A mathematical model for malaria transmission relating global warming and local socioeconomic conditions

OBJECTIVE: Sensitivity analysis was applied to a mathematical model describing malaria transmission relating global warming and local socioeconomic conditions. METHODS: A previous compartment model was proposed to describe the overall transmission of malaria. This model was built up on several parameters and the prevalence of malaria in a community was characterized by the values assigned to them. To assess the control efforts, the model parameters can vary on broad intervals. RESULTS: By performing the sensitivity analysis on equilibrium points, which represent the level of malaria infection in a community, the different possible scenarios are obtained when the parameters are changed. CONCLUSIONS: Depending on malaria risk, the efforts to control its transmission can be guided by a subset of parameters used in the mathematical model.

Fractional electrical impedances in botanical elements

Fractional calculus (FC) is no longer considered solely from a mathematical viewpoint, and is now applied in many emerging scientific areas, such as electricity, magnetism, mechanics, fluid dynamics, and medicine. In the field of dynamical systems, significant work has been carried out proving the importance of fractional order mathematical models. This article studies the electrical impedance of vegetables and fruits from a FC perspective. From this line of thought, several experiments are developed for measuring the impedance of botanical elements. The results are analyzed using Bode and polar diagrams, which lead to electrical circuit models revealing fractional-order behaviour.

Mathematical model for HIV dynamics in HIV-specific helper cells

In this paper we study a delay mathematical model for the dynamics of HIV in HIV-specific CD4 + T helper cells. We modify the model presented by Roy and Wodarz in 2012, where the HIV dynamics is studied, considering a single CD4 + T cell population. Non-specific helper cells are included as alternative target cell population, to account for macrophages and dendritic cells. In this paper, we include two types of delay: (1) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and; (2) virus production period for new virions to be produced within and released from the infected cells. We compute the reproduction number of the model, R0, and the local stability of the disease free equilibrium and of the endemic equilibrium. We find that for values of R0<1, the model approaches asymptotically the disease free equilibrium. For values of R0>1, the model approximates asymptotically the endemic equilibrium. We observe numerically the phenomenon of backward bifurcation for values of R0⪅1. This statement will be proved in future work. We also vary the values of the latent period and the production period of infected cells and free virus. We conclude that increasing these values translates in a decrease of the reproduction number. Thus, a good strategy to control the HIV virus should focus on drugs to prolong the latent period and/or slow down the virus production. These results suggest that the model is mathematically and epidemiologically well-posed.

A regulação endócrina dos balanços de cálcio

RESUMO: As concentrações circulantes de cálcio são notavelmente constantes a despeito das variações diárias na absorção intestinal e na eliminação renal deste elemento. A regulação da calcémia é um sistema complexo que compreende vários factores controladores (a calcémia, a fosforémia, as concentrações circulantes de paratormona (PTH) e calcitriol além de muitos outros factores como hormonas esteróides em geral, outros iões como o magnésio e outros factores hormonais) e vários órgãos alvo (glândulas paratiroideias, osso, rim e intestino). As respostas dos órgãos alvo também são muito variadas. No caso mais simples, a cristalização de sais de cálcio corresponde a uma mudança de fase em que participam moléculas orgânicas que a iniciam, aceleram ou inibem. Em geral a combinação de um factor controlador com o respectivo receptor de membrana (para polipeptídeos ou iões) ou intracelular (hormonas esteróides) é apenas o primeiro passo de uma cadeia bioquímica que introduz uma enorme amplificação na resposta. A esta variedade de mecanismos de resposta correspondem grandes diferenças nos tempos de resposta que podem ser de minutos a semanas. É hoje possível “observar” (medir) com apreciável rigor nos líquidos biológicos (sangue, urina, fezes, etc.) os factores mais importantes do sistema de regulação da calcémia (cálcio, fósforo, paratormona e calcitriol) assim como administrar estes factores em experiências agudas. Esta possibilidade reflecte – se na literatura neste campo que tem vindo a crescer. O advento das técnicas da biologia molecular tem permitido a caracterização molecular de algumas das disfunções da homeostase do cálcio e é de esperar um diagnóstico fisiopatológico cada vez mais rigoroso dessas disfunções. Com o avanço dos conhecimentos nesta área que não cessa de