Publication Date

2010-05-21

Availability

Open access

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2010-04-23

First Committee Member

Shigui Ruan - Committee Chair

Second Committee Member

Robert Stephen Cantrell - Committee Member

Third Committee Member

Chris Cosner - Committee Member

Fourth Committee Member

John Beier - Outside Committee Member

Abstract

Dengue (pronounced den'guee) Fever (DF) and Dengue Hemorrhagic Fever (DHF), collectively known as "dengue," are mosquito-borne, potentially mortal, flu-like viral diseases that affect humans worldwide. Transmitted to humans by the bite of an infected mosquito, dengue is caused by any one of four serotypes, or antigen-specific viruses. In this thesis, both the spatial and temporal dynamics of dengue transmission are investigated. Different chapters present new models while building on themes of previous chapters. In Chapter 2, we explore the temporal dynamics of dengue viral transmission by presenting and analyzing an ODE model that combines an SIR human host- with a multi-stage SI mosquito vector transmission system. In the case where the juvenile populations are at carrying capacity, juvenile mosquito mortality rates are sufficiently small to be absorbed by juvenile maturation rates, and no humans die from dengue, both the analysis and numerical simulations demonstrate that an epidemic will persist if the oviposition rate is greater than the adult mosquito death rate. In Chapter 3, we present and analyze a non-autonomous, non-linear ODE system that incorporates seasonality into the modeling of the transmission of the dengue virus. We derive conditions for the existence of a threshold parameter, the basic reproductive ratio, denoting the expected number of secondary cases produced by a typically infective individual. In Chapter 4, we present and analyze a non-linear system of coupled reaction-diffusion equations modeling the virus' spatial spread. In formulating our model, we seek to establish the existence of traveling wave solutions and to calculate spread rates for the spatial dissemination of the disease. We determine that the epidemic wave speed increases as average annual, and in our case, winter, temperatures increase. In Chapter 5, we present and analyze an ODE model that incorporates two serotypes of the dengue virus and allows for the possibility of both primary and secondary infections with each serotype. We obtain an analytical expression for the basic reproductive number, R_0, that defines it as the maximum of the reproduction numbers for each strain/serotype of the virus. In each chapter, numerical simulations are conducted to support the analytical conclusions.

Keywords

Mathematical Epidemiology; Dynamical Systems; Non-linear PDEs; Periodic Persistence

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