
Mathematical Theory of Differential Equations and Dynamical Systems.

Existence, nonexistence, and stability of certain type of solutions, such as
periodic solutions, quasiperiodic 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 largescale interconnected 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 interconnections among the vertex systems, and the
strength of interconnections can be described by the weights on edges. Largescale 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 graphtheorectic
approach that allows systematic construction of global Lyapunov functions for a very general class of
largescale 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 multigroup epidemic
models.
Other problems of interest are synchronization problems in coupled oscillators,
ranking problems for vertices on networks, and complexity problems in largescale 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 retroviruses 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 HTLVI (Human Tcell 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 ongoing interdisciplinary research projects that use mathematical models
to investigate transmission dynamics and related publichealth 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 healtheconomic 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.