aumentar temos cada vez maiores capacidades para fazer diagnósticos e é cada vez mais difícil interpretar com rigor os correspondentes quadros metabólicos. A análise ou síntese de sistemas complexos é a actividade mais nobre dos engenheiros que lhes permite desenhar pontes, diques, barcos, aviões ou automóveis. Com o aparecimento de computadores de médio ou grande porte foi – lhes possível utilizar descrições matemáticas não só para desenhar sistemas como ainda para interpretar eventuais falhas na sua operação. Essas descrições matemáticas consistem numa sequência de operações realizadas num computador segundo um “programa informático” que receberam a designação genérica de modelos, por analogia com as famosas leis (equações) da física que foram deduzidas a partir de um certo número de postulados e que permitem representar matematicamente processos físicos. As famosas leis de Newton são talvez os exemplos mais famosos de “modelos” de sistemas físicos. A introdução de modelos matemáticos em biologia e particularmente em medicina só se deu recentemente.MÉTODOS No trabalho que aqui se apresenta construiu - se um modelo simplificado da homeostase do cálcio destinado ao cálculo de variáveis observáveis (concentrações de cálcio, fósforo, PTH e calcitriol) de modo a poderem comparar-se valores calculados com valores observados. A escolha dos componentes do modelo foi determinada pela nossa experiência clínica e pela informação fisiopatológica e clínica publicada. Houve a preocupação de construir o modelo de forma modular de modo a ser possível a sua expansão sem grandes transformações na descrição matemática (e informática) já existente. Na sua fase actual o modelo não pode ser usado como instrumento de diagnóstico. É antes uma ferramenta destinada a esclarecer “em princípio” mecanismos fisiopatológicos. Usou – se o modelo para simular um certo número de observações publicadas e para exemplificar a sua eventual aplicação clínica na simulação de situações hipotéticas e na análise de possíveis mecanismos fisiopatológicos responsáveis por situações de hipo ou hipercalcémias. Simultaneamente fez – se uma análise dos dados acumulados relativos a doentes vistos no Serviço de Endocrinologia do Instituto Português de Oncologia de Francisco Gentil – Centro Regional Oncológico de Lisboa, S.A. CONCLUSÕES Numa população de 894 doentes com patologias variadas do Instituto Português de Oncologia de Lisboa os valores da calcémia tiveram uma distribuição normal unimodal com uma média de 9.56 mg/dl, e um erro padrão de 0.41 mg/dl. Estas observações sugerem que a calcémia está sujeita a regulação. A partir dos resultados publicados em que o metabolismo do cálcio foi perturbado por infusões de cálcio, calcitriol ou PTH, de estudos bioquímicos e fisiológicos sobre os mecanismos de acção de factores controladores da calcémia e do estudo do comportamento de órgãos alvo (paratiroideias, intestino, osso e rim) foi possível construir um modelo matemático de parâmetros concentrados do sistema de regulação da calcémia. As expressões analíticas usadas foram baseadas na cinética enzimática de modo a que os seus parâmetros tivessem um significado físico ou fisiológico simples. O modelo revelou apreciável robustez e flexibilidade. É estável quando não perturbado e transita entre estados estacionários quando perturbado. Na sua forma actual gera simulações que reproduzem satisfatoriamente um número apreciável de dados experimentais colhidos em doentes. Isto não significa que possa ser usado como instrumento de diagnóstico aplicável a doentes individuais. O desenho do modelo comporta a adição posterior de novas relações quando surgirem situações para as quais se revele insuficiente. A utilização exaustiva do modelo permitiu explicitar aspectos do metabolismo do cálcio que ou não estão contidas na sua formulação actual – o aparecimento de hipertrofia ou de adenomas das paratiroideias e as alterações na estrutura óssea , a participação de outros factores controladores – magnésio, ou estão insuficientemente descritas – alterações do metabolismo do fósforo nos hipoparatiroidismos. A análise dos dados relativos aos doentes do Serviço de Endocrinologia do IPO permitiu o início da caracterização dos tipos de patologia que representam e de possíveis mecanismos fisiopatológicos subjacentes. Estas observações são o ponto de partida para análises futuras. São exemplos das relações encontradas: a distribuição dos doentes por dois grandes grupos conforme a calcémia é determinada pelas concentrações circulantes de PTH ou estas são determinadas pela calcémia; a distribuição sazonal das concentrações de Vit. D25. no sangue; a correlação negativa entre estas e as concentrações de PTH no sangue. Também foi possível extrair a cinética do controlo da PTH sobre a síntese de calcitriol. O estudo dos níveis circulantes de PTH no pós-operatório imediato de doentes paratiroidectomizados permitiu determinar as suas taxas de degradação metabólica. O modelo permitiu simular as relações Ca/PTH no sangue, Ca/Fracção excretada da carga tubular, Ca/P no