Firstly, I proposed a very simple SIS/SIR model for a general vector-borne disease transmission considering constant population sizes over the season, where contact between the host and the vector responsible of the transmission is assumed to occur only during the summer of each year. I discussed two different types of threshold for pathogen persistence that I explicitly computed: a "short-term threshold" and a "long-term threshold". Later, I took into account the seasonality of the populations involved in the transmission. For a single season, the model consists of system of non linear differential equations considering the various stages of the infection transmission between the vector and the host population. Assuming the overwintering in the mosquito populations, I simulated the model for several years. Finally, I studied the spatial spread of a vector-borne disease throught an impusive reaction-diffusion model and I showed some simulations.

The works presented in this thesis are very different one from the other but they all deal with the mathematical modelling of emerging infectious diseases which, beyond being the leitmotiv of this thesis, is an important research area in the field of epidemiology and public health. A minor but significant part of the thesis has a theoretical flavour. This part is dedicated to the mathematical analysis of the competition model between two HIV subtypes in presence of vaccination and cross-immunity proposed by Porco and Blower (1998). We find the sharp conditions under which vaccination leads to the coexistence of the strains and using arguments from bifurcation theory, draw conclusions on the equilibria stability and find that a rather unusual behaviour of histeresis-type might emerge after repeated variations of the vaccination rate within a certain range. The most of this thesis has been inspired by real outbreaks occurred in Italy over the last 10 years and is about the modelling of the 1999-2000 H7N1 avian influenza outbreak and of the 2009-2010 H1N1 pandemic influenza. From an applied perspective, parameter estimation is a key part of the modelling process and in this thesis statistical inference has been performed within both a classical framework (i.e. by maximum likelihood and least square methods) and a Bayesian setting (i.e. by Markov Chain Monte Carlo techniques). However, my contribution goes beyond the application of inferential techniques to specific case studies. The stochastic, spatially explicit, between-farm transmission model developed for the transmission of the H7N1 virus has indeed been used to simulate different control strategies and asses their relative effectiveness. The modelling framework presented here for the H1N1 pandemic in Italy constitutes a novel approach that can be applied to a variety of different infections detected by surveillance system in many countries. We have coupled a deterministic compartmental model with a statistical description of the reporting process and have taken into account for the presence of stochasticity in the surveillance system. We thus tackled some statistical challenging issues (such as the estimation of the fraction of H1N1 cases reporting influenza-like-illness symptoms) that had not been addressed before. Last, we apply different estimation methods usually adopted in epidemiology to real and simulated school outbreaks, in the attempt to explore the suitability of a specific individual-based model at reproducing empirically observed epidemics in specific social contexts.

In this thesis we analyze the relationship between epidemiology and network theory, starting from the observation that the viral propagation between interacting agents is determined by intrinsic characteristics of the population contact network. We aim to investigate how a particular network structure can impact on the long-term behavior of epidemics. This field is way too large to be fully discussed; we limit ourselves to consider networks that are partitioned into local communities, in order to incorporate realistic contact structures into the model. The gross structure of hierarchical networks of this kind can be described by a quotient graph. The rationale of this approach is that individuals infect those belonging to the same community with higher probability than individuals in other communities. We describe the epidemic process as a continuous-time individual-based susceptible–infected–susceptible (SIS) model using a first-order mean-field approximation, both in homogeneous and in heterogeneous setting. For this mean-field model we show that the spectral radius of the smaller quotient graph, in connection with the infecting and curing rates, is related to the epidemic threshold, and it gives conditions in order to decide whether the overall healthy-state defines a globally asymptotically stable or an unstable equilibrium. Moreover we show that above the threshold another steady-state exists that can be computed using a lower-dimensional dynamical system associated with the evolution of the process on the quotient graph. Our investigations are based on the graph-theoretical notion of equitable partition and of its recent and rather flexible generalization, that of almost equitable partition. We also consider the important issue related to the control of the infectious disease. Taking into account the connectivity of the network, we provide a cost-optimal distribution of resources to prevent the disease from persisting indefinitely in the population; for a particular case of two-level immunization problem we report on the construction of a polynomial time complexity algorithm. In the second part of the thesis we include stochasticity in the model, considering the infection rates in the form of independent stochastic processes. This allows us to get stochastic differential equation for the probability of infection in each node. We report on the existence of the solution for all times. Moreover we show that there exist two regions, given in terms of the coefficients of the model, one where the system goes to extinction almost surely, and the other where it is stochastic permanent.

