Research Interests

 

Mathematical Theory of Differential Equations and Dynamical Systems.

  • Existence, non-existence, and stability of certain type of solutions, such as periodic solutions, quasi-periodic solutions, and almost periodic solutions. Our main approach is to study the evolution of various dimensional volumes under the nonlinear flow of the differential equation. This amounts to studying the dichotomies and other asymptotic behaviors of the solutions to associated compound differential equations of various orders.
  • Since 2006, due to my interests in epidemiological models in heterogeneous populations, I begin to study large-scale inter-connected systems of differential equations on networks. A network is represented by a weighted directed graph. At each vertex, a small system of differential equations is assigned, edges of the graph provide inter-connections among the vertex systems, and the strength of inter-connections can be described by the weights on edges. Large-scale systems on networks provide a mathematical framework to study many large scale from all fields of science and engineering.

    The first problem of interest is the existence, uniqueness and global stability of a positive steady state. In a result of my students and myself in 2010, we have discovered a graph-theorectic approach that allows systematic construction of global Lyapunov functions for a very general class of large-scale systems on networks. Among many new problems we have solved using this approach is the open problem of the uniqueness and global stability of the endemic equilibrium in multi-group epidemic models.

    Other problems of interest are synchronization problems in coupled oscillators, ranking problems for vertices on networks, and complexity problems in large-scale systems of coupled simple equations.

Mathematical Modelling of Immune System

    Human immune system is a very complex and dynamic that involves many different types of cells and immunological pathways. The immune response to infections also comes in different stages. Mathematical modeling using differential equations and dynamical systems has been used in the studies of immune response to infections from different type of pathogens, most notably the HIV.

  • My research interest in this area includes the modeling of the in vivo infection process of certain retro-viruses and how the immune system responds to the infection, as well the interaction of the infection, immune system and various treatment measures. One of my current projects studies the infection of HTLV-I (Human T-cell Lymphotropic Virus Type I), which is a retro virus and also an oncogenic virus.
  • My research group is also conducting interdisciplinary research projects with virologists at the Li Ka Shing Institute of Virology. In one of the recent projects, we are working with virologists, nephrologists and surgeons to investigate mechanisms and outcomes of infection from BKV and EBV among kidney transplant patients, using mathematical modeling. We expect that our results will lead to improved prognosis for transplant patients.

Mathematical Modeling in Public Health Sciences.

    My group has several on-going interdisciplinary research projects that use mathematical models to investigate transmission dynamics and related public-health issues for specific infectious diseases.

  • In one project, we are working with physicians, epidemiologists and public health researchers to model the transmission dynamics of Tuberculosis (TB) in aboriginal communities in Alberta. An objective of the modeling project is to quantitatively relate various social determinants for TB to TB incidences in a community or communities. Such research will enable health-economic analysis on potential TB intervention measures that are directed at social determinants, and provide research evidence to inform policies for TB control among aboriginal populations.
  • In anther project, we are developing a new methodology for estimation of HIV incidence using public health surveillance data and HIV transmission models. We are currently working with datasets from Alberta Health and China CDC. Our approach integrates the theories of dynamical systems and statistical inference.

    These projects offer students research and training opportunities for a diverse skill set: analytical, computational, and statistical skills. They also offer excellent opportunities for interdisciplinary research.

My Recent Publications.

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This page was last updated on November 15, 2012.