This paper addresses the optimal vaccination and treatment control problems for regional approximate controllability of a new spatiotemporal epidemic model that is developed by afterwards adding at the basic susceptible–infected–recovered (SIR) epidemic system with the Caputo time-fractional derivative of order $\alpha \in (0,1]$ and the diffusion term in each compartment. The obtained results can be used by policy-makers in any nation to optimally plan the limited vaccination and treatment resources ahead of ongoing outbreaks. Toward this aim the Sakawa-type controller is introduced, which is a finite-dimensional controller that prevents and controls the spreading of infection. Using the semigroup theory, we provide a framework to analyze the sufficient conditions under which the considered time-fractional reaction–diffusion SIR epidemic system is regionally approximately controllable with the proposed control actuation architecture. An approach on finding the optimal solution that minimizes the cost of corresponding vaccination and treatment control programs for the studied regional controllability problem is then presented. Finally, we finish with numerical results to illustrate our theoretical results.

本文解决了新的时空流行模型的区域近似可控性的最佳疫苗接种和治疗控制问题 $\alpha \in (0,1]$以及每个隔室中的扩散项。任何国家的政策制定者都可以使用获得的结果在持续爆发之前优化计划有限的疫苗接种和治疗资源。为此，引入了 Sakawa 型控制器，它是一种有限维控制器，可防止和控制感染的传播。使用半群理论，我们提供了一个框架来分析所考虑的时间分数反应扩散 SIR 流行系统在区域上可通过所提出的控制驱动架构大致可控的充分条件。然后提出了一种寻找最佳解决方案的方法，该解决方案使所研究的区域可控性问题的相应疫苗接种和治疗控制计划的成本最小化。最后，