sangue para valores normais ou altos de Ca. Foram feitas simulações de situações fisiopatológicas (em “doentes virtuais”): infusões crónicas de cálcio, PTH e calcitriol; alterações no comportamento de receptores. Estas simulações correspondem a experiências que não podem ser realizadas em humanos. São exemplos da utilização do modelo na exploração de possíveis mecanismos fisiopatológicos através da observação de resultados quantitativos inacessíveis à intuição. O modelo foi útil em duas fases do trabalho: Primeiro, durante a sua síntese implicou uma escolha criticamente selectiva de informação, sua análise quantitativa e processamento, uma explicitação rigorosa (analítica) das relações funcionais entre os controladores e as variáveis e da sua integração numa estrutura global; Segundo, a simulação de situações experimentais ou clínicas (dados do Serviço de Endocrinologia do IPO) em doentes obrigou a explicitar raciocínios fisiopatológicos habitualmente formulados em bases puramente intuitivas. Esta prática revelou comportamentos óbvios após as simulações – acção reduzida das infusões PTH (simulação de hiperparatiroidismos primários) enquanto não há inibição total da respectiva secreção, necessidade de aumento da massa secretora da paratiroideia nas insuficiências renais avançadas, etc. A síntese e utilização do modelo não implicaram uma preparação matemática avançada e foram possíveis mercê da disponibilidade de “software” interactivo especificamente desenhado para a simulação de sistemas dinâmicos em que os programas se escrevem em inglês usando a simbologia simples da álgebra elementar. A função nobre de modelos desta natureza é semelhante à dos modelos usados pelos físicos desde o século XVII: permitir explicações de carácter geral funcionando como uma ferramenta intelectual para manipulação de conceitos e para a realização de “experiências pensadas” (“thought experiments”) respeitando certos princípios físicos (princípios de conservação) que estabelecem as fronteiras da realidade. -------ABSTRACT: Calcium blood levels are remarkably constant despite great variations in calcium daily intake, intestinal absorption and renal excretion. The regulation of the calcium concentration in the blood is achieved by a complex system that includes several controller factors (mainly the serum levels of calcium, phosphorus, parathyroid hormone (PTH) and calcitriol but also of steroid hormones, ions such as magnesium and other hormonal factors) and several target organs (parathyroid glands, bone, kidney and intestine). The functional response to the controlling factors obeys a variety of kinetics. The precipitation of calcium salts is a simple phase transition in which organic molecules may provide nucleation centres or inhibit the process. The combination of a controller factor with its receptor located in the cell membrane (for peptides or ions) or in the nucleus (for steroid hormones) is only the first step of a biochemical chain that introduces a huge amplification in the response. To this great variability of response we have to add the times of response that vary from minutes to weeks. It is possible to “observe” (measure) with great accuracy in biological fluids (blood, urine, faeces, etc.) the most important factors intervening in the calcium regulation (calcium, phosphorus, PTH and calcitriol). The response of the system to acute infusions of the controlling factors has also been studied. Using molecular biology techniques it has been possible to characterize some calcium homeostasis dysfunctions and better physiopathological diagnosis are expected. With the increasingly new knowledge in this area we have better capacity to diagnose but it is harder to explain correctly the underlying metabolic mechanisms. The analysis or synthesis of complex systems is the noble activity of engineers that enables them to draw bridges, dams, boats, airplanes or cars. With the availability of medium-large frame computers it was possible to use mathematical descriptions not only to draw systems but also to explain flaws in its operations. These mathematical descriptions are generally known as models by analogy with the laws (equations) of physics that allow the mathematical description of physical processes. In practice it is not possible to find general solutions for the mathematical descriptions of complex systems but (numeric) computations for specific situations can be obtained with digital computers. The introduction of mathematical models in biology and particularly in medicine is a recent event. METHODS In this thesis a simplified model of calcium homeostasis was built that enables the computation of observable variables (concentrations of calcium, phosphorus, PTH and calcitriol) and allows the comparison between the simulation values and observed values. The choice of the model’s components was made according to our clinical experience and to the published clinical and physiopathological data. The model has a modular design that allows future expansions with minor alterations in its structure. In its present form the model cannot be used for diagnosis. It is a tool designed to enlighten physiopathological processes. To exemplify its possible clinical application in the simulation of hypothetical situations and in the analysis of possible mechanisms responsible for hypo or hypercalcemias the model was used to simulate a certain number of published observations. An analysis of clinical and laboratory data from the Endocrinology Department of the Portuguese Cancer Institute (I.P.O.F.G.