Development of a log-quadratic model to describe microbial inactivation, illustrated by thermal inactivation of Clostridium botulinum

In the commercial food industry, demonstration of microbiological safety and thermal process equivalence often involves a mathematical framework that assumes log-linear inactivation kinetics and invokes concepts of decimal reduction time (DT), z values, and accumulated lethality. However, many microbes, particularly spores, exhibit inactivation kinetics that are not log linear. This has led to alternative modeling approaches, such as the biphasic and Weibull models, that relax strong log-linear assumptions. Using a statistical framework, we developed a novel log-quadratic model, which approximates the biphasic and Weibull models and provides additional physiological interpretability. As a statistical linear model, the log-quadratic model is relatively simple to fit and straightforwardly provides confidence intervals for its fitted values. It allows a DT-like value to be derived, even from data that exhibit obvious "tailing." We also showed how existing models of non-log-linear microbial inactivation, such as the Weibull model, can fit into a statistical linear model framework that dramatically simplifies their solution. We applied the log-quadratic model to thermal inactivation data for the spore-forming bacterium Clostridium botulinum and evaluated its merits compared with those of popular previously described approaches. The log-quadratic model was used as the basis of a secondary model that can capture the dependence of microbial inactivation kinetics on temperature. This model, in turn, was linked to models of spore inactivation of Sapru et al. and Rodriguez et al. that posit different physiological states for spores within a population. We believe that the log-quadratic model provides a useful framework in which to test vitalistic and mechanistic hypotheses of inactivation by thermal and other processes. Copyright © 2009, American Society for Microbiology. All Rights Reserved.

Una secuencia didáctica para un concepto unificador en un curso de álgebra lineal de un programa de formación a la ingeniería

L’introduction aux concepts unificateurs dans l’enseignement des mathématiques privilégie typiquement l’approche axiomatique. Il n’est pas surprenant de constater qu’une telle approche tend à une algorithmisation des tâches pour augmenter l’efficacité de leur résolution et favoriser la transparence du nouveau concept enseigné (Chevallard, 1991). Cette réponse classique fait néanmoins oublier le rôle unificateur du concept et n’encourage pas à l’utilisation de sa puissance. Afin d’améliorer l’apprentissage d’un concept unificateur, ce travail de thèse étudie la pertinence d’une séquence didactique dans la formation d’ingénieurs centrée sur un concept unificateur de l’algèbre linéaire: la transformation linéaire (TL). La notion d’unification et la question du sens de la linéarité sont abordées à travers l’acquisition de compétences en résolution de problèmes. La séquence des problèmes à résoudre a pour objet le processus de construction d’un concept abstrait (la TL) sur un domaine déjà mathématisé, avec l’intention de dégager l’aspect unificateur de la notion formelle (Astolfi y Drouin, 1992). À partir de résultats de travaux en didactique des sciences et des mathématiques (Dupin 1995; Sfard 1991), nous élaborons des situations didactiques sur la base d’éléments de modélisation, en cherchant à articuler deux façons de concevoir l’objet (« procédurale » et « structurale ») de façon à trouver une stratégie de résolution plus sûre, plus économique et réutilisable. En particulier, nous avons cherché à situer la notion dans différents domaines mathématiques où elle est applicable : arithmétique, géométrique, algébrique et analytique. La séquence vise à développer des liens entre différents cadres mathématiques, et entre différentes représentations de la TL dans les différents registres mathématiques, en s’inspirant notamment dans cette démarche du développement historique de la notion. De plus, la séquence didactique vise à maintenir un équilibre entre le côté applicable des tâches à la pratique professionnelle visée, et le côté théorique propice à la structuration des concepts. L’étude a été conduite avec des étudiants chiliens en formation au génie, dans le premier cours d’algèbre linéaire. Nous avons mené une analyse a priori détaillée afin de renforcer la robustesse de la séquence et de préparer à l’analyse des données. Par l’analyse des réponses au questionnaire d’entrée, des productions des équipes et des commentaires reçus en entrevus, nous avons pu identifier les compétences mathématiques et les niveaux d’explicitation (Caron, 2004) mis à contribution dans l’utilisation de la TL. Les résultats obtenus montrent l’émergence du rôle unificateur de la TL, même chez ceux dont les habitudes en résolution de problèmes mathématiques sont marquées par une orientation procédurale, tant dans l’apprentissage que dans l’enseignement. La séquence didactique a montré son efficacité pour la construction progressive chez les étudiants de la notion de transformation linéaire (TL), avec le sens et les propriétés qui lui sont propres : la TL apparaît ainsi comme un moyen économique de résoudre des problèmes extérieurs à l’algèbre linéaire, ce qui permet aux étudiants d’en abstraire les propriétés sous-jacentes. Par ailleurs, nous avons pu observer que certains concepts enseignés auparavant peuvent agir comme obstacles à l’unification visée. Cela peut ramener les étudiants à leur point de départ, et le rôle de la TL se résume dans ces conditions à révéler des connaissances partielles, plutôt qu’à guider la résolution.

