Graduation Semester and Year




Document Type


Degree Name

Doctor of Philosophy in Mathematics



First Advisor

Christopher Kribs


The purpose of this dissertation is twofold: to deepen our understanding of the complex transmission routes of the Zika virus (ZIKV), and to study multiple pathogen interactions (specifically cocirculation of Zika and dengue and discrete-time coinfection models) through the lens of invasion reproductive numbers (IRNs) which measure the ability of a disease to invade a population endemic with another disease(s). In addition to being transmitted to humans through the bite of infected female Aedes aegypti mosquitoes, studies show that the ZIKV can also be sexually and vertically transmitted within both populations. We develop a new mathematical model of the ZIKV which incorporates sexual transmission in humans and mosquitos, vertical transmission in mosquitos, and mosquito to human transmission through bites. Analysis of this deterministic model shows that although the secondary transmission pathways cause minor qualitative changes on a Zika outbreak, they have important consequences for control strategies and estimates of Zika's basic reproductive number (BRN). Over the past five years, cocirculation of dengue and Zika has increased; however, very little is known about its epidemiological consequences. Using a dengue and Zika coinfection model that incorporates altered infectivity of mosquitoes (due to coinfection), and antibody-dependent enhancement (ADE) within the human population, we study the impact of cocirculation on the spread of both diseases. Central to our analysis is the derivation and interpretation of the basic reproductive number (BRN) and IRN of both pathogens. Our results identify threshold conditions under which one disease facilitates the spread of the other and show that ADE has a greater impact on disease persistence than altered vector infectivity. IRNs are utilized frequently in continuous-time models with multiple interacting pathogens; however, they are yet to be explored in discrete-time systems. We extend the concept of IRNs to discrete-time models by showing how to calculate them for a set of two-pathogen SIS models with coinfection. In our exploration, we address how sequencing events impacts the BRN and IRN, and analyze a formulation of the discrete-time model which assumes that events occur simultaneously. Results show that while the BRN is unaffected by variations in the order of events, the IRN differs. Furthermore, although the simultaneous model lacks the simplification property that other models possess and the mathematics involved in its analysis is complex, the model exhibits competitive exclusion under cross-immunity which is not observed in the sequential formulations.


Dengue, Coinfection, Discrete-time model, Invasion reproductive numbers, Basic reproductive number, Antibody-dependent enhancement


Mathematics | Physical Sciences and Mathematics


Degree granted by The University of Texas at Arlington

Included in

Mathematics Commons