Optimal Control for a SIR Epidemic Model with Nonlinear Incidence Rate

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
³ Centre de Recerca Matemática Campus de Bellaterra, Edifici C, 08193 Bellaterra, Barcelona, Spain
⁴ Departament de Matemátiques, Universitat Autónoma de Barcelona Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain

^* Corresponding author. E-mail: akorobeinikov@crm.cat

Abstract

The goal of this paper is to explore the impact of non-linearity of functional responses on the optimal control of infectious diseases. In order to address this issue, we consider a problem of minimization of the level of infection at the terminal time for a controlled SIR model, where the incidence rate is given by a non-linear unspecified function f(S,I). In this model we consider four distinctive control policies: the vaccination of the newborn and the susceptible individuals, isolation of the infected individuals, and an indirect policy aimed at reduction of the transmission. The Pontryagin maximum principle is used for the problem analysis. In this problem we prove that the optimal controls are bang-bang functions. Then, the maximum possible number of switchings of these controls is found. Based on this, we describe the possible behavior of the optimal controls.