Research Interests, Dr. Michael Li, University of Alberta
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
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
These research 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 and training in knowledge translation, and prepare trainees for a variety of career choices (academic research, industry, and government).
My research group always has opennings for motivated students pursuing graduate training or undergraduate summer research
in applied mathematics and matheamtical modeling.
My Recent Publications.
This page was last updated on November 15, 2016.