Optimal Control for a SIR Epidemic Model with Nonlinear Incidence Rate

• Published in: Mathematical modelling of natural phenomena Vol. 11; no. 4; pp. 89 - 104
• Main Authors: Grigorieva, E.V., Khailov, E.N., Korobeinikov, A.
• Format: Journal Article
• Language: English
• Published: Les Ulis EDP Sciences 01.01.2016
• Subjects: 49J15; 58E25; 92D30; Disease control; Epidemics; generalized Rolle's theorem; Incidence; infectious disease control; Infectious diseases; Linearity; Maximum principle; non-autonomous Riccati equation; nonlinear control system; nonlinear incidence; Nonlinearity; Optimal control; Optimization; Pontryagin maximum principle; SIR model; Vaccination
• ISSN: 1760-6101; 0973-5348; 1760-6101
• DOI: 10.1051/mmnp/201611407

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.