Spelling suggestions: "subject:"ordinary differential equations"" "subject:"rdinary differential equations""
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Evolutionary Game Theory and the Spread of InfluenzaBeauparlant, Marc A. January 2016 (has links)
Vaccination has been used to control the spread of infectious diseases for centuries with widespread success. Deterministic models studying the spread of infectious disease often use the assumption of mass vaccination; however, these models do not allow for the inclusion of human behaviour. Since current vaccination campaigns are voluntary in nature, it is important to extend the study of infectious disease models to include the effects of human behaviour. To model the effects of vaccination behaviour on the spread of influenza, we examine a series of models in which individuals vaccinate according to memory or individual decision-making processes based upon self-interest. Allowing individuals to vaccinate proportionally to an exponentially decaying memory function of disease prevalence, we demonstrate the existence of a Hopf bifurcation for short memory spans. Using a game-theoretic influenza model, we determine that lowering the perceived vaccine risk may be insufficient to increase coverage to established target levels. Utilizing evolutionary game theory, we examine models with imitation dynamics both with and without a decaying memory function and show that, under certain conditions, periodic dynamics occur without seasonal forcing. Our results suggest that maintaining diseases at low prevalence with voluntary vaccination campaigns could lead to subsequent epidemics following the free-rider dilemma and that future research in disease control reliant on individual-based decision-making need to include the effects of human behaviour.
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An algebraic - analytic framework for the study of intertwined families of evolution operatorsLee, Wha-Suck January 2015 (has links)
We introduce a new framework of generalized operators to handle vector valued distributions, intertwined evolution operators of B-evolution equations and Fokker Planck type evolution equations. Generalized operators capture these operators. The framework is a marriage between vector valued distribution theory and abstract harmonic analysis: a new convolution algebra is the offspring. The new algebra shows that convolution is more fundamental than operator composition. The framework is complete with a Hille-Yosida theorem for implicit evolution equations for generalized operators.
Feller semigroups and processes fit perfectly into the framework of generalized operators. Feller semigroups are intertwined by the Chapman Kolmogorov equation. Our framework handles more complex intertwinements which naturally arise from a dynamic boundary approach to an absorbing barrier of a fly trap model: we construct an entwined pseudo Poisson process which is a pair of stochastic processes entwined by the extended Chapman Kolmogorov equation. Similarly, we introduce the idea of an entwined Brownian motion. We show that the diffusion equation of an entwined Brownian motion involves an implicit evolution equation on a suitable scalar test space. We end off by constructing a new convolution of operator valued measures which generalizes the convolution of Feller convolution semigroups. / Thesis (PhD)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / Unrestricted
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Modeling and Analyzing the Progression of Retinitis PigmentosaJanuary 2020 (has links)
abstract: Patients suffering from Retinitis Pigmentosa (RP), the most common type of inherited retinal degeneration, experience irreversible vision loss due to photoreceptor degeneration. The preservation of cone photoreceptors has been deemed medically relevant as a therapy aimed at preventing blindness in patients with RP. Cones rely on aerobic glycolysis to supply the metabolites necessary for outer segment (OS) renewal and maintenance. The rod-derived cone viability factor (RdCVF), a protein secreted by the rod photoreceptors that preserves the cones, accelerates the flow of glucose into the cone cell stimulating aerobic glycolysis. This dissertation presents and analyzes ordinary differential equation (ODE) models of cellular and molecular level photoreceptor interactions in health and disease to examine mechanisms leading to blindness in patients with RP.
First, a mathematical model composed of four ODEs is formulated to investigate the progression of RP, accounting for the new understanding of RdCVF’s role in enhancing cone survival. A mathematical analysis is performed, and stability and bifurcation analyses are used to explore various pathways to blindness. Experimental data are used for parameter estimation and model validation. The numerical results are framed in terms of four stages in the progression of RP. Sensitivity analysis is used to determine mechanisms that have a significant affect on the cones at each stage of RP. Utilizing a non-dimensional form of the RP model, a numerical bifurcation analysis via MATCONT revealed the existence of stable limit cycles at two stages of RP.
