Analysis and Applications of Complex Dynamical Systems

许多复杂的自然、物理和人造系统可以在网络上定义的互连动力系统的框架中进行数学研究。用数学术语来说，网络是一个带权有向图，由大量顶点和顶点之间的有向边组成，边上有权重。在每个顶点上，定义了一个简单的动态系统，加权边对顶点系统之间的互连或交互进行编码。这产生了一个大规模的复杂动力系统。 PI研究的长期目标是发展数学理论来研究网络上相互关联的动力系统，并开发有效的数学工具来分析现实世界的复杂系统。拟议的研究包括两个主题：理论发展和实际应用。 主题一：理论发展。第一个主要研究问题是进一步扩展图论方法来构建由PI和他的研究生开发的网络动力系统的Lyapunov函数，该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卓有成效的应用。主要关注的是多重图上定义的动力系统，之前的理论只取得了有限的成功。这一进展预计将提供新技术来解决科学和工程领域复杂数学模型的全局稳定性方面的许多现有开放性问题。第二个主要研究问题是制定和分析一类新的国家结构模型，用于传染病在人口层面和宿主内部的传播。将研究离散和连续状态结构。对所得模型的分析可以带来对许多重要传染病的新认识，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。该主题的研究重点是整合数学模型和临床数据，以研究与传染病相关的重大现实问题。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实现艾滋病毒治愈的“休克和杀灭”方法。该研究将建立在与阿尔伯塔大学医学院临床研究人员现有的富有成效的合作基础上，对大脑的 HIV/SIV 感染进行建模，而大脑是 HIV 感染的天然储存库。第二个主要研究问题是促进对 HIV 感染免疫反应的关键差异的理解，这些差异将 HIV 感染的精英控制者与典型的 HIV 患者区分开来。将根据临床数据构建和校准针对 HIV 感染的免疫反应的新型宿主内数学模型。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.