Mathematical Sciences: Analysis and Parameter Estimation of Models with Internal Structure

年龄结构的流行病和人口问题的动力学导致了可用半群理论研究的非线性双曲方程组。在某些情况下，已获得了明确的阈值准则、解的渐近行为和感染项的非线性作用力的一般形式，这些结果将被用作系统地发展利用特定疾病的现有数据估计相关模型参数的方法的基础。将进行的生物数学研究的第二个领域涉及通过使用隔室反应扩散模型来研究真核细胞中DNA合成的触发。这些模型导致了具有时间延迟的耦合半线性抛物型方程，这些时间延迟是由于转录和翻译延迟以及细胞内生物化学物质的运动造成的。这里的主要分析方法涉及到这类反应扩散方程的局部和全局分支结果的发展和应用。在这些模型中，还将讨论基于现有数据的参数识别和估计。这项研究涉及两个具有生物学应用的数学问题。这项研究的第一部分将开发和研究数学模型，这些模型有助于预测感染过程，并形成决定如何最好地利用现有资源以控制疾病传播的基础。受影响人群的年龄结构在决定大多数疾病的病程和影响方面非常重要。将开发和研究模型，以考虑人口的年龄分布，并导致预测疾病进程和拟议控制战略的影响的方法。研究结果将应用于特定疾病的最优控制策略的研究。这项研究的第二部分将试图阐明触发动物细胞开始复制其遗传物质和繁殖的机制。这是一个基本的问题，对理解胚胎的生长发育以及由异常细胞增殖引起的癌症等疾病具有重要意义。要研究的数学模型旨在详细分析这一关键的DNA合成触发机制。

The dynamics of age-structured epidemic and population problems lead to systems of nonlinear hyperbolic equations that can be studied by the use of semigroup theory. Explicit threshold criteria, the asymptotic behavior of the solutions, and the general forms of the nonlinear force of infection terms have been obtained in certain cases, and these results will be used as the basis for a systematic development of methods for estimating the pertinent model parameters using available data on specific diseases. A second area of biomathematical research that will be pursued involves the study of the triggering of DNA synthesis in eucaryotic cells through the use of compartmental reaction-diffusion models. These models lead to coupled semi-linear parabolic equations with time delays that are due to transcription and translation delays and the movement of biochemicals in the cell. The main methods of analysis here involve the development and use of local and global bifurcation results for such reaction-diffusion equations. Parameter identification and estimation on the basis of available data will be addressed in these models also. This research is concerned with two mathematical problems with biological applications. The first part of this research will develop and study mathematical models that help predict the course of infections and which form the basis for making decisions on how to best use available resources in order to control the spread of diseases. The age structure of the affected population is very important in determining the course and effects of most diseases. Models will be developed and studied that account for the age distribution of the population and that lead to methods for predicting the course of the disease and the effects of proposed control strategies. The results will be applied to the study of optimal control strategies for specific diseases. The second part of this research will seek to shed light on the mechanism that triggers animal cells to start replicating their genetic material and reproducing. This is a fundamental question that has implications in the understanding of growth and development of embryos as well as in disorders such as cancer that result from abnormal cell proliferation. The mathematical models to be studied are designed to allow a detailed analysis of this crucial DNA synthesis triggering mechanism.