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

数学科学：艾滋病毒传播模型；

• 批准号: 8807478
• 负责人: Kenneth Cooke
• 金额: $ 13.55万
• 依托单位: Pomona College
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8807478&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Models; Spread; HIV

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.

该项目将重点关注三个科学领域。 第一个涉及艾滋病毒（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组。 它们将用常微分方程和延迟微分方程来表示。 将分析稳定性和分岔现象，进行模拟并对不同模型的预测进行比较。 第二个研究方向涉及具有分段连续延迟的微分方程。 这些方程出现在许多受记忆效应控制的物理环境中。 这项工作的目标是发展初始和边值问题的理论和应用。 将寻求稳定的条件；将研究具有有界和无界延迟的方程的周期和振荡现象。 还将考虑交替类型的一般泛函微分方程。 还将开展研究李雅普诺夫函数和离散化定性行为的工作。 本研究将利用李雅普诺夫直接法和不变性原理的扩展，重点研究与非线性差分方程相关的稳定性问题。 将探讨微分方程类的渐近动力学与这些方程的离散和半离散版本的动力学之间的关系。 将寻求在群体动力学和化学动力学中的应用。