Mathematical Investigation 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 behaviours 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-theoretic 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. The graph-theoretic approach has been successfully applied by researchers to problems in many different fields of science and engineering.

  Other problems of interest to us are synchronization problems in coupled oscillators, ranking problems for vertices on networks, and complexity problems in large-scale systems of coupled simple equations. Some new problems we are studying includes patterns in the distributions of quantities of interest among vertices at the the equilibrium, as well as factors that impact the patterns. Improved understanding of these problems will have important applications in science and engineering problems.

Mathematical Modelling of Viral Dynamics and Immune Responses

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 of viruses, 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. In another project, we study the HIV-1 infection in brain. The brain is a natural reservoir for the HIV-1 virus where the virus can hide in latent form in long-lived brain macrophages. One of the topics of our brain HIV-1 study is to using modeling to evaluate the effectiveness of the "shock and kill" strategy for the HIV-1 clearance.

• 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. We are also applying mathematical modeling tools to study the differences of immune responses between elite controllers of HIV-1, who can keep the HIV-1 viral load under control without having to taking anti-retroviral drug therapies, and regular HIV-1 patients. Understanding from this work will help the development of effective HIV-1 vaccines.

Mathematical Modeling of Real-Life Public Health Problems

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. We are also developing new and effective methods for detecting and diagnosing nonidentifiability issues in parameter estimation for differential equations models.

• During the COVID-19 epidemic, our group has been collaborating with Alberta Health in providing modeling support to inform evidence-based public health responses to the pandemic.

• These real-life modeling projects inevitably involve real disease data, and calibration of models by fitting the model outcomes to data in order to determine the right range for model parameters. This requires a different skill set from that for theoretical modeling studies. Parameter estimation using Bayesian inference and maximum likelihood methods and methods for model selection are standard tools. Efficiency in numerical simulations using software packages (Matlab, R, Python, etc) is crucial for real-life modeling studies.

• Real-life modeling is not all about data, parameter estimation, and model simulations. In fact, real-life modeling presents some of the most challenging mathematical problems. One of the difficult mathematical challenges is the "nonidentifiability" issue in parameter estimation from data: infinitely many parameter values can produce the same good-fit between model outcome and data, but they lead to drastically different predictions on important quantities that are not observed in the data. How to detect the existence of nonidentifiability and determine the degree of nonidentifiability, especially in large-scale and complex models, and how to effectively reduce or remove the nonidentifiability continue to challenge mathematical modelers.