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- Genre: Academic theses
Description
Solution methods for certain linear and nonlinear evolution equations are presented in this dissertation. Emphasis is placed mainly on the analytical treatment of nonautonomous differential equations, which are challenging to solve despite the existent numerical and symbolic computational software programs available. Ideas from the transformation theory are adopted allowing one to solve the problems under consideration from a non-traditional perspective. First, the Cauchy initial value problem is considered for a class of nonautonomous and inhomogeneous linear diffusion-type equation on the entire real line. Explicit transformations are used to reduce the equations under study to their corresponding standard forms emphasizing on natural relations with certain Riccati(and/or Ermakov)-type systems. These relations give solvability results for the Cauchy problem of the parabolic equation considered. The superposition principle allows to solve formally this problem from an unconventional point of view. An eigenfunction expansion approach is also considered for this general evolution equation. Examples considered to corroborate the efficacy of the proposed solution methods include the Fokker-Planck equation, the Black-Scholes model and the one-factor Gaussian Hull-White model. The results obtained in the first part are used to solve the Cauchy initial value problem for certain inhomogeneous Burgers-type equation. The connection between linear (the Diffusion-type) and nonlinear (Burgers-type) parabolic equations is stress in order to establish a strong commutative relation. Traveling wave solutions of a nonautonomous Burgers equation are also investigated. Finally, it is constructed explicitly the minimum-uncertainty squeezed states for quantum harmonic oscillators. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. It is shown that the product of the variances attains the required minimum value only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. Such explicit construction is possible due to the relation between the diffusion-type equation studied in the first part and the time-dependent Schrodinger equation. A modication of the radiation field operators for squeezed photons in a perfect cavity is also suggested with the help of a nonstandard solution of Heisenberg's equation of motion.
ContributorsVega-Guzmán, José Manuel, 1982- (Author) / Sulov, Sergei K (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Platte, Rodrigo (Committee member) / Chowell-Puente, Gerardo (Committee member) / Arizona State University (Publisher)
Created2013
Description
Extraordinary medical advances have led to significant reductions in the burden of infectious diseases in humans. However, infectious diseases still account for more than 13 million annual deaths. This large burden is partly due to some pathogens having found suitable conditions to emerge and spread in denser and more connected host populations, and others having evolved to escape the pressures imposed by the rampant use of antimicrobials. It is then critical to improve our understanding of how diseases spread in these modern landscapes, characterized by new host population structures and socio-economic environments, as well as containment measures such as the deployment of drugs. Thus, the motivation of this dissertation is two-fold. First, we study, using both data-driven and modeling approaches, the the spread of infectious diseases in urban areas. As a case study, we use confirmed-cases data on sexually transmitted diseases (STDs) in the United States to assess the conduciveness of population size of urban areas and their socio-economic characteristics as predictors of STD incidence. We find that the scaling of STD incidence in cities is superlinear, and that the percent of African-Americans residing in cities largely determines these statistical patterns. Since disparities in access to health care are often exacerbated in urban areas, within this project we also develop two modeling frameworks to study the effect of health care disparities on epidemic outcomes. Discrepant results between the two approaches indicate that knowledge of the shape of the recovery period distribution, not just its mean and variance, is key for assessing the epidemiological impact of inequalities. The second project proposes to study, from a modeling perspective, the spread of drug resistance in human populations featuring vital dynamics, stochasticity and contact structure. We derive effective treatment regimes that minimize both the overall disease burden and the spread of resistance. Additionally, targeted treatment in structured host populations may lead to higher levels of drug resistance, and if drug-resistant strains are compensated, they can spread widely even when the wild-type strain is below its epidemic threshold.
ContributorsPatterson-Lomba, Oscar (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Towers, Sherry (Thesis advisor) / Chowell-Puente, Gerardo (Committee member) / Arizona State University (Publisher)
Created2014
Description
Urban scaling analysis has introduced a new scientific paradigm to the study of cities. With it, the notions of size, heterogeneity and structure have taken a leading role. These notions are assumed to be behind the causes for why cities differ from one another, sometimes wildly. However, the mechanisms by which size, heterogeneity and structure shape the general statistical patterns that describe urban economic output are still unclear. Given the rapid rate of urbanization around the globe, we need precise and formal mathematical understandings of these matters. In this context, I perform in this dissertation probabilistic, distributional and computational explorations of (i) how the broadness, or narrowness, of the distribution of individual productivities within cities determines what and how we measure urban systemic output, (ii) how urban scaling may be expressed as a statistical statement when urban metrics display strong stochasticity, (iii) how the processes of aggregation constrain the variability of total urban output, and (iv) how the structure of urban skills diversification within cities induces a multiplicative process in the production of urban output.