Next, a novel eleven dimensional ODE model of molecular and cellular level interactions is described. The subsequent analysis is used to uncover mechanisms that affect cone photoreceptor functionality and vitality. Preliminary simulations show the existence of oscillatory behavior which is anticipated when all processes are functioning properly. Additional simulations are carried out to explore the impact of a reduction in the concentration of RdCVF coupled with disruption in the metabolism associated with cone OS shedding, and confirms cone-on-rod reliance. The simulation results are compared with experimental data. Finally, four cases are considered, and a sensitivity analysis is performed to reveal mechanisms that significantly impact the cones in each case. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020
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Lagrangeovský model pohybu kavitační bubliny / Lagrangian tracking of the cavitation bubbleBossio Castro, Alvaro Manuel January 2019 (has links)
In this thesis, the dynamics of an isolated cavitation bubble submerged in a steady flow is studied numerically. A Lagrangian-Eulerian approach is considered, in which properties of the fluid are computed first by means of Eulerian methods (in this study the commercial CFD software Ansys Fluent 19 was used) and the trajectory of the bubble is then computed in a Lagrangian fashion, i.e. the bubble is considered as a small particle moving relative to the fluid, due to the effect of several forces depending on fluid's pressure field, fluid's velocity field and bubble's radius. Bubble's radius dynamics, modeled by Rayleigh-Plesset equation, has a big influence on its kinetics, so a special attention is given to it. Two study cases are considered. The first one, motivated by acoustic cavitation is concerned with the response of the bubble's radius in a static flow under the influence of an oscillatory pressure field, the second one studies the trajectory of the bubble submerged in a fluid passing by a Venturi tube and a sharp-edged orifice plate.
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Convergence rates of adaptive algorithms for deterministic and stochastic differential equationsMoon, Kyoung-Sook January 2001 (has links)
NR 20140805
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Parameter optimization of linear ordinary differential equations with application in gene regulatory network inference problems / Parameteroptimering av linjära ordinära differentialekvationer med tillämpningar inom inferensproblem i regulatoriska gennätverkDeng, Yue January 2014 (has links)
In this thesis we analyze parameter optimization problems governed by linear ordinary differential equations (ODEs) and develop computationally efficient numerical methods for their solution. In addition, a series of noise-robust finite difference formulas are given for the estimation of the derivatives in the ODEs. The suggested methods have been employed to identify Gene Regulatory Networks (GRNs). GRNs are responsible for the expression of thousands of genes in any given developmental process. Network inference deals with deciphering the complex interplay of genes in order to characterize the cellular state directly from experimental data. Even though a plethora of methods using diverse conceptual ideas has been developed, a reliable network reconstruction remains challenging. This is due to several reasons, including the huge number of possible topologies, high level of noise, and the complexity of gene regulation at different levels. A promising approach is dynamic modeling using differential equations. In this thesis we present such an approach to infer quantitative dynamic models from biological data which addresses inherent weaknesses in the current state-of-the-art methods for data-driven reconstruction of GRNs. The method is computationally cheap such that the size of the network (model complexity) is no longer a main concern with respect to the computational cost but due to data limitations; the challenge is a huge number of possible topologies. Therefore we embed a filtration step into the method to reduce the number of free parameters before simulating dynamical behavior. The latter is used to produce more information about the network’s structure. We evaluate our method on simulated data, and study its performance with respect to data set size and levels of noise on a 1565-gene E.coli gene regulatory network. We show the computation time over various network sizes and estimate the order of computational complexity. Results on five networks in the benchmark collection DREAM4 Challenge are also presented. Results on five networks in the benchmark collection DREAM4 Challenge are also presented and show our method to outperform the current state of the art methods on synthetic data and allows the reconstruction of bio-physically accurate dynamic models from noisy data. / I detta examensarbete analyserar vi parameteroptimeringsproblem som är beskrivna med ordinära differentialekvationer (ODEer) och utvecklar beräkningstekniskt effektiva numeriska metoder för