American Institute of Mathematical Sciences

2015, 2015(special): 549-561. doi: 10.3934/proc.2015.0549

Optimal control for an epidemic in populations of varying size

 1 Department of Mathematics and Computer Sciences, Texas Woman's University, Denton, TX 76204 2 Department of Computer Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992 3 Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, 08193 Barcelona

Received  September 2014 Revised  February 2015 Published  November 2015

For a Susceptible-Infected-Recovered (SIR) control model with varying population size, the optimal control problem of minimization of the infected individuals at a terminal time is stated and solved. Three distinctive control policies are considered, namely the vaccination of the susceptible individuals, treatment of the infected individuals and an indirect policy aimed at reduction of the transmission. Such values of the model parameters and control constraints are used, for which the optimal controls are bang-bang. We estimated the maximal possible number of switchings of these controls, which task is related to the estimation of the number of zeros of the corresponding switching functions. Different approaches of estimating the number of zeros of the switching functions are applied. The found estimates enable us to reduce the optimal control problem to a considerably simpler problem of the finite-dimensional constrained minimization.
Citation: Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549
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References:
 [1] Roberta Fabbri, Russell Johnson, Carmen Núñez. On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 677-704. doi: 10.3934/dcds.2003.9.677 [2] 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 [3] 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 [4] 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 [5] Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3555-3565. doi: 10.3934/dcds.2013.33.3555 [6] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [7] Gastão S. F. Frederico, Delfim F. M. Torres. Noether's symmetry Theorem for variational and optimal control problems with time delay. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 619-630. doi: 10.3934/naco.2012.2.619 [8] 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 [9] 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 [10] 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 [11] María Anguiano, Tomás Caraballo. Asymptotic behaviour of a non-autonomous Lorenz-84 system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3901-3920. doi: 10.3934/dcds.2014.34.3901 [12] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [13] 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 [14] 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 [15] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [16] Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485 [17] 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 [18] Yuri Kifer. Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2687-2716. doi: 10.3934/dcds.2018113 [19] 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 [20] Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

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