Quasi-reaction systems are often modelled with stochastic differential equations in order to capture the inherent randomness of their dynamics. The traditional local linear approximation methods for the estimation of the reaction rates face significant challenges in certain conditions. When the system is observed at short intervals, high correlations between successive observations result in numerical instability, while measurements at wider intervals lead to biased estimates of the parameters. This thesis addresses these issues with the development of novel inferential procedures. First, we introduce a latent event history model, by formalizing a latent space of unobserved reactions and their connection with the observed states. Under this framework, the system parameters can be estimated via a modified Expectation-Maximisation algorithm, with an extended Kalman filter at the E-step for reconstructing the underlying latent states. The approach is shown to be more accurate than existing methods, particularly for observations measured at short time intervals. In order to model more complex scenarios, the method is further extended to account for external time-varying factors that may affect the reaction rates of the system. As a second inferential approach, we concentrate on the mean of the dynamics and propose a novel mean-field approximation method. The method exploits the analytical solution of the ordinary differential equations in the case of unitary systems to propose an approximate solution for a generic quasi-reaction system. Besides a high computational efficiency, the resulting approach is found to perform particularly well when the measurements of the system are taken at wide time intervals. Finally, we provide an illustration of the proposed inferential procedures on the modelling of COVID-19 transmission and cell differentiation.

Made available in DSpace on 2016-04-27T16:57:53Z (GMT). No. of bitstreams: 1 dissertacao_vania_andrade_luz.pdf: 1642035 bytes, checksum: 179a6afec6f9633eb81f44f31826cdea (MD5) Previous issue date: 2007-04-09 / This dissertation consists of a study focused on the teaching of geometrical transformations based on the analysis of exercises suggested in didactical books published since the 1960s in São Paulo State. The realization of this work is supported by an analysis with the following objectives: to examine the completion of the local mathematical organizations according to the Anthropological Theory of Didactics, by Yves Chevallard; to observe how the students main difficulties were approached, based on research by Jaime and Gutiérrez (1996) and Jahn (1998); to compare the teaching proposals of isometries and homoteties current at the time of the movement of reformation of Modern Mathematics to those which followed the publishing of the Parâmetros Curriculares Nationais (Brazilian National Curricular Standards). Initially, we proceeded to a study of the mathematical organization relative to geometrical transformations, a study in which will be examined the symmetries of elements of nature and the arts, the definitions of the main transformations that are part of the program of Ensino Fundamental (equivalent to an eight-year elementary school), besides the identification of historical elements related to the development of Geometry. After that, we will be moving on to the study of the didactical organization, in which researches about the teaching and learning of isometries were verified, as well as official texts as Curricular Guides, Proposals and Standards, besides a few aspects of the movement of reformation of Modern Mathematics in Brazil, and its relationship with the same movement in the United States and France. The analysis of the books was carried out from the grouping of exercises in types of tasks, a qualification used as much for the identification of the resolution techniques, as for the distinction of the technology appropriate to the justification of the former. Following that, we have performed a study of the set of proposed exercises, in which the above-mentioned was complied with, in the second paragraph of this abstract, as objectives of the analysis that was developed. Departing from the adopted theoretical basis, the conclusions of the research indicate that the orientations observed in each of the analyzed periods would not guarantee, per se, that important problems in the teaching of transformations were resolved. As it is known, the period correspondent to the movement of reformation of Modern Mathematics has emphasized the mathematical structure, whereas the period that followed the publishing of the Parâmetros Curriculares highlighted