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

这项研究将继续研究的模式， 生物和物理系统的联系 它们可以从功能和局部的角度来表述， 非局部微分方程主要三个领域 本研究的主要内容是：（1）年龄结构的影响 和其他内部变量的影响 流行病模型的行为; d）大小，形状和 细胞的分隔性对临界现象的影响， 控制细胞周期; c）空间的形成和稳定性 耦合退化扩散方程组的模式 模拟物理，化学以及一些人口过程。 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 这三个问题中的每一个都需要开发新的 数学分析技术，可以应用于广泛的 在某些分析中出现的方程类 生物和物理现象。一般的问题之一 与这项研究有关的兴趣是 传染病在人群中的垂直传播。 这个过程的数学模型可能有助于理解 更好的传播疾病的过程。

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.