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- All Subjects: Applied Mathematics
- Creators: Kuang, Yang
Description
Rabies disease remains enzootic among raccoons, skunks, foxes and bats in the United States. It is of primary concern for public-health agencies to control spatial spread of rabies in wildlife and its potential spillover infection of domestic animals and humans. Rabies is invariably fatal in wildlife if untreated, with a non-negligible incubation period. Understanding how this latency affects spatial spread of rabies in wildlife is the concern of chapter 2 and 3. Chapter 1 deals with the background of mathematical models for rabies and lists main objectives. In chapter 2, a reaction-diffusion susceptible-exposed-infected (SEI) model and a delayed diffusive susceptible-infected (SI) model are constructed to describe the same epidemic process -- rabies spread in foxes. For the delayed diffusive model a non-local infection term with delay is resulted from modeling the dispersal during incubation stage. Comparison is made regarding minimum traveling wave speeds of the two models, which are verified using numerical experiments. In chapter 3, starting with two Kermack and McKendrick's models where infectivity, death rate and diffusion rate of infected individuals can depend on the age of infection, the asymptotic speed of spread $c^\ast$ for the cumulated force of infection can be analyzed. For the special case of fixed incubation period, the asymptotic speed of spread is governed by the same integral equation for both models. Although explicit solutions for $c^\ast$ are difficult to obtain, assuming that diffusion coefficient of incubating animals is small, $c^\ast$ can be estimated in terms of model parameter values. Chapter 4 considers the implementation of realistic landscape in simulation of rabies spread in skunks and bats in northeast Texas. The Finite Element Method (FEM) is adopted because the irregular shapes of realistic landscape naturally lead to unstructured grids in the spatial domain. This implementation leads to a more accurate description of skunk rabies cases distributions.
ContributorsLiu, Hao (Author) / Kuang, Yang (Thesis advisor) / Jackiewicz, Zdzislaw (Committee member) / Lanchier, Nicolas (Committee member) / Smith, Hal (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2013
Description
Bacteriophage (phage) are viruses that infect bacteria. Typical laboratory experiments show that in a chemostat containing phage and susceptible bacteria species, a mutant bacteria species will evolve. This mutant species is usually resistant to the phage infection and less competitive compared to the susceptible bacteria species. In some experiments, both susceptible and resistant bacteria species, as well as phage, can coexist at an equilibrium for hundreds of hours. The current research is inspired by these observations, and the goal is to establish a mathematical model and explore sufficient and necessary conditions for the coexistence. In this dissertation a model with infinite distributed delay terms based on some existing work is established. A rigorous analysis of the well-posedness of this model is provided, and it is proved that the susceptible bacteria persist. To study the persistence of phage species, a "Phage Reproduction Number" (PRN) is defined. The mathematical analysis shows phage persist if PRN > 1 and vanish if PRN < 1. A sufficient condition and a necessary condition for persistence of resistant bacteria are given. The persistence of the phage is essential for the persistence of resistant bacteria. Also, the resistant bacteria persist if its fitness is the same as the susceptible bacteria and if PRN > 1. A special case of the general model leads to a system of ordinary differential equations, for which numerical simulation results are presented.
