Optimal control and free optimal time problem for a COVID-19 model with saturated vaccination function

Abstract

In this paper, we investigate a free terminal time optimal control applied to 6 ordinary differential equations which describe the spread of COVID-19 infection. We propose an extension of the classical Susceptible-Exposed-Infectious-Recovered (SEIR) model, where the infectious patients are divided into unreported (U) and reported cases (I). To have a more realistic model, we estimate the parameters of our model using real Moroccan data. We use Bootstrap as a statistical method to improve the reliability of the parameters estimates. The main goal of this work is to find the optimal control strategy and to determine the optimal duration of a vaccination campaign adequate to eradicate the infection in Morocco. For this, we introduce into the model a saturated vaccination function, which takes into account the limited resources on the COVID-19 vaccine, and we formulate a minimization problem where the final time is considered to be free. The existence of optimal control is investigated. The characterization of the sought optimal control and optimal final time is derived based on Pontryagin’s maximum principle. Using Matlab, we solve the optimality system with an iterative method based on the iterative Forward-Backward Sweep Method (FBSM). The numerical simulation results show the efficiency of a vaccination strategy on reducing the number of infectious individuals within an optimal period time which is approximately equal to 44 days.