Mathematical Sciences: Analysis of Models with Nonlocal Temporal and Spatial Dependece

这项研究将继续研究生物和物理系统的模型，因为它们可以用带有非局部项的泛函方程和偏微分方程组来表示。本研究主要涉及三个方面：a)种群的年龄结构和其他内部变量对流行病模型动态行为的影响：d)细胞的大小、形状和隔室性质对控制细胞周期的临界现象的影响；C)模拟物理、化学和一些种群过程的耦合退化扩散方程系统中空间模式的形成和稳定性。该项目将采用分析和数值模拟，旨在将数学结果与实验数据相匹配。这三个问题中的每一个都需要发展新的数学分析技术，这些技术可以应用于分析某些生物和物理现象中出现的广泛类型的方程。与这项研究相关的普遍感兴趣的问题之一是传染病在人群中的垂直传播过程。对这一过程进行数学建模可能有助于更好地理解疾病传播过程。

This investigation will continue the study of models of biological and physical systems which are related by the fact that they can be formulated in terms of functional and partial differential equations with nonlocal terms. The main three areas of this study are concerned with: a) the effect of age-structure and other internal variables of a population on the dynamic behavior of epidemic models; d) the effect of the size, shape and compartamental nature of a cell on the critical phenomena that control the cell cycle; c) the formation and stability of spatial patterns in systems of coupled degenerate diffusion equations modeling physical, chemical as well as some population processes. The project will employ both analytical and numerical simulations that will be aimed at fitting the mathematical results to experimental data. Each of the three problems requires the development of new mathematical techniques of analysis that can be applied to broad classes of equations that occur in the analysis of certain biological and physical phenomena. One of problems of general interest that is related to this research is the process of vertical transmission of infectious diseases in populations. Mathematical modeling of this process may help to understand better the process of disease transmission.