Date of Award

Spring 2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Zhilan Feng

Committee Chair

Zhilan Feng

Committee Member 1

John W Glasser

Committee Member 2

Greg Buzzard

Committee Member 3

NK Yip

Abstract

This dissertation presents epidemiological models that investigate synergy: synergy between HIV and HSV-2 or between humans and mosquitoes in a malaria study. Each of the three coupled disease models addresses different epidemiological questions with regard to gender or disease structure in the context of sexually-transmitted diseases (STDs), while the malaria model focuses on age-structure of the human population. ^ Mounting evidence indicates that HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the epidemiological synergy between HIV and HSV-2, we formulate, in Chapter 2, a deterministic compartmental model that describes the transmission dynamics of these diseases. Unlike earlier models, ours incorporates gender and heterogeneous mixing between activity groups. We derive explicit expressions for the basic and invasion reproduction numbers of HSV-2 and HIV. A qualitative analysis of the system includes the local and global behavior of the model. Simulations reinforce these analytical results and demonstrate epidemiological synergy between HSV-2 and HIV. Most importantly, we illustrate that the disease dynamics can be significantly influenced by the sexual structure of the population. ^ Chapter 3 deals with another model for HIV and HSV-2 using a different approach, which allows us to better capture the unique epidemiological feature of HSV-2 and its effect on the disease dynamics of HIV. As HSV-2 is an incurable viral infection, characterized by periodic reactivation, we construct a model of the co-infection dynamics by incorporating a time-since-infection variable to track the alternating periods of infectiousness of HSV-2. We also incorporate gender and activity groups in this model. The basic and invasion reproduction numbers are derived and interpreted. The calculations of the invasion reproduction numbers suggest a new interpretation: the class from which the initial disease carrier arises is important for understanding the invasion dynamics. Numerical simulations and a sensitivity analysis are conducted. Results illustrate the importance of including risk groups in the model. Treatment implications are discussed. ^ As the previous studies of HIV and HSV-2 focus only on a heterosexual population, Chapter 4 introduces a model with both heterosexual and homosexual interactions. We derive and interpret the basic and invasion reproduction numbers for HIV and HSV-2 using the approach of next-generation matrices. We then perform scenario analyses and conduct a sensitivity analysis to investigate the impact of model parameters on the reproduction numbers and disease prevalences. We conclude that homosexual transmission drastically changes the disease prevalences and that models that ignore homosexuality may greatly underestimate the disease burden; hence, it is important to account for this interaction. ^ In Chapter 5, an age-structured mathematical model for malaria is presented. The model includes the human and mosquito populations, structured by chronological age of humans. The infected human population is divided into symptomatic infectious, asymptomatic infectious, and asymptomatic chronically infected individuals. The original PDE model is reduced to an ODE model with multiple age groups coupled by aging. The basic reproduction number R0 is derived for both the PDE and ODE models. A major new feature of this model is that, based on biological observations, we assume that relapse of chronically infected individuals is triggered by mosquito bites. Our analysis indicates that this assumption contributes greatly to the magnitude of R 0 and, therefore, needs to be further studied and understood. Numerical simulations for n = 2 age groups and a sensitivity/uncertainty analysis suggest that it is important to consider asymptomatic individuals as a hidden source of malaria transmission, particularly those who are chronically infected. Age-dependent control measures are also investigated.

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