the relationship of isometries and homoteties with superior levels of mathematical determination. In the first case, the visibility of the mathematical work was not highlighted, while in the second one, the absence of technological elements might have compromised the completion of the mathematical organization / Esta dissertação consiste em um estudo focado no ensino de transformações geométricas e organizado com base na análise de exercícios propostos em livros didáticos publicados, a partir dos anos 60, no Estado de São Paulo. A realização deste trabalho apóia-se em uma análise com os seguintes objetivos: examinar a completude das organizações matemáticas locais conforme a Teoria Antropológica do Didático, de Yves Chevallard; observar como foram abordadas as principais dificuldades dos alunos, com base nas pesquisas de Jaime e Gutiérrez (1996) e de Jahn (1998); comparar as propostas de ensino de isometrias e homotetias vigentes à época do movimento de reforma da Matemática Moderna com aquelas que se seguiram à publicação dos Parâmetros Curriculares Nacionais. Inicialmente, procedemos a um estudo da organização matemática relativa a transformações geométricas, estudo no qual são examinadas as simetrias presentes em elementos da natureza e das artes, as definições das principais transformações que fazem parte do programa do Ensino Fundamental, além da identificação de elementos históricos relacionados ao desenvolvimento da geometria. Em seguida, passamos ao estudo da organização didática, no qual se averiguaram pesquisas sobre ensino e aprendizagem de isometrias, textos oficiais como Guias, Proposta e Parâmetros Curriculares, além de alguns aspectos do movimento de reforma da Matemática Moderna no Brasil, e sua relação com o mesmo movimento nos Estados Unidos e na França. A análise dos livros foi feita a partir do agrupamento dos exercícios em tipos de tarefas, qualificação essa utilizada tanto para a identificação da técnica de resolução, como para a distinção da tecnologia apropriada à justificação da primeira. Em seguida, efetuamos um estudo do conjunto dos exercícios propostos, no qual se encaminhou o cumprimento do acima exposto, no segundo parágrafo deste resumo, como objetivos da análise desenvolvida. A partir da base teórica adotada, as conclusões da pesquisa indicam que as orientações observadas em cada um dos períodos analisados não garantiriam, por si, que importantes problemas no ensino de transformações fossem resolvidos. Como sabido, a época correspondente ao movimento de reforma da Matemática Moderna enfatizou a estrutura matemática, ao passo que o período que se seguiu à publicação dos Parâmetros Curriculares destacou a relação de isometrias e homotetias com níveis superiores de determinação matemática. No primeiro caso, a visibilidade da obra matemática não foi destacada, enquanto, no segundo, a ausência de elementos tecnológicos pode ter comprometido a completude da organização matemática

Made available in DSpace on 2016-04-27T16:57:54Z (GMT). No. of bitstreams: 1 dissertacao_diana_maia.pdf: 2211549 bytes, checksum: 0b656e7207483294a3bdc1ff6f9fa699 (MD5) Previous issue date: 2007-05-17 / This research aims to supplement the studies that have already been conducted in regard to teaching quadratic functions as well as the utilization of software that support this teaching. It also aims to consider graphs of quadratic functions by making use of the global interpretation of the properties of images, as well as by adding a playful dynamic to initiate students into the notion of interval and function domain. The study has its theorectical roots based on the principles of Didactic Engineering and Raymond Duval´s Theory of Semiotic Representation Registers and Guy Brousseau s Theory of Didactical Situations. The didactic studies presents some brief analyses of some primary and middle school level didactic books and builds upon Benedetti s research (2003) about the utilization of graph software as well as Duval s research (1998) about the articulation problems between the graphical and algebraic registers. The computer-based tool used for the application of this study was Winplot. The study was carried out with private school students undergoing the 8th grade in the City of São Bernardo do Campo in the state of São Paulo. Data from four pairs that took part in seven research sessions were reviewed. Results reveal that an improvement was obtained and better retention was measured amongst students in regards of the square function concept, which was achieved by the articulation between visual variables and meaningful symbolic units / Esta pesquisa tem por objetivo, complementar estudos já