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.

在本文中，提出了SIRV流行病学模型的一般公式，它是相对于时间的分数阶导数系统，用于表征某些传染病以及u ₁和u ₂的比例，分别描述了疫苗接种和治疗。该分数数学公式基于分数阶Caputo导数。利用分数阶微分方程（FDEs）的稳定性定理研究平衡点（EPs）的稳定性，并计算该模型的基本重现次数。在没有接种疫苗和未进行治疗的情况下，分数欧拉法（FEM）用于获得建议模型的近似解。然后，我们制定了分数最优控制问题（FOCP），并使用Ponntryagin的最大原理推导了分数阶必要最优条件（NOC），其中以左Caputo分数阶导数（LCFD）的形式陈述了状态方程和伴随方程便于使用分数数值方法来求解此FOCP。通过使用有限元法开发向前-向后扫描方法（FBSM），可以对所得的最优系统进行数值求解。此外，我们讨论了最优结果的仿真，并在此FOCP中提出了三种控制策略，这取决于两个建议控制的组合。