Mathematical Sciences: Models for the Spread of HIV; Equations with Piecewise Continuous Argument; Dynamics of Discretizations

该项目将重点关注三个科学领域。第一个问题涉及艾滋病毒（人类免疫缺陷病毒）传播的流行病学模型的数学分析。将开发和评估病毒传播的数学模型。这些模型将根据性别、性行为和社会经济类别等标准将人口分成不同的群体。它们将用常微分方程和时滞微分方程来表示。分析了稳定性和分岔现象，进行了模拟，并对不同模型的预测结果进行了比较。第二项研究涉及到具有分段连续延迟的微分方程。这样的方程出现在许多由记忆效应控制的物理环境中。这项工作的目标是发展初值和边值问题的理论和应用。将寻求稳定的条件；将研究有界和无界延迟方程的周期和振荡现象。交替型泛函微分方程也将被考虑。还将研究李雅普诺夫函数和离散化的定性行为。本文将利用李亚普诺夫直接法的推广和不变性原理研究非线性差分方程的稳定性问题。将探讨微分方程类的渐近动力学与这些方程的离散化和半离散化版本的动力学之间的关系。将寻求在种群动力学和化学动力学方面的应用。

This project will focus on three scientific areas. The first concerns mathematical analysis of epidemiological models for the spread of HIV (human immunodeficiency virus). Mathematical models for the spread of the virus will be developed and evaluated. The models will be based on dividing the population into groups according to criteria such as sex, sexual behavior and socioeconomic categories. They will be represented in terms of ordinary and delay-differential equations. Stability and bifurcation phenomenon will be analyzed, simulations performed and comparisons made among the predictions of the different models. A second line of investigation concerns differential equations with piecewise continuous delays. Such equations arise in a number of physical contexts governed by memory effects. An objective of this work is to develop theory and applications of initial and boundary-value problems. Conditions for stability will be sought; periodic and oscillation phenomena will be studied for equations with bounded and unbounded delay. General functional differential equations of alternating type will also be considered. Work will also be done investigating Liapunov functions and the qualitative behavior of discretizations. This research will focus on stability questions related to nonlinear difference equations using an extension of Liapunov's direct method and the invariance principle. The relation between the asymptotic dynamics of classes of differential equations and the dynamics of discretized and semidiscretized versions of these equations will be explored. Applications to population dynamics and chemical kinetics will be sought.