Integration of chaos theory and mathematical models in building simulation: Part II: Conceptual frameworks

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasing complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I), published in the previous issue, reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develop conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to (1) building simulation scientists and designers (2) initiating a dialogue between scientists and engineers, and (3) stimulating future research on a wide range of issues involved in designing and managing building environmental systems.

Integrated Bayesian network frameworks for modelling complex ecological issues

Ecological problems are typically multi faceted and need to be addressed from a scientific and a management perspective. There is a wealth of modelling and simulation software available, each designed to address a particular aspect of the issue of concern. Choosing the appropriate tool, making sense of the disparate outputs, and taking decisions when little or no empirical data is available, are everyday challenges facing the ecologist and environmental manager. Bayesian Networks provide a statistical modelling framework that enables analysis and integration of information in its own right as well as integration of a variety of models addressing different aspects of a common overall problem. There has been increased interest in the use of BNs to model environmental systems and issues of concern. However, the development of more sophisticated BNs, utilising dynamic and object oriented (OO) features, is still at the frontier of ecological research. Such features are particularly appealing in an ecological context, since the underlying facts are often spatial and temporal in nature. This thesis focuses on an integrated BN approach which facilitates OO modelling. Our research devises a new heuristic method, the Iterative Bayesian Network Development Cycle (IBNDC), for the development of BN models within a multi-field and multi-expert context. Expert elicitation is a popular method used to quantify BNs when data is sparse, but expert knowledge is abundant. The resulting BNs need to be substantiated and validated taking this uncertainty into account. Our research demonstrates the application of the IBNDC approach to support these aspects of BN modelling. The complex nature of environmental issues makes them ideal case studies for the proposed integrated approach to modelling. Moreover, they lend themselves to a series of integrated sub-networks describing different scientific components, combining scientific and management perspectives, or pooling similar contributions developed in different locations by different research groups. In southern Africa the two largest free-ranging cheetah (Acinonyx jubatus) populations are in Namibia and Botswana, where the majority of cheetahs are located outside protected areas. Consequently, cheetah conservation in these two countries is focussed primarily on the free-ranging populations as well as the mitigation of conflict between humans and cheetahs. In contrast, in neighbouring South Africa, the majority of cheetahs are found in fenced reserves. Nonetheless, conflict between humans and cheetahs remains an issue here. Conservation effort in South Africa is also focussed on managing the geographically isolated cheetah populations as one large meta-population. Relocation is one option among a suite of tools used to resolve human-cheetah conflict in southern Africa. Successfully relocating captured problem cheetahs, and maintaining a viable free-ranging cheetah population, are two environmental issues in cheetah conservation forming the first case study in this thesis. The second case study involves the initiation of blooms of Lyngbya majuscula, a blue-green algae, in Deception Bay, Australia. L. majuscula is a toxic algal bloom which has severe health, ecological and economic impacts on the community located in the vicinity of this algal bloom. Deception Bay is an important tourist destination with its proximity to Brisbane, Australia’s third largest city. Lyngbya is one of several algae considered to be a Harmful Algal Bloom (HAB). This group of algae includes other widespread blooms such as red tides. The occurrence of Lyngbya blooms is not a local phenomenon, but blooms of this toxic weed occur in coastal waters worldwide. With the increase in frequency and extent of these HAB blooms, it is important to gain a better understanding of the underlying factors contributing to the initiation and sustenance of these blooms. This knowledge will contribute to better management practices and the identification of those management actions which could prevent or diminish the severity of these blooms.

Computation of Mathematical Models for Complex Industrial Processes

Designed for undergraduate and postgraduate students, academic researchers and industrial practitioners, this book provides comprehensive case studies on numerical computing of industrial processes and step-by-step procedures for conducting industrial computing. It assumes minimal knowledge in numerical computing and computer programming, making it easy to read, understand and follow. Topics discussed include fundamentals of industrial computing, finite difference methods, the Wavelet-Collocation Method, the Wavelet-Galerkin Method, High Resolution Methods, and comparative studies of various methods. These are discussed using examples of carefully selected models from real processes of industrial significance. The step-by-step procedures in all these case studies can be easily applied to other industrial processes without a need for major changes and thus provide readers with useful frameworks for the applications of engineering computing in fundamental research problems and practical development scenarios.

