Analysis and Applications of Complex Dynamical Systems

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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.