-C.R.O.L.,S.A.) is also presented. CONCLUSIONS In a population of 188 patients without an identifiable disease of the calcium metabolism at the Portuguese Cancer Institute the calcemia levels had a unimodal distribution with an average of 9.56 mg/dL and a S.E.M of 0.41 mg/dL. This observation confirms that serum calcium is regulated. Using published data; in which calcium metabolism was disrupted by calcium, PTH or calcitriol infusions; from biochemical and physiological studies of the action of controller factors on the calcemia; in which the response of target organs (parathyroid glands, intestine, bone, kidney) was studied it was possible to build a mathematical model of concentrated parameters of the calcium homeostasis. Analytical expressions used were based on enzymatic kinetics. The model is flexible and robust. It is stable when not disturbed and changes between steady states when disturbed. In its present form it provides simulations that reproduce closely a number of experimental clinical data. This does not mean that it can be used as a diagnostic tool for individual patients. The exhaustive utilisation of the model revealed the need of future expansions to include aspects of the calcium metabolism not included in its present form –hypertrophy or adenomas of the parathyroid glands, bone structure changes, participation of other controller factors such as magnesium – or insufficiently described – phosphate metabolism in hypoparathyroidism. The analysis of the data collected from the I.P.O.’s Endocrinology Department allowed the initial characterization of the different pathologies represented and of their possible physiopathological mechanisms. These observations are a starting point for future analysis. As examples of the relations found were: the distribution of patients in two groups according to the dependency of calcium by PTH levels or PTH levels by calcium concentration; the seasonal distribution of the serum concentrations of D25; its negative correlation with PTH concentration. It was also possible to extract the kinetics of the control of the synthesis of calcitriol by PTH. The analysis of immediate post-surgical levels of PTH in parathyroidectomized patients allowed the determination of its metabolic clearance. The model also allowed the simulation of the relations between Ca/PTH in blood, serum Ca/Fraction of tubular load excreted and Ca/P in blood for normal and high values of calcium. Simulations were made of pathological situations (in “virtual patients”): chronic infusions of calcium, PTH and calcitriol; changes in the characteristics of receptors. These simulations are not possible in real persons. They are an example of the use of this model in exploring possible mechanisms of disease through the observation of quantitative results not accessible to simple intuition. This model was useful in two phases: Firstly, its construction required a careful choice of data, its quantitative analysis and processing, an analytical description of the relations between controller factors and variables and their integration in a global structure. Secondly, the simulation of experimental or clinical (I.P.O.’s Endocrinology Department) data implied testing physiopathological explanations that previously were based on intuition. The construction and utilisation of the model didn’t demand an advanced mathematical preparation since user-friendly interactive software was used. This software was specifically designed for the simulation of dynamic systems. The programs are written in English using elementary algebra symbols. The essential function of this type of models is identical to that of those used by physicists since the XVII century which describe quantitatively natural processes and are an intellectual tool for the manipulation of concepts and the performance of “thought experiments” based in certain physical principles (conservation principles) that are the frontiers of reality.