Matrices in the service of epidemic models

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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.