Mathematical Sciences: Analysis and Parameter Estimation of Models with Internal Structure

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The dynamics of age-structured epidemic and population problems lead to systems of nonlinear hyperbolic equations that can be studied by the use of semigroup theory. Explicit threshold criteria, the asymptotic behavior of the solutions, and the general forms of the nonlinear force of infection terms have been obtained in certain cases, and these results will be used as the basis for a systematic development of methods for estimating the pertinent model parameters using available data on specific diseases. A second area of biomathematical research that will be pursued involves the study of the triggering of DNA synthesis in eucaryotic cells through the use of compartmental reaction-diffusion models. These models lead to coupled semi-linear parabolic equations with time delays that are due to transcription and translation delays and the movement of biochemicals in the cell. The main methods of analysis here involve the development and use of local and global bifurcation results for such reaction-diffusion equations. Parameter identification and estimation on the basis of available data will be addressed in these models also. This research is concerned with two mathematical problems with biological applications. The first part of this research will develop and study mathematical models that help predict the course of infections and which form the basis for making decisions on how to best use available resources in order to control the spread of diseases. The age structure of the affected population is very important in determining the course and effects of most diseases. Models will be developed and studied that account for the age distribution of the population and that lead to methods for predicting the course of the disease and the effects of proposed control strategies. The results will be applied to the study of optimal control strategies for specific diseases. The second part of this research will seek to shed light on the mechanism that triggers animal cells to start replicating their genetic material and reproducing. This is a fundamental question that has implications in the understanding of growth and development of embryos as well as in disorders such as cancer that result from abnormal cell proliferation. The mathematical models to be studied are designed to allow a detailed analysis of this crucial DNA synthesis triggering mechanism.