att beräkna lösningen. Dessutom härleder vi brusrobusta finita-differens approximationer för uppskattning av derivator i ODEn. De föreslagna metoderna har tillämpats för regulatoriska gennätverk (RGN). RGNer är ansvariga för uttrycket av tusentals gener. Nätverksinferens handlar om att identifiera den komplicerad interaktionen mellan gener för att kunna karaktärisera cellernas tillstånd direkt från experimentella data. Tillförlitlig nätverksrekonstruktion är ett utmanande problem, trots att många metoder som använder många olika typer av konceptuella idéer har utvecklats. Detta beror på flera olika saker, inklusive att det finns ett enormt antal topologier, mycket brus, och komplexiteten av genregulering på olika nivåer. Ett lovande angreppssätt är dynamisk modellering från biologiska data som angriper en underliggande svaghet i den för tillfället ledande metoden för data-driven rekonstruktion. Metoden är beräkningstekniskt billig så att storleken på nätverket inte längre är huvudproblemet för beräkningen men ligger fortfarande i databegränsningar. Utmaningen är ett enormt antal av topologier. Därför bygger vi in ett filtreringssteg i metoder för att reducera antalet fria parameterar och simulerar sedan det dynamiska beteendet. Anledningen är att producera mer information om nätverkets struktur. Vi utvärderar metoden på simulerat data, och studierar dess prestanda med avseende på datastorlek och brusnivå genom att tillämpa den på ett regulartoriskt gennätverk med 1565-gen E.coli. Vi illustrerar beräkningstiden över olika nätverksstorlekar och uppskattar beräkningskomplexiteten. Resultat på fem nätverk från DREAM4 är också presenterade och visar att vår metod har bättre prestanda än nuvarande metoder när de tillämpas på syntetiska data och tillåter rekonstruktion av bio-fysikaliskt noggranna dynamiska modeller från data med brus.
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A Physiologically-Based Pharmacokinetic Model for the Antibiotic ErtapenemJoyner, Michele L., Forbes, Whitney, Maiden, Michelle, Nikas, Ariel N. 01 February 2016 (has links)
Ertapenem is an antibiotic commonly used to treat a broad spectrum of infections, which is part of a broader class of antibiotics called carbapenem. Unlike other carbapenems, ertapenem has a longer half-life and thus only has to be administered once a day. A physiologically-based pharmacokinetic (PBPK) model was developed to investigate the uptake, distribution, and elimination of ertapenem following a single one gram dose. PBPK modeling incorporates known physiological parameters such as body weight, organ volumes, and blood ow rates in particular tissues. Furthermore, ertapenem is highly bound in human blood plasma; therefore, nonlinear binding is incorporated in the model since only the free portion of the drug can saturate tissues and, hence, is the only portion of the drug considered to be medicinally effective. Parameters in the model were estimated using a least squares inverse problem formulation with published data for blood concentrations of ertapenem for normal height, normal weight males. Finally, an uncertainty analysis of the parameter estimation and model predictions is presented.
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Seasonal Variation in a Predator-Predator-Prey ModelBolohan, Noah 31 August 2020 (has links)
Seasonal shifts in predation habits, from a generalist in the summer to a specialist in the winter, have been documented for the great horned owl (Bulbo virginialis) in the boreal forest. This shift occurs largely due to varying prey availability. There is little study of this switching behaviour in the current literature. Since season length is predicted to change under future climate scenarios, it is important to understand resulting effects on species dynamics. Previous work has been done on a two-species seasonal model for the great horned owl and its focal prey, the snowshoe hare (Lepus americanus). In this thesis, we extend the model by adding one of the hare's most important predators, the Canadian lynx (Lynx canadensis). We study the qualitative behaviour of this model as season length changes using tools and techniques from dynamical systems. Our main approach is to determine when the lynx and the owl may invade the system at low density and ask whether mutual invasion of the predators implies stable coexistence in the three-species model. We observe that, as summer length increases, mutual invasion is less likely, and we expect to see extinction of the lynx. However, in all cases where mutual invasion was satisfied, the three species stably coexist.
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Moving in time: a neural network model of rhythm-based motor sequence performanceZeid, Omar Mohamed 05 September 2019 (has links)
Many complex actions are precomposed, by sequencing simpler motor actions. For such a complex action to be executed accurately, those simpler actions must be planned in the desired order, held in working memory, and then enacted one-by-one until the sequence is complete. Examples of this phenomenon include writing, typing, and speaking.