Mathematical models for collective cell spreading

Collective cell spreading is frequently observed in development, tissue repair and disease progression. Mathematical modelling used in conjunction with experimental investigation can provide key insights into the mechanisms driving the spread of cell populations. In this study, we investigated how experimental and modelling frameworks can be used to identify several key features underlying collective cell spreading. In particular, we were able to independently quantify the roles of cell motility and cell proliferation in a spreading cell population, and investigate how these roles are influenced by factors such as the initial cell density, type of cell population and the assay geometry.

Integration of chaos theory and mathematical models in building simulation: Part I: Literature review

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasingly complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I) reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develops conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to building simulation scientists, initiates a dialogue and builds bridges between scientists and engineers, and stimulates future research about a wide range of issues on building environmental systems.

Hops, steps and jumps : mathematical progress in the early years

While curriculum frameworks ale major influences on learning, teachers know that children progress at different rates. Sometimes this is evident within a particular topic. and at other times more obvious across different topics. In this paper. we present the hops, steps, and jumps of numeracy learning of some 3000 Australian children. All were assessed using I Can Do Maths, and their achievements mapped to provide a detailed picture of how children hop, step and jump on their numeracy journey. This mapping provides teachers with infonnation about key hurdles to numeracy learning for Australian children.

Design and implementation of a mathematical videogame

This paper presents videogames as a very useful tool in high studies with respect to mathematical matters. It describes the implementation of a videogame developed by its authors which makes it possible for students to reinforce mathematical concepts in a motivating environment. With this work we intend to contribute to the process of engaging a bigger number of university teaching professionals and researchers in the use of serious games and the study of their theoretical frameworks, design, development and application of scientific education. With this idea the authors of the present paper have created and developed the videogame “The Math Castle” which consists in a series of tests through which various aspects of Mathematics are dealt with, especially in the areas of Discrete Mathematics, which due to its nature can be particularly well adapted to this kind of activity, Analysis or Geometry. In this paper there lies a complete description of the game developed and the results obtained with it.

Optimization and Machine Learning Frameworks for Complex Network Analysis

Thesis (Ph.D.)--University of Washington, 2016-06

Classification and analysis of integrating frameworks in multiscale modelling

Chemical engineers are turning to multiscale modelling to extend traditional modelling approaches into new application areas and to achieve higher levels of detail and accuracy. There is, however, little advice available on the best strategy to use in constructing a multiscale model. This paper presents a starting point for the systematic analysis of multiscale models by defining several integrating frameworks for linking models at different scales. It briefly explores how the nature of the information flow between the models at the different scales is influenced by the choice of framework, and presents some restrictions on model-framework compatibility. The concepts are illustrated with reference to the modelling of a catalytic packed bed reactor. (C) 2004 Elsevier Ltd. All rights reserved.

Mapping variation in children’s mathematical reasoning: the case of ‘what else belongs?’

Explaining appears to dominate primary teachers’ understanding of mathematical reasoning when it is not confused with problem solving. Drawing on previous literature of mathematical reasoning, we generate a view of the critical aspects of reasoning that may assist primary teachers when designing and enacting tasks to elicit and develop mathematical reasoning. The task used in this study of children’s reasoning is a number commonality problem. We analysed written and verbal samples of reasoning gathered from children in grades 3 and 4 from three primary schools in Australia and one elementary school in Canada to map the variation in their reasoning. We found that comparing and contrasting was a critical aspect of forming conjectures when generalising in this context, an action not specified in frameworks for generalising in early algebra. The variance in children’s reasoning elicited through this task also illuminated the difference between explaining and justifying.

Mathematical models for cellular aggregation: the chemotactic instability and clustering formation

In this thesis we present a mathematical formulation of the interaction between microorganisms such as bacteria or amoebae and chemicals, often produced by the organisms themselves. This interaction is called chemotaxis and leads to cellular aggregation. We derive some models to describe chemotaxis. The first is the pioneristic Keller-Segel parabolic-parabolic model and it is derived by two different frameworks: a macroscopic perspective and a microscopic perspective, in which we start with a stochastic differential equation and we perform a mean-field approximation. This parabolic model may be generalized by the introduction of a degenerate diffusion parameter, which depends on the density itself via a power law. Then we derive a model for chemotaxis based on Cattaneo's law of heat propagation with finite speed, which is a hyperbolic model. The last model proposed here is a hydrodynamic model, which takes into account the inertia of the system by a friction force. In the limit of strong friction, the model reduces to the parabolic model, whereas in the limit of weak friction, we recover a hyperbolic model. Finally, we analyze the instability condition, which is the condition that leads to aggregation, and we describe the different kinds of aggregates we may obtain: the parabolic models lead to clusters or peaks whereas the hyperbolic models lead to the formation of network patterns or filaments. Moreover, we discuss the analogy between bacterial colonies and self gravitating systems by comparing the chemotactic collapse and the gravitational collapse (Jeans instability).