标题： 涉及Caputo分数阶时间导数和非线性扩散算子的反应-扩散SIR模型的分数阶微分方程 显示英文标题

标题： Fractional differential equations of a reaction-diffusion SIR model involving the Caputo-fractional time-derivative and a nonlinear diffusion operator

摘要： 本研究的主要目的是通过使用非局部的Caputo分数阶时间分数导数并采用$p$-Laplacian算子，来分析一个反应扩散的分数阶抛物线SIR流行病模型。 通过疫苗接种计划来实现免疫，该计划被视为控制变量。 寻找能够减少患病人数、相关疫苗接种和治疗费用的最优控制对，在有限的时间和空间内是我们的主要研究内容。 建立了时空SIR模型的非负解的存在性和唯一性。 还证明了最优控制的存在性。 此外，我们得到了以状态函数和伴随函数形式描述的最优控制。 然后，通过一个离散迭代方案解决最优性系统，在适当的测试后收敛，类似于前向-后向扫算法。 最后，给出了数值近似结果，以展示所提出的控制方案的有效性，该方案分别使用不同的分数阶值和$p$，即Caputo导数的阶数和$p$-Laplacian算子的阶数，得到了有意义的结果。 显示英文摘要

摘要： The main aim of this study is to analyze a fractional parabolic SIR epidemic model of a reaction-diffusion, by using the nonlocal Caputo fractional time-fractional derivative and employing the $p$-Laplacian operator. The immunity is imposed through the vaccination program, which is regarded as a control variable. Finding the optimal control pair that reduces the number of sick people, the associated vaccination, and treatment expenses across a constrained time and space is our main study. The existence and uniqueness of the nonnegative solution for the spatiotemporal SIR model are established. It is also demonstrated that an optimal control exists. In addition, we obtain a description of the optimal control in terms of state and adjoint functions. Then, the optimality system is resolved by a discrete iterative scheme that converges after an appropriate test, similar to the forward-backward sweep method. Finally, numerical approximations are given to show the effectiveness of the proposed control program, which provides meaningful results using different values of the fractional order and $p$, respectively the order of the Caputo derivative and the $p$-Laplacian operators.