Delays are inherent in many physical, biological, economic and engineering systems. My research in this area is focused on both finding solutions to these equations and finding stability manifolds, with an emphasis on developing
methods that are practical and useful for others.
In Engineering, pure delays are often used to ideally represent the effects of transmission, transportation, and inertial phenomena. Delay differential equations (DDEs) constitute basic mathematical models for such real phenomena. The principal difficulty in studying DDEs lies in their special transcendental character. Delay problems always lead to an infinite spectrum of frequencies. Hence, they are often solved using numerical methods, asymptotic solutions, approximations (e.g., Padé) and graphical approaches. I now
collaborate with Professor Galip Ulsoy, Henry Ford Professor of Mechanical Engineering. Professor Ulsoy developed a new analytic approach, based on the matrix Lambert function, for the complete solution of a system of linear constant coefficient DDEs. We are expanding this theory to account for systems with multiple time delays, time-varying coefficients, and specific nonlinearities. We have
validated the method for stability, free and forced response, by comparison to numerical integration for selected examples. I will discuss this method as applied to an engineering problem where delay is significant: regenerative chatter in a machining operation on a lathe and a biological problem: control of drug therapies. The matrix Lambert function based solution approach for DDEs is analogous to the use of the matrix exponential for the free and forced solution of linear constant coefficient ordinary differential equations. Systems with multiple time delays and nonlinearities arise quite naturally in engineering and biology and yet little attention has been paid to their analyses. Our method should provide a framework for others to use in studying these complicated systems.
I will also discuss the application of time delays to study HBV infections. Acute hepatitis B infection (HBV) is cleared in 85\% to 95\% of infected adults, while the rest progress towards chronic infection. Why some people clear the virus, while others do not is still not well understood. The quality and the dynamics of the immune response have been implicated, but a precise quantitative understanding of this response is still lacking. Here we analyze data from a set of individuals identified during acute HBV infection, and develop mathematical models to test the role of immune responses in various stages of early HBV infection. Fitting the models to the viral load data we are able to separate the kinetics of the non-cytolytic and the cytolytic immune responses thus explaining the relative contribution of these two processes.
The non-cytolytic phase occurs around the peak of viral load and helps reduce it, whereas the cytolytic processes are crucial for a second phase of viral decay, which eventually may lead to control of the virus. The model also demonstrates the difficulty of controlling the infection in a setting of regeneration of uninfected hepatocytes, and requires us to introduce a class of cells refractory to viral production that confer protection against re-infection of the liver. Taken together these results contribute to a clearer picture of acute HBV dynamics.
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