Mathematical Analysis and Statistical Inference of Complex Dynamical Systems

本研究的目标是：（1）发展新的数学理论和分析工具，用于研究科学和工程领域中出现的大规模和复杂的动力系统，使用定义在由有向图表示的网络上的非线性微分方程互联系统的数学框架;及（2）发展一套严谨的数学模式，以评估公众健康。*该研究将解决有关复杂和大规模动态系统的理论和实践方面的重大挑战，例如：*-相互连接的简单顶点系统如何引起复杂系统行为？*-如何对不同类型的复杂系统行为进行数学分类和描述？*-每种类型的系统行为如何与几何网络拓扑相关？*-互联系统的复杂性如何影响其稳定性？* * 制定严格的理论方法，将数学模型、疾病监测数据和统计推断方法结合起来，以分析过去的疾病数据，评估过去的疾病动态，并评价潜在疾病控制和预防战略的成本效益;*-开发基于模型的卫生经济学分析工具，以便对疫苗接种计划进行更准确的成本和效果分析。制定有效的方法来检测，评估和解决疾病数据参数估计和模型拟合中的不可识别性这一重要问题。开发方便用户的计算机软件包，使公共卫生最终用户的基于模型的评估工具箱自动化。将稳定性和分岔分析与现代图论相结合，将推动和丰富复杂大系统动力学理论的发展，并对科学和工程领域复杂大系统的分析产生直接影响。通过与加拿大和其他国家的传染病专家，流行病学家，公共卫生研究人员和公共卫生机构的跨学科合作，研究人员将进一步开发和应用数学模型作为有效的评估和调查工具，以解决与传染病有关的重要公共卫生问题。研究结果有望为公共卫生政策提供信息，并为公众带来健康益处。

The objectives of the proposed research are: (1) to develop new mathematical theories and analytic tools for the investigation of large--scale and complex dynamical systems arising from many fields of science and engineering, using a mathematical framework of interconnected systems of nonlinear differential equations defined over a network represented by a directed graph; and (2) to develop a rigorous mathematical modelling approach to public health assessment.***The research will address significant challenges regarding both theoretic and practical aspects of complex and large--scale dynamical systems such as:***- How do interconnected simple vertex systems give rise to complex system behaviours?***- How to classify and describe mathematically different types of complex system behaviours?***- How does each type of system behaviours correlate to the network topology of geometry?***- How does complexity in an interconnected system influence its stability?***- Develop realistic mathematical models to investigate the complex temporal and spatial dynamics of the infectious diseases at different scales;***- Develop a rigorous theoretical approach that integrate mathematical models, disease surveillance data, and methodologies of statistical inference to analyze past disease data, assess past disease dynamics, and evaluate the cost--effectiveness of potential disease control and prevention strategies;***- Develop model-based health economics analysis tools for more accurate cost and effects analysis of vaccination programs.***- Develop an effective approach to detect, evaluate, and resolve the important issue of non-identifiability in the parameter estimation and model fitting from disease data.***- Develop user-friendly computer software packages that automate the model-based assessment toolbox for public health end-users.******The research will advance and enrich the theory of large--scale and complex dynamical systems by integrating stability and bifurcation analysis with the modern graph theory, and have direct impact on the analysis of complex and large--scale systems from science and engineering. Through interdisciplinary collaborations with infectious disease specialist, epidemiologists, public heath researchers, and public health agencies in Canada and other countries, the researcher will further develop and apply mathematical modelling as an effective assessment and investigative tool to address important public health issues related to infectious diseases. Research results are expected to inform public health policies and bring health benefits to the general public.**