Chaos, Solitons & Fractals ( IF 3.764 ) Pub Date : 2020-05-22 , DOI: 10.1016/j.chaos.2020.109892
I. Ameen; Dumitru Baleanu; Hegagi Mohamed Ali

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 u1 and u2, 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.

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