Analysis of Models in Mathematical Biology and of Sign Pattern Matrices

我的研究建议集中在应用数学的两个领域：I.数学生物学，II.矩阵分析。*I.在数学生物学中，我的建议主要涉及传染病传播模型的数学分析。每种疾病的模型制定取决于疾病的生物学特征、所处理的问题和可用的数据。我的重点是稳定性分析，包括解决方案的分歧、敏感性和疾病控制。*一个目标是确定有助于推荐疾病控制策略(例如，流感疫苗接种和抗病毒药物)的临界量。基本繁殖数是找到这个阈值的标准方法，但也有其他的量，即类型和目标繁殖数。*我建议研究如何使这些量适应离散时间生态模型。最近我开发了网络上的疾病模型，我将把这些模型扩展到更现实的动态网络。在海地霍乱数据的推动下，一个社区网络模型最近产生了一个域基本繁殖数。当水的运动速度快于病原体的腐烂时，这个数字是通过劳伦级数计算的。我建议从理论上研究疾病传播率大的区域聚集在一起促进疾病侵袭的数值观察。*II.符号模式矩阵的研究是组合矩阵理论的一个分支，其中只知道矩阵条目的符号。我提出的研究主要是关于给定符号模式的矩阵实现所允许的特征值。特别有趣的是允许(要求)所有本征值具有对应于势(符号)稳定性的负实部的情况。对于稳定性重要的特征值的位置由经典的三维惯性矢量给出。我引入了精化惯量的概念，这是一种与其他纯虚数本征值不同的具有零本征值的四维向量，我建议进一步研究这个概念。特别是，3个精化惯性对于检测潜在动力系统中由Hopf分叉引起的周期行为非常重要。*因此，本研究在许多领域的线性化系统的应用中具有重要意义，例如生物化学、种群生物学、经济学和流行病学。这个项目还涉及潜在稳定性，目标是刻画潜在稳定的符号模式(一个长期悬而未决的问题)，至少对于具有某种有向图结构的模式子集是这样。在矩阵稳定性中，主子式起着重要的作用，最近的兴趣集中在确定对称矩阵是否存在每一阶的非零主子式。这导致了一个主要的等级特征序列，并且是主要等级分配问题的一个变种，我打算对模式进行研究。

My research proposal is focussed on two areas of applied mathematics: I. Mathematical biology, II. Matrix analysis.***I. In mathematical biology, my proposal mainly addresses mathematical analysis of infectious disease transmission models. Model formulation for each disease is governed by the biological features of the disease, the question being addressed and available data. My focus is on stability analysis, including bifurcation of solutions, sensitivity and disease control.*One aim is to determine threshold quantities that help recommend disease control strategies (e.g., vaccination and antivirals for influenza). The basic reproduction number is the standard method for finding this threshold, but other quantities, namely type and target reproduction numbers have been formulated.*I propose to investigate how these can be adapted to discrete time ecological models. Recently I have developed disease models on networks, and I will extend these models to more realistic dynamic networks. Motivated by data on cholera in Haiti, a community network model recently led to a domain basic reproduction number. When water movement is fast compared with pathogen decay, this number is computed through a Laurent series. I propose to investigate theoretically the numerical observation that regions of large disease transmissibility clustered together facilitate disease invasion.***II. The study of sign pattern matrices is a branch of combinatorial matrix theory, in which only the sign of matrix entries is known. My proposed research is mainly related to eigenvalues allowed by matrix realizations of a given sign pattern. Of special interest are the cases in which all eigenvalues are allowed (required) to have negative real parts, corresponding to potential (sign) stability. The location of eigenvalues important for stability is given by the classical 3-d inertia vector. I introduced the concept of refined inertia, a 4-d vector with zero eigenvalues distinguished from other pure imaginary ones, and I propose to further investigate this concept. In particular 3 refined inertias are important in detecting periodic behavior arising from Hopf bifurcation in underlying dynamical systems.*Thus this research has relevance in applications to linearized systems in many areas, e.g., biochemistry, population biology, economics, and epidemiology. This project also relates to potential stability, and an objective is to characterize potentially stable sign patterns (a long standing open problem), at least for a subset of patterns having some digraph structure. In matrix stability, principal minors play an important role, and a recent interest focusses on determining for a symmetric matrix whether or not there is a nonzero principal minor of each order. This leads to a principal rank characteristic sequence, and is a variant on the principal rank assignment problem, which I intend to pursue for patterns.********