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，这些模型与它们可以用非局部项的泛函和偏微分方程来表示的事实有关。本研究主要涉及三个方面：a)人口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关键现象的影响；C)模拟物理、化学和某些种群过程的耦合简并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，目的是使数学结果与实验数据相吻合。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时出现的大类方程。与本研究相关的一个普遍关注的问题是传染病在人群中的垂直传播过程。这一过程的数学建模可能有助于更好地理解疾病传播的过程。

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.