realizados a respeito do ensino da função quadrática e da utilização de software para este fim. Com o intuito de abordar a questão da construção gráfica da função quadrática utilizando o procedimento de interpretação global das propriedades figurais e, ainda, inserir uma dinâmica lúdica para introduzir as noções de intervalo e domínio da função. Fundamentada nos princípios da Engenharia Didática e embasada na Teoria dos Registros de Representação Semiótica de Raymond Duval e na Teoria das Situações de Guy Brousseau. A seqüência didática apresentada orienta-se nas análises de alguns livros didáticos do Ensino Fundamental e Médio, nas pesquisas de Benedetti (2003), sobre a utilização de software gráficos e sobre o trabalho de Duval (1988) sobre a articulação entre os registros gráfico e algébrico. A ferramenta computacional utilizada na aplicação da seqüência foi o software Winplot, além do uso de papel e lápis. A seqüência foi aplicada com alunos da oitava série do Ensino Fundamental de uma escola particular na cidade de São Bernardo do Campo no estado de São Paulo. Foram analisados os protocolos de quatro duplas, que participaram das sete sessões. Os resultados obtidos levam a concluir que houve um avanço por parte dos alunos, na apreensão do conceito de função quadrática, propiciado pela compreensão e articulação entre as variáveis visuais e unidades simbólicas significativas

Made available in DSpace on 2016-04-27T16:58:14Z (GMT). No. of bitstreams: 1 Maria Sueli Gomes Saldanha.pdf: 608041 bytes, checksum: 1bdc94c03284d28bbaa8e5048a8c11dd (MD5) Previous issue date: 2007-05-22 / The purpose of this research, done according to the qualitative approach, was to analyze the teaching-learning process, involving the teacher, the student and the mathematical knowledge in the solving of problems, focusing on inequalities and logarithmic inequations. In order to reach this goal, six didactic situations were elaborated, based on the dialectic notion tool-object of the researcher Régina Douady. These same notions were used in the data analysis. The target-public was constituted by students and a teacher-researcher from the 10th grade of a private school in the city of São Paulo. The main instruments used in the data collection were recordings on tape of the dialogues among the teacher-researcher and the group of students, dialogues among teacher-researcher and classmatesgroup and the written work of the students in the solving of the proposed problems. The conclusion obtained was the approach developed by the proposed didactic sequence that favored the learning of inequalities and logarithmic inequations and, also, made possible to reflect about the teaching practice of the teacherresearcher / Esta pesquisa realizada na abordagem qualitativa, teve como objetivo analisar o processo ensino-aprendizagem, envolvendo professor, aluno e saber matemático na resolução de problemas, enfocando as desigualdades e inequações logarítmicas. Para atingir este objetivo, foram elaboradas seis situações didáticas, com base na noção dialética-ferramenta-objeto da pesquisadora francesa Régine Douady. Essas mesmas noções foram utilizadas nas análises dos dados. O público-alvo constitui-se de alunos e professora-pesquisadora de uma classe da 2ª série do Ensino Médio de uma escola particular de Vila Nova Conceição da cidade de São Paulo. Os principais instrumentos utilizados na coleta dos dados foram gravações em fita cassete dos diálogos entre professora-pesquisadora e o grupo de alunos, professora-pesquisadora e o grupo sala e produções escritas dos alunos na resolução dos problemas propostos. A conclusão obtida foi que a abordagem desenvolvida pela seqüência didática proposta favoreceu a aprendizagem das desigualdades e inequações logarítmicas e, também, oportunizou uma reflexão sobre a prática de ensino da professora-pesquisadora

Made available in DSpace on 2016-04-27T16:58:14Z (GMT). No. of bitstreams: 1 Ricardo Cardoso.pdf: 519569 bytes, checksum: 93a997cb00e05e91088b9b8f83a07660 (MD5) Previous issue date: 2007-05-24 / Secretaria da Educação do Estado de São Paulo / Statistics has stand out lately due to its utility in almost every area of human knowledge. Existing research and essay on the subject suggest the need for deepening knowledge on the difficulties with teaching process of this discipline. Our main issue is to investigate whether Brazilian public school teachers develop the Descriptive Statistics in high school and whether they are able to use, efficiently, the basic statistics concepts Data Organization, Measures of Central Tendency, Separatrixes and Dispersion to solve daily practical problems. The aim of this