An efficient algorithm for solving the fractional optimal control of SIRV epidemic model with a combination of vaccination and treatment

Abstract

• The problem of the fractional SIRV epidemic model has been presented. • The stability of the equilibrium points are studied. • The fractional optimal control problem with two controls efforts is formulated and studied. • Different control strategies are proposed to eradicate the disease in a specific period of time. • Numerical simulations are given to show the effect of vaccination and treatment. In this paper, a general formulation for the SIRV epidemiological model is presented as a system of fractional order derivatives with respect to time to characterize some infectious diseases alongside the proportion of u 1 and u 2 , that describe of vaccination and treatment, respectively. This fractional mathematical formulation is based on the fractional-order Caputo derivative. The stability of Equilibrium Points (EPs) is studied using the stability theorem of the Fractional Differential Equations (FDEs), and the basic reproduction number of this model is computed. The Fractional Euler Method (FEM) is used to obtain an approximate solution of the suggested model in the absence of vaccination and treatment. Then, we formulated a Fractional Optimal Control Problem (FOCP) and derived a fractional-order Necessary Optimality Conditions (NOCs) by using Ponntryagin’s maximum principle, where we stated the state and adjoint equations in the form of the Left Caputo Fractional Derivative (LCFD) to facilitate the use of fractional numerical methods to solve this FOCP. The resulting optimality system is solved numerically by developing the Forward-Backward Sweep Method (FBSM) using the FEM. Moreover, we discuss the simulation of optimality results and propose three control strategies in this FOCP depend on a combination of two suggested controls.