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

该项目将侧重于三个科学领域。 的 首先是流行病学模型的数学分析 艾滋病病毒（HIV）的传播。 病毒传播的数学模型 开发和评估。 模型将基于划分 根据诸如性别、 性行为和社会经济类别。 他们将 表示为普通和延迟微分 方程 稳定性和分岔现象将是 分析，进行模拟和比较， 不同模型的预测。 第二条调查路线涉及微分 具有分段连续时滞的方程 这样的方程出现了 在许多受记忆效应支配的物理环境中。 一个 这项工作的目标是发展理论和应用 初值和边值问题。 稳定性条件 将寻求;周期性和振荡现象将是 研究了有界和无界时滞的方程。 一般 交替型泛函微分方程也将 被考虑。 工作也将完成调查李雅普诺夫函数和 离散化的定性行为。 这项研究将 重点研究了与非线性差分有关的稳定性问题 方程使用Liapunov的直接方法的扩展和 不变性原理 渐近性之间的关系 微分方程类的动力学和 这些方程的离散化和半离散化版本将 被探索。 种群动力学和化学的应用 将寻求动力学。

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.