------------------RESUMÉE: Les concentrations circulantes de calcium sont constantes même pendant des variations de l’absorption intestinale et de l’élimination rénale de cet élément. La régulation de la calcémie est un système complexe qui comprend plusieurs éléments contrôleurs (la calcémie, la phosphorémie, les concentrations circulantes de l’hormone parathyroïdienne (PTH) e du calcitriol et d’autres comme les hormones stéroïdes ou des ions comme le magnésium) et plusieurs organes (glandes parathyroïdiennes, l’os, le rein et l’intestin). Les réponses de ces organes sont variées. Dans le cas plus simple, la cristallisation des sels de calcium correspond à un changement de phase dans lequel y participent des molécules organiques que la débutent, l’accélèrent ou l’inhibent. Généralement la combinaison d’un élément contrôleur avec leur récepteur de membrane (pour les peptides ou les ions) ou intracellulaire (pour les hormones stéroïdes) n’est que le premier pas d’une chaîne biochimique qu’introduit une grande amplification de la réponse. A cette variété de réponses correspondent des grandes différences des temps de réponses qu’y vont des minuits a semaines. Il est possible « observer » (mesurer) dans les fluides biologiques (sang, urine, fèces, etc.) les éléments plus importants du système de régulation de la calcémie (calcium, phosphate, PTH et le calcitriol) et les administrer en expérimentes aigus. Cette possibilité est visible dans la littérature publiée dans ce domaine qui est en croissance permanente. L’avenir des techniques de biologie moléculaire a permis caractériser des nombreuses dysfonctions de la régulation de la calcémie et on attend un diagnostique physiopathologique de ces dysfonctions chaque fois plus rigoureuses. Les connaissances dans ce domaine s’agrandissent et on a de plus de capacités pour faire des diagnostiques et il est chaque fois plus difficile les interpréter. L’analyse ou synthèse de systèmes complexes est l’activité plus noble des ingénieurs qui les permit dessiner des ponts, bateaux, avions ou automobiles. Avec des ordinateurs de médium ou grand port il les est possible utiliser descriptions mathématiques pour dessiner les systèmes et interpréter des éventuelles fautes d’opération. Ces descriptions mathématiques sont une séquence d’opérations réalisées dans un ordinateur selon « un programme informatique » qui ont reçu la désignation générique de modèles, pour analogie avec les équations de la physique qui ont été déduits d’un nombre de postulées et qu’ont permit représenter des processus physiques en équations mathématiques. Les fameuses équations de Newton sont peut-être les exemples plus connus des systèmes physiques. L’introduction des modèles mathématiques en biologie et en particulier en médecine est un évènement récent. Dans ce travaille, on a construit un modèle simplifié de l’homéostasie du calcium pour calculer les variables observables (concentrations de calcium, phosphate, PTH et calcitriol) pour les comparer. Les choix des components a été déterminés par notre expérience clinique et par l’information physiopathologique et clinique publiée. Le modèle a été construit de façon modulaire ce que permit leur postérieur expansion sans des grandes altérations dans la description mathématique et informatique déjà existante. Dans cette forme le modèle ne peut être utilisé comme un instrument de diagnostique. Il est un outil pour éclairer la physiopathologie. Le modèle a été utilisé pour simuler un certain nombre d’observations publiées et pour exemplifier leur possible utilisation clinique dans la simulation des hypothèses et de la physiopathologie des situations d’hypo ou hypercalcémie. On a fait une analyse des éléments des procès cliniques des malades observées dans le Service d’Endocrinologie de l’IPOFG-CROL, SA. Dans une population de 894 malades avec des différentes pathologies les valeurs de calcémie on une distribution uni modale avec une Médie de 9.56 mg/dL et une erreur standard de 0.41 mg/dL. Ces observations suggèrent que la calcémie soit sujette de régulation. En utilisant des résultats de travaux publiés dans lesquels le métabolisme du calcium a été changé par des infusions de calcium, calcitriol ou PTH, des études biochimiques et physiologiques sur des mécanismes d’action des éléments contrôleurs de la calcémie et de l’étude du comportement des organes cible (parathyroïdes, intestin, rein, os), il a été possible de construire un modèle mathématique de paramètres concentrés du système de régulation de la calcémie. Les expressions analytiques utilisées ont été basées sur la cinétique enzymatique de façon à que les paramètres aient eu une signification physique ou biologique. Le modèle est stable quand il n’est pas perturbé et transit entre états stationnaires quand il est sujet a des perturbations. A ce moment il fait des simulations qui reproduisent de façon satisfaisant un nombre d’observations expérimentales. La construction du modèle permit l’addiction de nouvelles relations dans les cas ou il est insuffisant. L’utilisation exhaustive du modèle a permit expliciter des aspects du métabolisme du calcium qui y ne sont pas compris – l’hyperplasie ou la formation des adénomes des parathyroïdes, les altérations de la structure des os, la participation d’outres éléments régulateurs (magnésium), ou sont insuffisamment décrites – les altérations du métabolisme des phosphates dans l’hypoparathyroidism. L’analyse de l’information des malades du Service d’Endocrinologie a permit caractériser les pathologies représentées et leurs possibles mécanismes physiopathologiques. Ces observations sont le point de départ pour les analyses futures. Sont des exemples