Mathematical Sciences: Periodic Solutions of Functional Differential and Difference Equations. Mathematical Models in Biology

库克DMS-9502922 库克将研究系统的方程多个延迟。 对于某些满足适当定义的负反馈条件的系统，将证明周期解的存在性。 一个新的方法来寻找差分方程的周期解，或建立没有这样的解决方案，将被利用。 该项目的第二部分将是研究用于描述生物现象的数学模型。 一个模型描述了传染病在人群中的传播。 将对其进行分析，以确定孵育期和感染期的影响。 最后，将研究免疫系统与人体组织中入侵病毒相互作用的一般模型。 微分方程是生物科学中数学建模的基础。 涉及连续变化的现象，如在运动、物质和能量中看到的现象，都服从某些一般定律，这些定律可以用偏导数之间的相互作用和关系来表达。 数学的关键作用不仅在于陈述关系，而且在于从中提取定性和定量的意义。

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.