With increased longevity in a nowadays society, Alzheimer’s disease (AD), an incurable neurodegenerative disease, becomes a serious threat to human health. To address the challenges to understanding and predicting AD development and treatment, we focus on the development of a stochastic reaction-diffusion model with time-varying delay and impulsive perturbations, which is driven by L$\acute{\mathrm{e}}$vy jump process to model the in vivo progression of AD. Moreover, based on an bounded impulsive interval method, certain sufficient conditions of finite-time stability are provided. Further, from a cost-benefit perspective, an optimal control problem of AD is formulated to minimize the pathogenic proteins and control cost. Several illustrative examples are presented to demonstrate and verify theoretical results of this study.

随着当今社会寿命的延长，阿尔茨海默病 (AD) 是一种无法治愈的神经退行性疾病，已成为对人类健康的严重威胁。为了解决理解和预测 AD 发展和治疗的挑战，我们专注于开发具有时变延迟和脉冲扰动的随机反应扩散模型，该模型由 L 驱动$\acute{\mathrm{电子}}$vy 跳跃过程来模拟 AD 的体内进展。此外，基于有界脉冲区间方法，给出了有限时间稳定性的某些充分条件。此外，从成本效益的角度来看，制定了 AD 的最优控制问题，以最大限度地减少致病蛋白和控制成本。给出了几个说明性的例子来证明和验证本研究的理论结果。