Analysis of Models in Mathematical Biology and of Sign Pattern Matrices

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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.********