ContributorsHan, Zhun (Author) / Smith, Hal (Thesis advisor) / Armbruster, Dieter (Committee member) / Kawski, Matthias (Committee member) / Kuang, Yang (Committee member) / Thieme, Horst (Committee member) / Arizona State University (Publisher)
Created2012
Description
This dissertation is intended to tie together a body of work which utilizes a variety of methods to study applied mathematical models involving heterogeneity often omitted with classical modeling techniques. I posit three cogent classifications of heterogeneity: physiological, behavioral, and local (specifically connectivity in this work). I consider physiological heterogeneity using the method of transport equations to study heterogeneous susceptibility to diseases in open populations (those with births and deaths). I then present three separate models of behavioral heterogeneity. An SIS/SAS model of gonorrhea transmission in a population of highly active men-who-have-sex-with-men (MSM) is presented to study the impact of safe behavior (prevention and self-awareness) on the prevalence of this endemic disease. Behavior is modeled in this examples via static parameters describing consistent condom use and frequency of STD testing. In an example of behavioral heterogeneity, in the absence of underlying dynamics, I present a generalization to ``test theory without an answer key" (also known as cultural consensus modeling or CCM). CCM is commonly used to study the distribution of cultural knowledge within a population. The generalized framework presented allows for selecting the best model among various extensions of CCM: multiple subcultures, estimating the degree to which individuals guess yes, and making competence homogenous in the population. This permits model selection based on the principle of information criteria. The third behaviorally heterogeneous model studies adaptive behavioral response based on epidemiological-economic theory within an $SIR$ epidemic setting. Theorems used to analyze the stability of such models with a generalized, non-linear incidence structure are adapted and applied to the case of standard incidence and adaptive incidence. As an example of study in spatial heterogeneity I provide an explicit solution to a generalization of the continuous time approximation of the Albert-Barabasi scale-free network algorithm. The solution is found by recursively solving the differential equations via integrating factors, identifying a pattern for the coefficients and then proving this observed pattern is consistent using induction. An application to disease dynamics on such evolving structures is then studied.
ContributorsMorin, Benjamin (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Hiebeler, David (Thesis advisor) / Hruschka, Daniel (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2012
Description
In vertebrate outer retina, changes in the membrane potential of horizontal cells affect the calcium influx and glutamate release of cone photoreceptors via a negative feedback. This feedback has a number of important physiological consequences. One is called background-induced flicker enhancement (BIFE) in which the onset of dim background enhances the center flicker response of horizontal cells. The underlying mechanism for the feedback is still unclear but competing hypotheses have been proposed. One is the GABA hypothesis, which states that the feedback is mediated by gamma-aminobutyric acid (GABA), an inhibitory neurotransmitter released from horizontal cells. Another is the ephaptic hypothesis, which contends that the feedback is non-GABAergic and is achieved through the modulation of electrical potential in the intersynaptic cleft between cones and horizontal cells. In this study, a continuum spine model of the cone-horizontal cell synaptic circuitry is formulated. This model, a partial differential equation system, incorporates both the GABA and ephaptic feedback mechanisms. Simulation results, in comparison with experiments, indicate that the ephaptic mechanism is necessary in order for the model to capture the major spatial and temporal dynamics of the BIFE effect. In addition, simulations indicate that the GABA mechanism may play some minor modulation role.
ContributorsChang, Shaojie (Author) / Baer, Steven M. (Thesis advisor) / Gardner, Carl L (Thesis advisor) / Crook, Sharon M (Committee member) / Kuang, Yang (Committee member) / Ringhofer, Christian (Committee member) / Arizona State University (Publisher)
Created2012
Description
Dividing the plane in half leaves every border point of one region a border point of both regions. Can we divide up the plane into three or more regions such that any point on the boundary of at least one region is on the border of all the regions? In fact, it is possible to design a dynamical system for which the basins of attractions have this Wada property. In certain circumstances, both the Hénon map, a simple system, and the forced damped pendulum, a physical model, produce Wada basins.
