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Dissertation/Thesis Abstract

Finite Difference Schemes for a Structured Model of Mycobacterium marinum
by Delcambre, Mark Lane, Ph.D., University of Louisiana at Lafayette, 2014, 126; 3622932
Abstract (Summary)

This work presents a novel size-structured model to mathematically describe the transmission dynamics of Mycobacterium marinum in an aquatic environment. Biological background on the formation of the model is discussed in Chapter 1. In Chapter 2, the model is developed and consisted of a system on nonlinear partial differential equations coupled to three nonlinear ordinary differential equations. The weak solution is defined and current numerical work on structured models is discussed.

In Chapter 3, a first-order method is developed to approximate the solution to the model, and in Chapter 4, a second-order high resolution method is developed. Theoretical foundations for both methods are established. Also, convergence to the unique weak solution is verified for both methods.

In Chapters 3 and 4, the numerical results begin with showing each method is in fact of the appropriate order for a simple version of the model, and then with the full nonlinear version. Chapter 3 continues the numerical results section with preliminary studies on the key features of this model, such as various forms of growth rates (indicative of possible theories of development), and conditions for competitive exclusion or coexistence as determined by reproductive fitness and genetic spread in the population. In Chapter 4, we compare the first and second-order methods to show the computational benefits that come with a second-order method. We also demonstrate that the model can be a tool to understand surprising or nonintuitive phenomena regarding competitive advantage in the context of biologically realistic growth, birth, and death rates.

Indexing (document details)
Advisor: Ackleh, Azmy S.
Commitee: Deng, Keng, Sutton, Karyn
School: University of Louisiana at Lafayette
Department: Sciences
School Location: United States -- Louisiana
Source: DAI-B 75/10(E), Dissertation Abstracts International
Subjects: Applied Mathematics, Epidemiology
Keywords: Existence-uniqueness, Finite difference, Mycobacterial solutions, Partial differential equations, Structured models
Publication Number: 3622932
ISBN: 978-1-303-95088-9
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