ContributorsGómez-Liévano, Andrés (Author) / Lobo, Jose (Thesis advisor) / Muneepeerakul, Rachata (Thesis advisor) / Bettencourt, Luis M. A. (Committee member) / Chowell-Puente, Gerardo (Committee member) / Arizona State University (Publisher)
Created2014
Description
Diseases have been part of human life for generations and evolve within the population, sometimes dying out while other times becoming endemic or the cause of recurrent outbreaks. The long term influence of a disease stems from different dynamics within or between pathogen-host, that have been analyzed and studied by many researchers using mathematical models. Co-infection with different pathogens is common, yet little is known about how infection with one pathogen affects the host's immunological response to another. Moreover, no work has been found in the literature that considers the variability of the host immune health or that examines a disease at the population level and its corresponding interconnectedness with the host immune system. Knowing that the spread of the disease in the population starts at the individual level, this thesis explores how variability in immune system response within an endemic environment affects an individual's vulnerability, and how prone it is to co-infections. Immunology-based models of Malaria and Tuberculosis (TB) are constructed by extending and modifying existing mathematical models in the literature. The two are then combined to give a single nine-variable model of co-infection with Malaria and TB. Because these models are difficult to gain any insight analytically due to the large number of parameters, a phenomenological model of co-infection is proposed with subsystems corresponding to the individual immunology-based model of a single infection. Within this phenomenological model, the variability of the host immune health is also incorporated through three different pathogen response curves using nonlinear bounded Michaelis-Menten functions that describe the level or state of immune system (healthy, moderate and severely compromised). The immunology-based models of Malaria and TB give numerical results that agree with the biological observations. The Malaria--TB co-infection model gives reasonable results and these suggest that the order in which the two diseases are introduced have an impact on the behavior of both. The subsystems of the phenomenological models that correspond to a single infection (either of Malaria or TB) mimic much of the observed behavior of the immunology-based counterpart and can demonstrate different behavior depending on the chosen pathogen response curve. In addition, varying some of the parameters and initial conditions in the phenomenological model yields a range of topologically different mathematical behaviors, which suggests that this behavior may be able to be observed in the immunology-based models as well. The phenomenological models clearly replicate the qualitative behavior of primary and secondary infection as well as co-infection. The mathematical solutions of the models correspond to the fundamental states described by immunologists: virgin state, immune state and tolerance state. The phenomenological model of co-infection also demonstrates a range of parameter values and initial conditions in which the introduction of a second disease causes both diseases to grow without bound even though those same parameters and initial conditions did not yield unbounded growth in the corresponding subsystems. This results applies to all three states of the host immune system. In terms of the immunology-based system, this would suggest the following: there may be parameter values and initial conditions in which a person can clear Malaria or TB (separately) from their system but in which the presence of both can result in the person dying of one of the diseases. Finally, this thesis studies links between epidemiology (population level) and immunology in an effort to assess the impact of pathogen's spread within the population on the immune response of individuals. Models of Malaria and TB are proposed that incorporate the immune system of the host into a mathematical model of an epidemic at the population level.
ContributorsSoho, Edmé L (Author) / Wirkus, Stephen (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Chowell-Puente, Gerardo (Committee member) / Arizona State University (Publisher)
Created2011
Description
Yersinia enterocolitica is a major foodborne pathogen found worldwide that causes approximately 87,000 human cases and approximately 1,100 hospitalizations per year in the United States. Y. enterocolitica is a very unique pathogen with the domesticated pig acting as the main animal reservoir for pathogenic bio/serotypes, and as the primary source of human infection. Similar to other gastrointestinal infections, Yersinia enterocolitica is known to trigger autoimmune responses in humans. The most frequent complication associated with Y. enterocolitica is reactive arthritis - an aseptic, asymmetrical inflammation in the peripheral and axial joints, most frequently occurring as an autoimmune response in patients with the HLA-B27 histocompatability antigen. As a foodborne illness it may prove to be a reasonable explanation for some of the cases of arthritis observed in past populations that are considered to be of unknown etiology. The goal of this dissertation project was to study the relationship between the foodborne illness -Y. enterocolitica, and the incidence of arthritis in individuals with and without contact with the domesticated pig.