Under most circumstances, the ability to learn and reproduce novel motor sequences is hindered when additional information is presented. However, in cases where the motor sequence is musical in nature (e.g. a choreographed dance or a piano melody), one must learn two sequences at the same time, one of motor actions and one of the time intervals between actions. Despite this added complexity, humans learn and perform rhythm-based motor sequences regularly. It has been shown that people can learn motoric and rhythmic sequences separately and then combine them with little trouble (Ullén & Bengtsson 2003). Also, functional MRI data suggest that there are distinct sets of neural regions responsible for the two different sequence types (Bengtsson et al. 2004).
Although research on musical rhythm is extensive, few computational models exist to extend and inform our understanding of its neural bases. To that end, this dissertation introduces the TAMSIN (Timing And Motor System Integration Network) model, a systems-level neural network model designed to replicate rhythm-based motor sequence performance. TAMSIN utilizes separate Competitive Queuing (CQ) modules for motoric and temporal sequences, as well as modules designed to coordinate these sequence types into a cogent output performance consistent with a perceived beat and tempo.
Chapters 1-4 explore prior literature on CQ architectures, rhythmic perception/production, and computational modeling, thereby illustrating the need for a model to tie those research areas together. Chapter 5 details the structure of the TAMSIN model and its mathematical specification. Chapter 6 presents and discusses the results of the model simulated under various circumstances. Chapter 7 compares the simulation results to behavioral and imaging results from the experimental literature. The final chapter discusses future modifications that could be made to TAMSIN to simulate aspects of rhythm learning, rhythm perception, and disordered productions, such as those seen in Parkinson’s disease.
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Amélioration de la rapidité d'exécution des systèmes EDO de grande taille issus de Modelica / Improvement of execution speed of large scale ODE systems from ModelicaGallois, Thibaut-Hugues 03 December 2015 (has links)
L'étude des systèmes aux équations différentielles ordinaires vise à prédire le futur des systèmes considérés. La connaissance de l'évolution dans le temps de toutes les variables d' état du modèle permet de prédire de possibles changements radicaux des variables ou des défaillances, par exemple, un moteur peut exploser, un pont peut s'écrouler, une voiture peut se mettre à consommer plus d'essence. De plus, les systèmes dynamiques peuvent contenir des dérivées spatiales et leur discrétisation peut ajouter un très grand nombre d'équations. La résolution des équations différentielles ordinaires est alors une étape essentielle dans la construction des systèmes physiques en terme de dimensionnement et de faisabilité. Le solveur de tels systèmes EDOs doit être rapide, précis et pertinent.En pratique, il n'est pas possible de trouver une fonction continue qui soit solution exacte du problème EDO. C'est pourquoi, des méthodes numériques sont utilisées afin de donner des solutions discrèes qui approchent la solution continue avec une erreur contrôlable. La gestion précise de ce contrôle est très importante afin d'obtenir une solution pertinente en un temps raisonnable.Cette thèse développe un nouveau solveur qui utilise plusieurs méthodes d'amélioration de la vitesse d'exécution des systèmes EDOs. La première méthode est l'utilisation d'un nouveau schéma numérique. Le but est de minimiser le coût de l'intégration en produisant une erreur qui soit le plus proche possible de la tolérance maximale permise par l'utilisateur du solveur. Une autre méthode pour améliorer la vitesse d'exécution est de paralléliser le solveur EDO en utilisant une architecture multicoeur et multiprocesseur. Enfin, le solveur a été testé avec différentes applications d'OpenModelica. / The study of systems of Ordinary Differential Equations aims at predicting the future of the considered systems. The access to the evolution of all states of a system's model allows us to predict possible drastic shifts of the states or failures, e.g. an engine blowing up, a bridge collapsin, a car consuming more gasoline etc. Solving ordinary differential equations is then an essential step of building industrial physical systems in regard to dimensioning and reliability. The solver of such ODE systems needs to be fast, accurate and relevant.In practice, it is not possible to find a continuous function as the exact solution of the real ODE problem. Consequently numerical methods are used to give discrete solutions which approximates the continuous one with a controllable error. The correct handline of this control is very important to get a relevant solution within an acceptable recovery time. Starting from existing studies of local and global errors, this thesis work goes more deeply and adjusts the time step of the integration time algorithm and solves the problem in a very efficient manner.A new scheme is proposed is this thesis, to minimize the cost of integration. Another method to improve the execution speed is to parallelize the ODE solver by using a multicore and a multiprocessor architecture. Finally, the solver has been tested with different applications from OpenModelica.
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