ContributorsWhitehurst, Ryan David (Author) / Kostelich, Eric (Thesis director) / Jones, Donald (Committee member) / Armbruster, Dieter (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Chemistry and Biochemistry (Contributor)
Created2013-05
Description
There is a need in the ecology literature to have a discussion about the fundamental theories from which population dynamics arises. Ad hoc model development is not uncommon in the field often as a result of a need to publish rapidly and frequently. Ecologists and statisticians like Robert J. Steidl and Kenneth P Burnham have called for a more deliberative approach they call "hard thinking". For example, the phenomena of population growth can be captured by almost any sigmoid function. The question of which sigmoid function best explains a data set cannot be answered meaningfully by statistical regression since that can only speak to the validity of the shape. There is a need to revisit enzyme kinetics and ecological stoichiometry to properly justify basal model selection in ecology. This dissertation derives several common population growth models from a generalized equation. The mechanistic validity of these models in different contexts is explored through a kinetic lens. The behavioral kinetic framework is then put to the test by examining a set of biologically plausible growth models against the 1968-1995 elk population count data for northern Yellowstone. Using only this count data, the novel Monod-Holling growth model was able to accurately predict minimum viable population and life expectancy despite both being exogenous to the model and data set. Lastly, the elk/wolf data from Yellowstone was used to compare the validity of the Rosenzweig-MacArthur and Arditi-Ginzburg models. They both were derived from a more general model which included both predator and prey mediated steps. The Arditi-Ginzburg model was able to fit the training data better, but only the Rosenzweig-MacArthur model matched the validation data. Accounting for animal sexual behavior allowed for the creation of the Monod-Holling model which is just as simple as the logistic differential equation but provides greater insights for conservation purposes. Explicitly acknowledging the ethology of wolf predation helps explain the differences in predictive performances by the best fit Rosenzweig-MacArthur and Arditi-Ginzburg models. The behavioral kinetic framework has proven to be a useful tool, and it has the ability to provide even further insights going forward.
ContributorsPringle, Jack Andrew McCracken (Author) / Anderies, John M (Thesis advisor) / Kuang, Yang (Committee member) / Milner, Fabio (Committee member) / Arizona State University (Publisher)
Created2022
Description
The mutual inhibition between synthetic gene circuits and cell growth produces growth feedback in the host-circuit system. Previous studies have demonstrated that the growth feedback has an marked impact on the molecular dynamics of the host-circuit system. However, the complexity of the growth feedback effect is not fully understood. A theoretical framework was developed to study the dynamics of the coupling between growth feedback and synthetic gene circuits. The study’s results reveal three major points about the impact of growth feedback. First, a nonlinear emergent behavior mediated by growth feedback. The unexpected behavior depends on the dynamic ribosome allocation between gene circuit expression and host cell growth. Second, the emergence and loss of unexpected qualitative states on the host-circuit system generated by ultrasensitive growth feedback. Third, the growth feedback-induced cooperativity behavior in synthetic gene modules competing for resources. In addition, growth feedback attenuated the winner-takes-all rules on resource competition between the two self-activating modules. These results demonstrate that growth feedback plays an important role in the host-circuit system’s molecular dynamics. Characterizing general principles from the effect of growth facilitates the ability to minimize or even harness unexpected gene expression behaviors derived from the effect of growth feedback.
ContributorsMelendez-Alvarez, Juan Ramon (Author) / Tian, Xiaojun (Thesis advisor) / Wang, Xiao (Committee member) / Kuang, Yang (Committee member) / Arizona State University (Publisher)
Created2022
Description
The representation of a patient’s characteristics as the parameters of a model is a key component in many studies of personalized medicine, where the underlying mathematical models are used to describe, explain, and forecast the course of treatment. In this context, clinical observations form the bridge between the mathematical frameworks and applications. However, the formulation and theoretical studies of the models and the clinical studies are often not completely compatible, which is one of the main obstacles in the application of mathematical models in practice. The goal of my study is to extend a mathematical framework to model prostate cancer based mainly on the concept of cell-quota within an evolutionary framework and to study the relevant aspects for the model to gain useful insights in practice. Specifically, the first aim is to construct a mathematical model that can explain and predict the observed clinical data under various treatment combinations. The second aim is to find a fundamental model structure that can capture the dynamics of cancer progression within a realistic set of data. Finally, relevant clinical aspects such as how the patient's parameters change over the course of treatment and how to incorporate treatment optimization within a framework of uncertainty quantification, will be examined to construct a useful framework in practice.
