Matrices in the service of epidemic models

该提案涵盖了确定性应用数学的两个领域，这两个领域由稳定性的共同主题交织在一起，每个领域都相互补充。它们是数学生物学和矩阵分析。生物模型，特别是与疾病传播相关的模型，将以数学方式制定和分析。目的通常是确定阈值数量，以帮助决定控制策略（例如，针对 SARS 的隔离、针对流感的疫苗接种和抗病毒药物）。其他目标是确定什么机制导致传染病（例如麻疹）数据中常见的周期性，并预测新出现的疾病（例如北美的西尼罗河病毒）的传播。 公共卫生政策制定者可以利用这些模型的结果来指导与疾病控制相关的决策。在许多此类模型的分析中，矩阵分析起着关键作用。 确定系统的稳态在受到小扰动时是否会返回到该状态尤为重要，这涉及矩阵稳定性。 将研究具有与稳定性相关的重要性质的矩阵的定性类别。此类的示例包括允许稳定性的矩阵符号模式以及允许通过 Hopf 分岔产生振荡解的频谱的矩阵符号模式。这些方面的进展不仅在组合矩阵分析中具有重要意义，而且对于生物、化学和经济系统也具有重要意义。

This proposal spans two areas of deterministic applied mathematics that are woven together by the common theme of stability, with each area complementing the other. They are mathematical biology and matrix analysis. Biological models, especially those relevant for disease transmission, will be formulated mathematically and analyzed. The aim is often to determine threshold quantities to aid in deciding control strategies (e.g., isolation for SARS, vaccination and antivirals for influenza). Other aims are to determine what mechanisms lead to periodicity as is often seen in data on infectious diseases (e.g., measles) and to predict the spread of newly emerging diseases (e.g., West Nile virus in N. America). Public health policy makers could use results of these models to guide decisions relating to disease control. In the analyses of many of these models, matrix analysis plays a key role. It is especially important to determine whether a stationary state of a system returns to that state when given a small perturbation, and this involves matrix stability. Qualitative classes of matrices that have important properties related to stability will be investigated. Examples of such classes are matrix sign patterns that allow stability and those that allow spectra that could lead to oscillatory solutions through Hopf bifurcation. Advances on these will be significant not only in combinatorial matrix analysis but also for biological, chemical and economic systems.