Analysis and Applications of Complex Dynamical Systems

许多复杂的自然、物理和人工系统都可以在定义在网络上的互联动力系统的框架下进行数学研究。在数学术语中，网络是一个加权有向图，它由大量的顶点和边上有权的顶点之间的有向边组成。在每个顶点上定义了一个简单的动力系统，加权边编码了顶点系统之间的相互联系或相互作用。这就产生了一个大规模的复杂动力系统。PI研究的长期目标是为研究网络上相互关联的动力系统发展数学理论，并为分析现实世界的复杂系统开发有效的数学工具。提出的研究包括两个主要主题：理论发展和现实应用。主题一：理论发展。第一个主要研究问题是进一步推广图论方法来构造网络上动力系统的Lyapunov函数，这一方法已经在许多科学和工程领域的数学模型分析中得到了丰硕的应用。主要的焦点是定义在多重图上的动力系统，以前的理论对这些系统只取得了有限的成功。这一进展有望为解决科学和工程中复杂数学模型的全局稳定性问题提供新的技术。第二个主要研究问题是建立和分析一类新的国家结构的传染病传播模型，无论是在人群水平上还是在宿主内部。我们将研究离散和连续的状态结构。对结果模型的分析可以导致对许多重要传染病的新理解，包括艾滋病毒感染、麻疹、结核病和黄热病。主题二：现实世界的应用。这一主题的研究集中于整合数学模型和临床数据，以调查与传染病有关的重大现实问题。第一个主要研究问题涉及艾滋病毒感染的潜伏病毒库以及消除潜伏病毒库以治愈艾滋病毒的“休克和杀死”方法。这项研究将建立在与艾伯塔大学医学院临床研究人员现有的卓有成效的合作基础上，对艾滋病毒/SIV感染的脑进行建模，脑是艾滋病毒感染的自然宿主。第二个主要的研究问题是促进对艾滋病毒感染免疫反应的关键差异的理解，这些差异将艾滋病毒感染的精英控制者与典型的艾滋病毒患者区分开来。将根据临床数据构建和校准针对艾滋病毒感染的免疫反应的新的宿主内数学模型。这项研究的新见解可以帮助艾滋病毒疫苗的开发。

Many complex natural, physical and manmade systems can be investigated mathematically in the framework of interconnected dynamical systems defined on networks. In mathematical terms, a network is a weighted directed graph that consists of a large number of vertices and directed edges among vertices with weights on the edges. On each vertex, a simple dynamical system is defined and the weighted edges encodes interconnections or interactions among the vertex systems. This gives rise to a large-scale complex dynamical system. The long-term objective of PI's research is to develop mathematical theories for the investigation of interconnected dynamical systems on networks and develop effective mathematical tools for the analysis of real-world complex systems. The proposed research consists of two main themes: theoretical development and real-world applications. Theme I: Theoretical development. The first main research problem is to further extend the graph-theoretic approach to the construction of Lyapunov functions for dynamical systems on networks developed by the PI and his graduate students, which has already seen fruitful applications in the analysis of mathematical models in many areas of science and engineering. The main focus is on dynamical systems defined on multigraphs for which the previous theories only had limited success. This development is expected to provide new techniques to resolve many existing open questions on the global stability for complex mathematical models from science and engineering. The second main research problem is to formulate and analyze a new class of state-structured models for the transmission of infectious diseases, both at the population level and within host. Both discrete and continuous state structures will be investigated. The analysis of the resulting models can lead to new understandings of many important infectious diseases including HIV infection, Measles, Tuberculosis, and Yellow fever. Theme II: Real-world applications. Research in this theme focuses on integrating mathematical models and clinical data to investigate significant real-world problems related to infectious diseases. The first main research problem deals with the latent viral reservoirs of HIV infection and the "shock and kill" approach to the elimination of latent reservoirs to achieve a cure of HIV. The research will be built on existing fruitful collaborations with clinical researchers at the University of Alberta Faculty of Medicine on modeling the HIV/SIV infection of brain, a natural reservoir for HIV infection. The second main research problem is to advance the understanding of key differences in immune responses to HIV infection that differentiate elite controllers of HIV infection from typical HIV patients. New classes of within-host mathematical models for the immune responses to the HIV infection will be constructed and calibrated from clinical data. New insights from this research can help HIV vaccine development.