Matrices in the service of epidemic models

该提案跨越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编织在一起，每个领域都是互补的。它们是数学生物学和矩阵分析。生物学模型，特别是那些与疾病传播有关的模型，将用数学公式表述和分析。其目的往往是确定阈值，以帮助决定控制策略（例如，对SARS进行隔离，对流感进行疫苗接种和抗病毒药物）。其他目标是确定导致传染病（如麻疹）数据中经常出现的周期性的机制，并预测新出现的疾病（如西尼罗河病毒在北美洲的传播）的传播。公共卫生决策者可以利用这些模型的结果来指导与疾病控制有关的决策。在对这些模型的分析中，矩阵分析起着关键作用。当给定一个小的扰动时，确定系统的定态是否返回到定态是特别重要的，这涉及到矩阵稳定性。我们将研究具有与稳定性相关的重要性质的矩阵的定性类。这类的例子是允许稳定的矩阵符号模式和允许通过Hopf分岔导致振荡解的谱。这些方面的进展不仅在组合矩阵分析方面，而且在生物、化学和经济系统方面都具有重要意义。

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.