Stamov Gani, Stamova Ivanka, Spirova Cvetelina
Department of Mathematical Physics, Technical University of Sofia, 8800 Sliven, Bulgaria.
Department of Mathematics, University of Texas at San Antonio, San Antonio, TX 78249, USA.
Entropy (Basel). 2021 Dec 3;23(12):1631. doi: 10.3390/e23121631.
In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds' existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology-the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.
在本文中，我们研究了一个应用于生物学的脉冲时滞反应扩散模型。所引入的模型将现有的反应扩散时滞流行病模型推广到了脉冲情形。在考虑的模型中引入了积分流形的概念。这个概念扩展了单状态概念，并且在多稳态系统的研究中有重要应用。通过扩展李雅普诺夫方法，建立了积分流形的存在性结果。基于李雅普诺夫函数技术并结合庞加莱型不等式，还给出了与积分流形的有界性、持久性和稳定性相关的定性准则。所提出的脉冲控制模型的应用与数学生物学中一个非常重要的问题——流行病模型的最优控制问题密切相关。所考虑的脉冲效应可被流行病学家用作一种非常有效的治疗控制策略。此外，由于积分流形方法在各种背景下都相关，我们的结果可应用于对各种不同感兴趣的流行病学问题的定性研究。
