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- All Subjects: Applied Mathematics
- Creators: Castillo-Chavez, Carlos
Description
Infectious diseases are a leading cause of death worldwide. With the development of drugs, vaccines and antibiotics, it was believed that for the first time in human history diseases would no longer be a major cause of mortality. Newly emerging diseases, re-emerging diseases and the emergence of microorganisms resistant to existing treatment have forced us to re-evaluate our optimistic perspective. In this study, a simple mathematical framework for super-infection is considered in order to explore the transmission dynamics of drug-resistance. Through its theoretical analysis, we identify the conditions necessary for the coexistence between sensitive strains and drug-resistant strains. Farther, in order to investigate the effectiveness of control measures, the model is extended so as to include vaccination and treatment. The impact that these preventive and control measures may have on its disease dynamics is evaluated. Theoretical results being confirmed via numerical simulations. Our theoretical results on two-strain drug-resistance models are applied in the context of Malaria, antimalarial drugs, and the administration of a possible partially effective vaccine. The objective is to develop a monitoring epidemiological framework that help evaluate the impact of antimalarial drugs and partially-effective vaccine in reducing the disease burden at the population level. Optimal control theory is applied in the context of this framework in order to assess the impact of time dependent cost-effective treatment efforts. It is shown that cost-effective combinations of treatment efforts depend on the population size, cost of implementing treatment controls, and the parameters of the model. We use these results to identify optimal control strategies for several scenarios.
ContributorsUrdapilleta, Alicia (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Wang, Xiaohong (Thesis advisor) / Wirkus, Stephen (Committee member) / Camacho, Erika (Committee member) / Arizona State University (Publisher)
Created2011
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
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
Olfaction is an important sensory modality for behavior since odors inform animals of the presence of food, potential mates, and predators. The fruit fly, Drosophila melanogaster, is a favorable model organism for the investigation of the biophysical mechanisms that contribute to olfaction because its olfactory system is anatomically similar to but simpler than that of vertebrates. In the Drosophila olfactory system, sensory transduction takes place in olfactory receptor neurons housed in the antennae and maxillary palps on the front of the head. The first stage of olfactory processing resides in the antennal lobe, where the structural unit is the glomerulus. There are at least three classes of neurons in the antennal lobe - excitatory projection neurons, excitatory local neurons, and inhibitory local neurons. The arborizations of the local neurons are confined to the antennal lobe, and output from the antennal lobe is carried by projection neurons to higher regions of the brain. Different views exist of how circuits of the Drosophila antennal lobe translate input from the olfactory receptor neurons into projection neuron output. We construct a conductance based neuronal network model of the Drosophila antennal lobe with the aim of understanding possible mechanisms within the antennal lobe that account for the variety of projection neuron activity observed in experimental data. We explore possible outputs obtained from olfactory receptor neuron input that mimic experimental recordings under different connectivity paradigms. First, we develop realistic minimal cell models for the excitatory local neurons, inhibitory local neurons, and projections neurons based on experimental data for Drosophila channel kinetics, and explore the firing characteristics and mathematical structure of these models. We then investigate possible interglomerular and intraglomerular connectivity patterns in the Drosophila antennal lobe, where olfactory receptor neuron input to the antennal lobe is modeled with Poisson spike trains, and synaptic connections within the antennal lobe are mediated by chemical synapses and gap junctions as described in the Drosophila antennal lobe literature. Our simulation results show that inhibitory local neurons spread inhibition among all glomeruli, where projection neuron responses are decreased relatively uniformly for connections of synaptic strengths that are homogeneous. Also, in the case of homogeneous excitatory synaptic connections, the excitatory local neuron network facilitates odor detection in the presence of weak stimuli. Excitatory local neurons can spread excitation from projection neurons that receive more input from olfactory receptor neurons to projection neurons that receive less input from olfactory receptor neurons. For the parameter values for the network models associated with these results, eLNs decrease the ability of the network to discriminate among single odors.
ContributorsLuli, Dori (Author) / Crook, Sharon (Thesis advisor) / Baer, Steven (Committee member) / Castillo-Chavez, Carlos (Committee member) / Smith, Brian (Committee member) / Arizona State University (Publisher)
Created2013
Description
In complex consumer-resource type systems, where diverse individuals are interconnected and interdependent, one can often anticipate what has become known as the tragedy of the commons, i.e., a situation, when overly efficient consumers exhaust the common resource, causing collapse of the entire population. In this dissertation I use mathematical modeling to explore different variations on the consumer-resource type systems, identifying some possible transitional regimes that can precede the tragedy of the commons. I then reformulate it as a game of a multi-player prisoner's dilemma and study two possible approaches for preventing it, namely direct modification of players' payoffs through punishment/reward and modification of the environment in which the interactions occur. I also investigate the questions of whether the strategy of resource allocation for reproduction or competition would yield higher fitness in an evolving consumer-resource type system and demonstrate that the direction in which the system will evolve will depend not only on the state of the environment but largely on the initial composition of the population. I then apply the developed framework to modeling cancer as an evolving ecological system and draw conclusions about some alternative approaches to cancer treatment.