ContributorsPhan, Tin (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric J (Committee member) / Crook, Sharon (Committee member) / Maley, Carlo (Committee member) / Bryce, Alan (Committee member) / Arizona State University (Publisher)
Created2021
Description
Synthetic biology (SB) has become an important field of science focusing on designing and engineering new biological parts and systems, or re-designing existing biological systems for useful purposes. The dramatic growth of SB throughout the past two decades has not only provided us numerous achievements, but also brought us more timely and underexplored problems. In SB's entire history, mathematical modeling has always been an indispensable approach to predict the experimental outcomes, improve experimental design and obtain mechanism-understanding of the biological systems. \textit{Escherichia coli} (\textit{E. coli}) is one of the most important experimental platforms, its growth dynamics is the major research objective in this dissertation. Chapter 2 employs a reaction-diffusion model to predict the \textit{E. coli} colony growth on a semi-solid agar plate under multiple controls. In that chapter, a density-dependent diffusion model with non-monotonic growth to capture the colony's non-linear growth profile is introduced. Findings of the new model to experimental data are compared and contrasted with those from other proposed models. In addition, the cross-sectional profile of the colony are computed and compared with experimental data. \textit{E. coli} colony is also used to perform spatial patterns driven by designed gene circuits. In Chapter 3, a gene circuit (MINPAC) and its corresponding pattern formation results are presented. Specifically, a series of partial differential equation (PDE) models are developed to describe the pattern formation driven by the MINPAC circuit. Model simulations of the patterns based on different experimental conditions and numerical analysis of the models to obtain a deeper understanding of the mechanisms are performed and discussed. Mathematical analysis of the simplified models, including traveling wave analysis and local stability analysis, is also presented and used to explore the control strategies of the pattern formation. The interaction between the gene circuit and the host \textit{E. coli} may be crucial and even greatly affect the experimental outcomes. Chapter 4 focuses on the growth feedback between the circuit and the host cell under different nutrient conditions. Two ordinary differential equation (ODE) models are developed to describe such feedback with nutrient variation. Preliminary results on data fitting using both two models and the model dynamical analysis are included.
ContributorsHe, Changhan (Author) / Kuang, Yang (Thesis advisor) / Wang, Xiao (Committee member) / Kostelich, Eric (Committee member) / Tian, Xiaojun (Committee member) / Gumel, Abba (Committee member) / Arizona State University (Publisher)
Created2021
Description
A description of numerical and analytical work pertaining to models that describe the growth and progression of glioblastoma multiforme (GBM), an aggressive form of primary brain cancer. Two reaction-diffusion models are used: the Fisher-Kolmogorov-Petrovsky-Piskunov equation and a 2-population model that divides the tumor into actively proliferating and quiescent (or necrotic) cells. The numerical portion of this work (chapter 2) focuses on simulating GBM expansion in patients undergoing treatment for recurrence of tumor following initial surgery. The models are simulated on 3-dimensional brain geometries derived from magnetic resonance imaging (MRI) scans provided by the Barrow Neurological Institute. The study consists of 17 clinical time intervals across 10 patients that have been followed in detail, each of whom shows significant progression of tumor over a period of 1 to 3 months on sequential follow up scans. A Taguchi sampling design is implemented to estimate the variability of the predicted tumors to using 144 different choices of model parameters. In 9 cases, model parameters can be identified such that the simulated tumor contains at least 40 percent of the volume of the observed tumor. In the analytical portion of the paper (chapters 3 and 4), a positively invariant region for our 2-population model is identified. Then, a rigorous derivation of the critical patch size associated with the model is performed. The critical patch (KISS) size is the minimum habitat size needed for a population to survive in a region. Habitats larger than the critical patch size allow a population to persist, while smaller habitats lead to extinction. The critical patch size of the 2-population model is consistent with that of the Fisher-Kolmogorov-Petrovsky-Piskunov equation, one of the first reaction-diffusion models proposed for GBM. The critical patch size may indicate that GBM tumors have a minimum size depending on the location in the brain. A theoretical relationship between the size of a GBM tumor at steady-state and its maximum cell density is also derived, which has potential applications for patient-specific parameter estimation based on magnetic resonance imaging data.
ContributorsHarris, Duane C. (Author) / Kuang, Yang (Thesis advisor) / Kostelich, Eric J. (Thesis advisor) / Preul, Mark C. (Committee member) / Crook, Sharon (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2023