L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, John Wiley & Sons, New York, 1962. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  F. P. vasilev, Optimization Methods, Factorial Press, Moscow, 2002. Google Scholar 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). 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).
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) $\alpha$ $0.083333$ $0.100000$ $0.111111$ $0.083333$ $0.142857$ $\gamma$ $0.060802$ $0.047815$ $0.500000$ $0.154762$ $0.134000$ $\delta$ $0.026767$ $0.010038$ $0.005900$ $0.018550$ $0.006600$ $\sigma$ $0.030165$ $0.058020$ $0.289100$ $0.020562$ $0.027500$ $\rho$ $0.049652$ $0.119808$ $0.014880$ $0.041186$ $0.053692$ $\chi$ $0.031486$ $0.015743$ $0.001120$ $0.036524$ $0.012594$ $\mu$ $0.497512$ $0.222222$ $0.300000$ $0.500000$ $0.500000$ $u_{\max} = \beta_{I}$ $0.160000$ $0.128000$ $0.315000$ $3.532000$ $0.588000$ $u_{\min}$ $0.123077$ $0.098462$ $0.242308$ $2.716923$ $0.452308$ $v_{\max} = \beta_{H}$ $0.062000$ $0.080000$ $0.016500$ $0.012000$ $0.794000$ $v_{\min}$ $0.047692$ $0.061538$ $0.012692$ $0.009231$ $0.610769$ $w_{\max} = \beta_{F}$ $0.489000$ $0.111000$ $0.160000$ $0.462000$ $7.653000$ $w_{\min}$ $0.376154$ $0.085385$ $0.123077$ $0.355385$ $5.886923$ $\eta_{\max}$ $0.797512$ $0.522222$ $0.600000$ $0.800000$ $0.800000$ $\eta_{\min} = \mu$ $0.497512$ $0.222222$ $0.300000$ $0.500000$ $0.500000$ $\lambda = \rho + \chi$ $0.081138$ $0.135551$ $0.016000$ $0.077710$ $0.066286$ $\nu = \gamma + \delta + \sigma$ $0.117734$ $0.115873$ $0.795000$ $0.193874$ $0.168100$
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) $B_{1}^{2} - 4A_{1}C_{1}$ $3.521008$ $2.109150$ $3.623030$ $32.461802$ $083.275796$ $B_{2}^{2} - 4A_{2}C_{2}$ $5.670857$ $3.829651$ $5.286568$ $62.011891$ $165.813664$ $B_{3}^{2} - 4A_{3}C_{3}$ $1.528994$ $0.923566$ $3.221486$ $86.006805$ $220.629650$ $B_{4}^{2} - 4A_{4}C_{4}$ $3.609691$ $1.779715$ $3.068885$ $64.706358$ $323.796477$
  Roberta Fabbri, Russell Johnson, Carmen Núñez. On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 677-704. doi: 10.3934/dcds.2003.9.677  Guy Barles, Ariela Briani, Emmanuel Trélat. Value function for regional control problems via dynamic programming and Pontryagin maximum principle. Mathematical Control & Related Fields, 2018, 8 (3&4) : 509-533. doi: 10.3934/mcrf.2018021  Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110  Julia Amador, Mariajesus Lopez-Herrero. Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3137-3151. doi: 10.3934/dcdsb.2017211  Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022  Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61  Joaquim P. Mateus, Paulo Rebelo, Silvério Rosa, César M. Silva, Delfim F. M. Torres. Optimal control of non-autonomous SEIRS models with vaccination and treatment. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1179-1199. doi: 10.3934/dcdss.2018067  IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial & Management Optimization, 2018, 14 (2) : 427-446. doi: 10.3934/jimo.2017054  Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018  Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161  María Anguiano, Tomás Caraballo. Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 3901-3920. doi: 10.3934/dcds.2014.34.3901  Xiao-Li Ding, Iván Area, Juan J. Nieto. Controlled singular evolution equations and Pontryagin type maximum principle with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021059  M. H. A. Biswas, L. T. Paiva, MdR de Pinho. A SEIR model for control of infectious diseases with constraints. Mathematical Biosciences & Engineering, 2014, 11 (4) : 761-784. doi: 10.3934/mbe.2014.11.761  Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023  Birgit Jacob, Hafida Laasri. Well-posedness of infinite-dimensional non-autonomous passive boundary control systems. Evolution Equations & Control Theory, 2021, 10 (2) : 385-409. doi: 10.3934/eect.2020072  Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195  H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77  M. Predescu, R. Levins, T. Awerbuch-Friedlander. Analysis of a nonlinear system for community intervention in mosquito control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 605-622. doi: 10.3934/dcdsb.2006.6.605  Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4959-4985. doi: 10.3934/dcds.2021064  Thorsten Hüls. A model function for non-autonomous bifurcations of maps. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 351-363. doi: 10.3934/dcdsb.2007.7.351

2020 Impact Factor: 2.425