Date of Award

Summer 2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Jin Wang

Committee Member

Ruhai Zhou

Committee Member

Yan Peng

Committee Member

Chung-Hao Chen

Abstract

The theory of optimal control, a modern extension of the calculus of variations. has found many applications in a wide range of scientific fields, particularly in epidemiology with respect to disease prevention and intervention. In this dissertation. we conduct optimal control modeling, simulation and analysis to cholera dynamics. Cholera is a severe intestinal infectious disease that remains a serious public health threat in developing countries. Transmission of cholera involves complex interactions between the human host, the pathogen, and the environment. The worldwide cholera outbreaks and their increasing severity, frequency and duration in recent years underscore the gap between the complex mechanism of cholera transmission and our current quantitative understanding and control strategies for this disease.

We incorporate multiple time-dependent intervention strategies, including vaccination, antibiotic treatment, and water sanitation, into cholera epidemiological models and seek solutions that best balance the costs and gains of the controls. Pontryagin's Maximum/Minimum principle allows us to construct the optimal control system that involves the state equations, the adjoint equations, and the optimality condition that characterizes the controls. The system is then numerically solved using an iterative procedure based on the Forward-Backward Sweep Method. We discuss in detail the mathematical models and numerical results for various scenarios and their implications to public health administration on disease control.

In the last part of this dissertation, we investigate new iterative algorithms with improved convergence properties compared to the original Forward-Backward Sweep Method. We discuss the applications of such numerical algorithms to optimal control problems as well as other types of constrained dynamical systems. We conduct careful error analysis and present several numerical examples to validate the analytic results.

DOI

10.25777/hpam-v235

ISBN

9781303528927

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