research is to verify the level of knowledge mobilization by high-school teachers. According to LINS (2004, p.54) we know that the general impression not registered systematically by researches that the licenciate s mathematical formation, in great part similar to the future graduate, does not contribute substantially to the formation of this future professional, unless to reinforce the expositive classes routines . We will try to diagnose in which level of knowledge the high-school teacher is in school curriculum of Statistics. Based on BIFI s questionnaire (2006, p.54), we will verify whether the high-school Mathematics teacher is able to calculate, justify and relate the described measures. The answers of the activities will be analyzed through C.H.I.C. software / A Estatística tem se destacado ultimamente por sua utilidade em praticamente todas as áreas do conhecimento humano. Pesquisas e dissertações existentes sobre o assunto sugerem a necessidade de aprofundar os conhecimentos sobre as dificuldades no processo de ensino da disciplina. Nossa questão principal é investigar se os Professores da Rede Pública desenvolvem o ensino da Estatística Descritiva no ensino médio, e se estão aptos a utilizar, de forma eficaz, as noções estatísticas de base Organização de dados, Medidas de Tendência Central , Separatrizes e Dispersão, para resolver problemas práticos de seu cotidiano. O intuito dessa pesquisa é verificar o nível de mobilização dos conhecimentos por parte dos Professores do Ensino Médio. Segundo LINS (2004, p.54) sabemos que persiste a impressão geral não documentada de forma sistemática por pesquisas de que a formação matemática do licenciado, em boa parte similar a do futuro bacharel, não contribui de modo substancial para a formação daquele futuro profissional, a não ser ao reforçar as rotinas de aulas expositivas . Tentaremos diagnosticar qual o nível de conhecimento em que o Professor do Ensino Médio se encontra no conteúdo curricular de Estatística. Baseando-se no questionário de BIFI (2006, p. 54), verificar se o professor de matemática do ensino médio é capaz de calcular, justificar e relacionar as medidas descritas. As respostas das atividades serão analisadas com o auxílio do software C.H.I.C

Made available in DSpace on 2016-04-27T17:13:05Z (GMT). No. of bitstreams: 1 GIVANILDO FARIAS DA SILVA.pdf: 3620616 bytes, checksum: 2b513f4eebb2312f779ebbe02a0a6c12 (MD5) Previous issue date: 2008-11-28 / Made available in DSpace on 2016-08-25T17:25:37Z (GMT). No. of bitstreams: 2 GIVANILDO FARIAS DA SILVA.pdf.jpg: 2104 bytes, checksum: c4715912a635b5fbde63d2a9b070733f (MD5) GIVANILDO FARIAS DA SILVA.pdf: 3620616 bytes, checksum: 2b513f4eebb2312f779ebbe02a0a6c12 (MD5) Previous issue date: 2008-11-28 / Secretaria da Educação do Estado de São Paulo / The target of this work is investigating the change s process that School Mathematic for Secondary School, nowadays High School, during decades of 1950 and 1960. This target was gained on through legislation on the time, as Programas do Ensino Secundário, laid out by Portaria 966 on the 2nd of October 1951, the Law 4024/61 that created the Diretrizes e Bases da Educação Nacional besides Sugestões para um roteiro de Programa para a cadeira de Matemática proposed by GEEM in 1965. It had used set of didactic books Matemática Curso Colegial Moderno, by Scipione Di Pierro Neto, Luiz Mauro Rocha and Ruy Madsen Barbosa, published on time of Modern Mathematic s Movement that have been considered at first hand publishing in this period to High School. To substantiate these analyses, it was used the contribution by André Chervel (1990) with the History of School Subjects and Alain Choppin (2004) he have searched and to reflect about didactic books and he have consided these books a complex production. Like this, it is possible to observe the changes on the introduction of the subjects that it would be taught on High School ever since Portaria de 51 even Sugestões by GEEM, in 1965, besides assimilation these Sugestões on set Matemática Curso Colegial Moderno. However, the main change had been a new language for content s introduction begining from Teory of Conjunct, from Logical and from Algebraical Structures, besides incorporation of news contents, as Matrix, Transformation s Geometry, Probability and an analyse by elementary and trigonometricals Function and Analitic Geometry / O objetivo deste trabalho é investigar os processos de transformação que a Matemática Escolar, ensinada no Ensino Colegial, atual Ensino Médio, sofreu nas décadas de 1950 a 1960. Para alcançar tal objetivo foram utilizadas nessa pesquisa as legislações vigentes na época, como os Programas