Modeling, Estimation and Approximation in Structured Models

• Main Author: Zhao, Lihong
• Language: EN
• Published: University of Louisiana at Lafayette 2018
• Subjects: Mathematics
• Online Access: http://pqdtopen.proquest.com/#viewpdf?dispub=10681468

<p> Structured models provide a powerful tool to study the dynamics of complex systems, and are studied actively. This dissertation contains two specific applications of structured models. </p><p> The first application focuses on modeling and estimation of population dynamics of a recent invasive species, the island apple snail <i>Pomacea maculata.</i> As growth and vital rates are thought to be size-dependent, and due to the differences observed between female and male snails from our data, a sex-specific model in which individuals are structured by weight is applied to study their population dynamics. Since efforts to directly compute the growth rates from individual data do not yield statistically supported estimates, a population-level approach is used, in which growth rate functions are estimated via an inference-based parameter estimation approach. A residual-based model comparison test is used to confirm that the growth dynamics exhibit two distinct growth phases from juvenile to adult. This model is further calibrated based on information gained from additional experiments, hypothesized biological relationships, and estimates from related literature. The calibrated model is then used to investigate population projections under different scenarios, inform future data collecting strategies, and to explore possible control measures. </p><p> A large amount of variation on growth and other vital processes was observed in our data sets, and in other data sets reported in the literature. Further modeling efforts would necessarily take this into account to improve population projections. The phenomenon of intra-individual and inter-individual variability is common in biological systems. For example, in some infectious diseases, it may be important to develop a model that accounts for the variability seen in the disease outcomes. One such example with this key feature of variability in the structure variable, is a bacteria load structured model of the transmission dynamics of <i>Mycobacterium marinum</i> in aquatic animals. In that model, the infected fish population is a mixture of subpopulations with different intra-host progression rates. A second order finite difference scheme is established to approximate the solution. Convergence of the finite difference approximation to the weak solution is proven. Second order convergence and key features of the model are explored.</p><p>