Optimal control problem for a seir type model of ebola epidemics

Abstract

A Susceptible, Exposed, Infectious, and Recovered (SEIR) type control model describing the Ebola epidemic in a population of constant size is considered over a fixed time interval. This model is an extension of the well-known SEIR model and is more suitable to the study of the control mechanism of Ebola epidemics. Along with the traditional SEIR compartments, this model contains an isolated infectious compartment representing the number of infected and exposed individuals that have been isolated from the susceptible individuals. The model has two intervention controls reflecting efforts to protect susceptible individuals from infected and exposed individuals. Additionally,there are two control functions that define efforts for the detection and isolation of infected and exposed individuals. The minimization problem of the sum of total fractions of infected and exposed individuals and total weighted costs of control constraints over a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. Accordingly, the controls are bang-bang functions determined by the corresponding switching functions. In order to estimate the number of zeros of the switching functions, a new approach is proposed based on the analys is of the Cauchy problems for the derivatives of these functions. It is found that theoptimal controls of the original problem have at most one switching. This allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of three variables. Results of the numerical solution to this problem and their analysis are provided.