Mathematical Analysis and Statistical Inference of Complex Dynamical Systems

拟议研究的目标是：(1)开发新的数学理论和分析工具，用于研究科学和工程许多领域产生的大型复杂动力系统，使用由有向图表示的网络上定义的相互关联的非线性微分方程组的数学框架；以及(2)开发用于公共健康评估的严格的数学建模方法。 这项研究将解决复杂和大规模动力系统的理论和实践方面的重大挑战，例如： -相互关联的简单顶点系统如何产生复杂的系统行为？ -如何在数学上对不同类型的复杂系统行为进行分类和描述？ -每种类型的系统行为如何与几何图形的网络拓扑相关联？ -互联系统的复杂性如何影响其稳定性？ -开发现实的数学模型，以调查不同规模的传染病的复杂时间和空间动态； -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，并评估潜在的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，以便更准确地分析疫苗接种方案的成本和效果。 -开发一种有效的方法来检测、评估和解决从疾病数据进行参数估计和模型拟合中的不可辨识性这一重要问题。 -开发用户友好的计算机软件包，使公共卫生最终用户的基于模型的评估工具箱自动化。 该研究将稳定性和分叉分析与现代图论相结合，将推进和丰富大型复杂动力系统的理论，对科学和工程上的复杂大系统分析具有直接的影响。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一种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。预计研究结果将为公共卫生政策提供信息，并为普通公众带来健康益处。

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 non­identifiability 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.