do Ensino Secundário, dispostos pela Portaria n° 966 de 2 de outubro de 1951, a Lei n° 4024/61 que instituiu as Diretrizes e Bases da Educação Nacional, além das Sugestões para um roteiro de Programa para a cadeira de Matemática, propostos pelo GEEM em 1965. Utilizou-se também a Coleção de livros didáticos Matemática Curso Colegial Moderno, de Scipione Di Pierro Neto, Luiz Mauro Rocha e Ruy Madsen Barbosa, publicada em tempos do Movimento da Matemática Moderna, considerada uma das primeiras publicações nesse período para o Ensino Colegial. Para fundamentar as análises foram utilizadas as contribuições de André Chervel (1990), com a História das Disciplinas escolares, e de Alain Choppin (2004), que procurou fazer uma reflexão sobre os livros didáticos ao considerá-los produções complexas. Com isso, podem-se observar as mudanças nas apresentações dos conteúdos ensinados no Colegial a partir da Portaria de 51 passando pelas Sugestões do GEEM (1965), assim como as assimilações dessas Sugestões na Coleção Matemática-Curso Colegial Moderno. A principal mudança, no entanto, se deu por meio de uma nova linguagem na apresentação dos conteúdos, a partir da Teoria dos Conjuntos, da Lógica e das Estruturas Algébricas, além da inclusão de novos conteúdos, tais como Matrizes, Geometria das Transformações, Probabilidade e um estudo aprofundado das Funções elementares e trigonométricas e da Geometria Analítica

Made available in DSpace on 2016-04-27T16:58:19Z (GMT). No. of bitstreams: 1 alexandre notari.pdf: 286199 bytes, checksum: 591269af2d5afccbd3071b0fac3cc7e1 (MD5) Previous issue date: 2002-10-30 / This research has as objective to get a systematic diagnosis of the main errors and difficulties revealed for pupils of Basic Ensino and Average Ensino in the simplification of arithmetical and algebraic fractions. They had been selected, for the inquiry, one 8ª series of Basic Ensino and one 1ª series of Average Ensino of two public schools of the Region West, of the city of São Paulo. We assume as referencial theoretician the happened ones of research where Algebra, in these levels of education, must search its beddings in the laws of the Arithmetic. However, it does not reduce it to a generalization of these laws, a time that algebraic the conceptual system incorporates elements and Inter-relations with proper characteristics. The data had been gotten by means of period of training in a classroom of each one of the mentioned series, application of a test and interviews. The main conclusions point with respect to one high number of errors in the simplification of algebraic fractions that disclose a incompreensão of the formal rules that regulate these transformations. Between these, a predominance of errors due to a generalization of rules of a situation for another one occurs, without an analysis of the conditions that validate this generalization. They disclose in the treatment of the arithmetical expressions a predominance of automatically carried through computational procedures, without a reflection on the nature of the task proposal; they indicate, still, an absence of integration between the arithmetical and algebraic conceptual domínios / Esta pesquisa tem como objetivo obter um diagnóstico sistemático dos principais erros e dificuldades manifestados por alunos do Ensino Fundamental e do Ensino Médio na simplificação de frações aritméticas e algébricas. Foram selecionadas, para a investigação, uma 8ª série do Ensino Fundamental e uma 1ª série do Ensino Médio de duas escolas públicas da Região Oeste, da cidade de São Paulo. Assumimos como referencial teórico os advindos de pesquisas em que a Álgebra, nesses níveis de ensino, deve buscar seus fundamentos nas leis da Aritmética. Entretanto, não o reduz a uma generalização dessas leis, uma vez que o sistema conceitual algébrico incorpora elementos e inter-relações com características próprias. Os dados foram obtidos por meio de estágio em uma classe de cada uma das séries mencionadas, de aplicação de uma prova e de entrevistas. As principais conclusões apontam para um elevado número de erros na simplificação de frações algébricas que revelam uma incompreensão das regras formais que regulamentam essas transformações. Entre esses, ocorre uma predominância de erros devidos a uma generalização de regras de uma situação para outra, sem uma análise das condições que validam essa generalização. Revelam no tratamento das expressões aritméticas um predomínio de procedimentos computacionais realizados automaticamente, sem uma reflexão sobre a natureza da tarefa proposta; indicam, ainda, uma ausência de integração entre os domínios conceituais aritméticos e algébricos