Cooke DMS-9502922 Cooke will study systems of equations with multiple delays. Existence of periodic solutions will be proved for some systems satisfying a suitably defined negative feedback condition. A new method for finding periodic solutions of difference equations, or establishing that there are no such solutions, will be exploited. A second part of the project will be the study of mathematical models for the description of biological phenomena. One model describes the transmission of infectious diseases within a population. It will be analyzed to determine the effect of incubation and infectious periods. Lastly, a general model for the interaction of the immune system with an invading virus in human tissue will be studied. Differential equations form the backbone of mathematical modeling in the biological sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not only to state the relationships, but also to extract qualitative and quantitative meaning from them.
