Research Interests

 

Qualitative Theory of Differential Equations and Dynamical Systems.

  • Existence, non-existence, and stability of certain type of solutions, suchas 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.
  • Recently, due to my interests in epidemiological models, I begin to studya class of reaction diffusion systems that arise in biological models.Of interest to me are problems related to global stability of positive equilibria, existence and non-existence of periodic solutions, and existence of traveling waves.
  • Also motivated by the analysis of a class of epidemic models, I am interestedin the method of singular perturbations, especially in the presenceof "turning points", and the method of relaxation oscillations .

Mathematical Modelling of Immune System

    Human immune system is a very complex and dynamic system 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 hhas been used in the studies of immune response to various infections, most notably that of the HIV. My research interest in this area is the modeling of the in vivo infection process of certain 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 project studies the infection of HTLV-I (Human T-cell Lymphotropic Virus Type I), which is a retro virus and also an oncogenic virus.

Mathematical Theory of Epidemiology.

    The goal here is to use mathematical means, differential or difference equations in particular, to model the transmission process of an infectious disease in a human or animal population. In my point of view, mathematical models can be very useful in identifying the underlying mechanisms foran epidemic or endemic process. I study certain class of very basic epidemic models, the so called SEIRS models. A good understanding of thebasic mechanisms in these models  allow other workers in this fieldto modify these basic models in order to study a specific disease. It mayalso help a field epidemiologist to estimate certain important parametersthat would be otherwise difficult to estimate. This research has been continuously supported by research grants from the National Science Foundation (NSF).

My Recent Publications.

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This page was last updated on November 15, 2002.