ContributorsKareva, Irina (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Collins, James (Committee member) / Nagy, John (Committee member) / Smith, Hal (Committee member) / Arizona State University (Publisher)
Created2012
Description
There have been many studies on the dynamics of infectious diseases considering the age structure of the population. This study analyzes the dynamics when the population is stratified by size. This kind of models are useful in the spread of a disease in fisheries where size matters, for microorganism populations or even human diseases that are driven by weight. A simple size structured SIR model is introduced for which a threshold condition, R0, equilibria and stability are established in special cases. Hethcote's approach is used to derive, from first principles, a parallel ODE size-structure system involving n-size classes.The specific case of n = 2 is partially analyzed. Constant effort harvesting is added to this model with the purpose of exploring the role of controls and harvesting. Different harvesting policies are proposed and analyzed through simulations.
ContributorsTorres García, Griselle (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Feng, Zhilan (Thesis advisor) / Lee, Sunmi (Committee member) / Arizona State University (Publisher)
Created2012
Description
Dopamine (DA) is a neurotransmitter involved in attention, goal oriented behavior, movement, reward learning, and short term and working memory. For the past four decades, mathematical and computational modeling approaches have been useful in DA research, and although every modeling approach has limitations, a model is an efficient way to generate and explore hypotheses. This work develops a model of DA dynamics in a representative, single DA neuron by integrating previous experimental, theoretical and computational research. The model consists of three compartments: the cytosol, the vesicles, and the extracellular space and forms the basis of a new mathematical paradigm for examining the dynamics of DA synthesis, storage, release and reuptake. The model can be driven by action potentials generated by any model of excitable membrane potential or even from experimentally induced depolarization voltage recordings. Here the model is forced by a previously published model of the excitable membrane of a mesencephalic DA neuron in order to study the biochemical processes involved in extracellular DA production. After demonstrating that the model exhibits realistic dynamics resembling those observed experimentally, the model is used to examine the functional changes in presynaptic mechanisms due to application of cocaine. Sensitivity analysis and numerical studies that focus on various possible mechanisms for the inhibition of DAT by cocaine provide insight for the complex interactions involved in DA dynamics. In particular, comparing numerical results for a mixed inhibition mechanism to those for competitive, non-competitive and uncompetitive inhibition mechanisms reveals many behavioral similarities for these different types of inhibition that depend on inhibition parameters and levels of cocaine. Placing experimental results within this context of mixed inhibition provides a possible explanation for the conflicting views of uptake inhibition mechanisms found in experimental neuroscience literature.
ContributorsTello-Bravo, David (Author) / Crook, Sharon M (Thesis advisor) / Greenwood, Priscilla E (Thesis advisor) / Baer, Steven M. (Committee member) / Castaneda, Edward (Committee member) / Castillo-Chavez, Carlos (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
A sequence of models is developed to describe urban population growth in the context of the embedded physical, social and economic environments and an urban disease are developed. This set of models is focused on urban growth and the relationship between the desire to move and the utility derived from city life. This utility is measured in terms of the economic opportunities in the city, the level of human constructed amenity, and the level of amenity caused by the natural environment. The set of urban disease models is focused on examining prospects of eliminating a disease for which a vaccine does not exist. It is inspired by an outbreak of the vector-borne disease dengue fever in Peru, during 2000-2001.
ContributorsMurillo, D (Author) / Castillo-Chavez, Carlos (Thesis advisor) / Anderies, John M (Thesis advisor) / Boone, Christopher (Committee member) / Arizona State University (Publisher)
Created2012
Description
In the traditional setting of quantum mechanics, the Hamiltonian operator does not depend on time. While some Schrödinger equations with time-dependent Hamiltonians have been solved, explicitly solvable cases are typically scarce. This thesis is a collection of papers in which this first author along with Suslov, Suazo, and Lopez, has worked on solving a series of Schrödinger equations with a time-dependent quadratic Hamiltonian that has applications in problems of quantum electrodynamics, lasers, quantum devices such as quantum dots, and external varying fields. In particular the author discusses a new completely integrable case of the time-dependent Schrödinger equation in R^n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. The author also considers several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schrödinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator. Finally, the author constructs integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrödinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application.
ContributorsCordero-Soto, Ricardo J (Author) / Suslov, Sergei (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Engman, Martin (Committee member) / Herrera-Valdez, Marco (Committee member) / Arizona State University (Publisher)
Created2011