# 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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