Functional Differential Equations in Biology and Epidemiology

生物学和流行病学中的泛函微分方程

• 批准号: RGPIN-2017-04257
• 负责人: Yuan, Yuan
• 金额: $ 1.46万
• 依托单位: Memorial University of Newfoundland
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759183
• 关键词: Functional; Differential; Equations; Biology; Epidemiology

A distinguishing feature of the functional differential equations (FDEs) is that the evolution rate of the process described by such equations depends on the past history. During the past three decades, study on the theory and application of FDEs has been very active, the FDE related mathematical models are applicable to phenomena of different natures, such as, in biology, ecology, epidemiology, engineering and economy.The long-term goals of the research include, understanding in-depth the dynamical behavior and the reasons that govern such behavior in biological and epidemiological systems that can be illustrated by FDEs, especially how time delay and the system structure affect the evolution of the species, whether the infectious disease can spread and persist, or lead to an infectious disease outbreak. Mathematical techniques based on nonlinear analysis, FDEs and dynamical systems including existence, non-existence, stability of certain type of periodic/quasi-periodic solutions, persistence, bifurcations will be mainly approached.More realistic models may relate to the variation of environment which result in non-autonomous systems. In the literature, although some work has been done for autonomous and non-autonomous FDEs, there is a lack of theoretical analysis, especially for the stability and bifurcations. In the proposed research I will seek to extend and develop the stability and bifurcation theories to non-monotonic autonomous, non-autonomous, spatial and temporal involved FDEs with delays . And I will apply the theoretical results to tackle problems of qualitative analysis of dynamical behaviors arising in biological and epidemiological systems with stage/age-structures, or even disease transmission network models, with particular emphasis on establishing strong links on the effect of time delay, system construction and the dynamical behavior. I am also flexible enough to explore promising avenues of research as they emerge.This proposed research will increase our knowledge of the qualitative properties in applied FDEs, from mathematical and biological/epidemiological points of view. The theories and methodologies developed in the proposal will have very high impact on the development in the nonlinear dynamics community. They will not only strengthen the foundation for theoretical expansion in a large class of FDEs, but will also provide a practical view to support biological and epidemiological decision making. The interdisciplinary collaboration and the application to the fishery will have great potential to provide novel insights to strength marine population dynamics, optimize harvesting rates, and maintain ecosystem structure. The anticipated outcomes are likely to have impact by leading to advancements in the field of applied dynamics and will enhance HQP training and collaboration benefiting in Canada.

泛函微分方程（FDES）的一个显着特点是，这种方程所描述的过程的演化速率依赖于过去的历史。在过去的三十年里，关于FDE的理论和应用的研究非常活跃，FDE相关的数学模型适用于不同性质的现象，如生物学、生态学、流行病学、工程学和经济学。理解-深入的动力学行为和原因，支配这种行为在生物和流行病学系统，可以说明的FDES，特别是时滞和系统结构如何影响物种的进化，传染病是否能够传播和持续，或者导致传染病爆发。本课程主要探讨以非线性分析、有限差分方程及动力系统为基础的数学技巧，包括周期/拟周期解的存在性、不存在性、稳定性、持久性、分支等，更实际的模型可能与环境变化有关，从而导致非自治系统。在文献中，虽然已经对自治和非自治的FDE做了一些工作，但缺乏理论分析，特别是对稳定性和分支的分析。在拟议的研究中，我将寻求将稳定性和分歧理论扩展和发展到非单调自治、非自治、空间和时间相关的具有延迟的FDE。我将应用这些理论结果来解决具有阶段/年龄结构的生物和流行病学系统，甚至疾病传播网络模型中出现的动力学行为的定性分析问题，特别强调建立时滞，系统结构和动力学行为的影响之间的强联系。我也有足够的灵活性，探索有前途的研究途径，因为他们emerging.This拟议的研究将增加我们的知识，在应用FDES的定性属性，从数学和生物/流行病学的观点。该提案中提出的理论和方法将对非线性动力学社区的发展产生非常高的影响。它们不仅将加强在一个大类的FDES的理论扩展的基础，但也将提供一个实用的观点，以支持生物学和流行病学的决策。跨学科的合作和渔业的应用将有很大的潜力提供新的见解，以加强海洋种群动态，优化捕捞率，并保持生态系统结构。预期的结果可能会产生影响，导致应用动力学领域的进步，并将加强HQP培训和合作，使加拿大受益。