ContributorsBrown, Starletta (Author) / Hurtado, Ana M (Thesis advisor) / Chowell-Puente, Gerardo (Committee member) / Hill, Kim (Committee member) / Arizona State University (Publisher)
Created2015
Description
The increased number of novel pathogens that potentially threaten the human population has motivated the development of mathematical and computational modeling approaches for forecasting epidemic impact and understanding key environmental characteristics that influence the spread of diseases. Yet, in the case that substantial uncertainty surrounds the transmission process during a rapidly developing infectious disease outbreak, complex mechanistic models may be too difficult to be calibrated quick enough for policy makers to make informed decisions. Simple phenomenological models that rely on a small number of parameters can provide an initial platform for assessing the epidemic trajectory, estimating the reproduction number and quantifying the disease burden from the early epidemic phase.
Chapter 1 provides background information and motivation for infectious disease forecasting and outlines the rest of the thesis.
In chapter 2, logistic patch models are used to assess and forecast the 2013-2015 West Africa Zaire ebolavirus epidemic. In particular, this chapter is concerned with comparing and contrasting the effects that spatial heterogeneity has on the forecasting performance of the cumulative infected case counts reported during the epidemic.
In chapter 3, two simple phenomenological models inspired from population biology are used to assess the Research and Policy for Infectious Disease Dynamics (RAPIDD) Ebola Challenge; a simulated epidemic that generated 4 infectious disease scenarios. Because of the nature of the synthetically generated data, model predictions are compared to exact epidemiological quantities used in the simulation.
In chapter 4, these models are applied to the 1904 Plague epidemic that occurred in Bombay. This chapter provides evidence that these simple models may be applicable to infectious diseases no matter the disease transmission mechanism.
Chapter 5, uses the patch models from chapter 2 to explore how migration in the 1904 Plague epidemic changes the final epidemic size.
The final chapter is an interdisciplinary project concerning within-host dynamics of cereal yellow dwarf virus-RPV, a plant pathogen from a virus group that infects over 150 grass species. Motivated by environmental nutrient enrichment due to anthropological activities, mathematical models are employed to investigate the relevance of resource competition to pathogen and host dynamics.
Chapter 1 provides background information and motivation for infectious disease forecasting and outlines the rest of the thesis.
In chapter 2, logistic patch models are used to assess and forecast the 2013-2015 West Africa Zaire ebolavirus epidemic. In particular, this chapter is concerned with comparing and contrasting the effects that spatial heterogeneity has on the forecasting performance of the cumulative infected case counts reported during the epidemic.
In chapter 3, two simple phenomenological models inspired from population biology are used to assess the Research and Policy for Infectious Disease Dynamics (RAPIDD) Ebola Challenge; a simulated epidemic that generated 4 infectious disease scenarios. Because of the nature of the synthetically generated data, model predictions are compared to exact epidemiological quantities used in the simulation.
In chapter 4, these models are applied to the 1904 Plague epidemic that occurred in Bombay. This chapter provides evidence that these simple models may be applicable to infectious diseases no matter the disease transmission mechanism.
Chapter 5, uses the patch models from chapter 2 to explore how migration in the 1904 Plague epidemic changes the final epidemic size.
The final chapter is an interdisciplinary project concerning within-host dynamics of cereal yellow dwarf virus-RPV, a plant pathogen from a virus group that infects over 150 grass species. Motivated by environmental nutrient enrichment due to anthropological activities, mathematical models are employed to investigate the relevance of resource competition to pathogen and host dynamics.
ContributorsPell, Bruce (Author) / Kuang, Yang (Thesis advisor) / Chowell-Puente, Gerardo (Committee member) / Nagy, John (Committee member) / Kostelich, Eric (Committee member) / Gardner, Carl (Committee member) / Arizona State University (Publisher)
Created2016