Date of Award

12-2018

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Xin Liu, Committee Chair

Committee Member

Robert Lund

Committee Member

Peter Kiessler

Committee Member

Brian Fralix

Abstract

This dissertation focuses on the study of some environmental applications of stochastic processes. In the first Chapter, we study an extreme value time series, where the extremes are derived using the block maxima/minima method. In extreme value analyses, the block maxima/minima are typically assumed to be statistically independent from block to block. However, extreme data arising from applications often present significant correlations. The main result of the chapter is to develop a simple and efficient method to estimate extreme model parameters and compute standard errors for them that account for correlation. More precisely, we develop a likelihood for sequences of extremes when observations are dependent in time. Our likelihood allows the series to have temporal correlation, but also keeps a generalized extreme value marginal distribution at each point in time. Using the likelihood, we obtain more realistic standard errors of the generalized extreme value parameters. An analysis of the Faraday/Vernadsky annual minimum temperatures is conducted. While the standard error of the estimated trend increases when dependence is taken into account, it does not change the correlation-ignored inference that annual minimum temperatures at the station are increasing. The main theoretical result shows the consistency and asymptotical normality of the maximum likelihood estimators. In the second chapter, we study a stochastic vector-borne epidemic model. Vector-Borne Diseases (VBDs) are infections transmitted by the bite of infected arthropod species, such as mosquitoes, ticks, triatomine bugs, which were responsible for more human deaths in the 20th centuries than all other causes combined. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R0, the initial size of infectives, and the type of scaling in terms of host population size. The quantity R0 for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model.

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