des relations trouvées: la distribution des malades par deux groupes: ceux dans lequel la calcémie est déterminée par la PTH ou ceux dans lesquels la PTH est déterminée par la calcémie; la distribution sazonale de la concentration de la vitamine D; la corrélation négative entre la vitamine D et la PTH. On a eu la possibilité de déduire la cinétique de control de la PTH sur la synthèse du calcitriol. L’étude des niveaux circulants de PTH sur des sujets parathyroidectomisées a permit déduire leur taux de dégradation métabolique. Le modèle a permit simuler les relations Ca/PTH dans le sang, Ca/fraction éliminée par le rein, Ca/P dans le sang pour des valeurs normales ou hautes de calcium. On a fait des simulations de situations physiopathologiques (dans “malades virtuelles”): Infusions chroniques de calcium, PTH ou calcitriol; altérations des récepteurs. Ces simulations ne peuvent pas être réalisées dans les humains. Sont des exemples d’utilisation du modèle dans l’exploration des possibles mécanismes de la physiopathologie en observant des résultats quantitatifs inaccessibles à l’intuition. Le modèle a été utile pendant deux étapes des travaux: La première, dans sa construction on a choisi l’information disponible, son analyse quantitative, l’explicitation rigoureuse (analytique) des relations fonctionnelles entre les contrôleurs et les variables et sa intégration dans une structure globale. La deuxième, la simulation de situations expérimentales ou cliniques (du Service d’Endocrinologie) a obligé d’expliciter des raisonnements physiopathologiques généralement formulés utilisant l’intuition. Cette pratique a montré des comportements – action réduite des infusions de PTH (jusqu’à l’inhibition totale de leur respective sécrétion), nécessité d’augmenter la masse sécréteuse de la parathyroïde dans les insuffisants rénales, etc. La synthèse et utilisation du modèle n’ont pas besoin d’une formation avancée en mathématique et sont possibles grâce à un programme interactif qui a été conçu pour la simulation des systèmes dynamiques dans lesquels le programme se construit en anglais en utilisant la symbolique élémentaire de l’algèbre. La fonction noble de ces modèles est semblable à celles des physiques du XVII siècle: Permettre établir explications générales en fonctionnant comme un outil intellectuel pour manipuler des concepts et pour la réalisation d’expérimentes pensées en respectant certains principes de la physique (principe de la conservation) qu’établissent les frontières de la réalité.

Study and implementation of some quantitative trading models

Quantitative or algorithmic trading is the automatization of investments decisions obeying a fixed or dynamic sets of rules to determine trading orders. It has increasingly made its way up to 70% of the trading volume of one of the biggest financial markets such as the New York Stock Exchange (NYSE). However, there is not a signi cant amount of academic literature devoted to it due to the private nature of investment banks and hedge funds. This projects aims to review the literature and discuss the models available in a subject that publications are scarce and infrequently. We review the basic and fundamental mathematical concepts needed for modeling financial markets such as: stochastic processes, stochastic integration and basic models for prices and spreads dynamics necessary for building quantitative strategies. We also contrast these models with real market data with minutely sampling frequency from the Dow Jones Industrial Average (DJIA). Quantitative strategies try to exploit two types of behavior: trend following or mean reversion. The former is grouped in the so-called technical models and the later in the so-called pairs trading. Technical models have been discarded by financial theoreticians but we show that they can be properly cast into a well defined scientific predictor if the signal generated by them pass the test of being a Markov time. That is, we can tell if the signal has occurred or not by examining the information up to the current time; or more technically, if the event is F_t-measurable. On the other hand the concept of pairs trading or market neutral strategy is fairly simple. However it can be cast in a variety of mathematical models ranging from a method based on a simple euclidean distance, in a co-integration framework or involving stochastic differential equations such as the well-known Ornstein-Uhlenbeck mean reversal ODE and its variations. A model for forecasting any economic or financial magnitude could be properly defined with scientific rigor but it could also lack of any economical value and be considered useless from a practical point of view. This is why this project could not be complete without a backtesting of the mentioned strategies. Conducting a useful and realistic backtesting is by no means a trivial exercise since the \laws" that govern financial markets are constantly evolving in time. This is the reason because we make emphasis in the calibration process of the strategies' parameters to adapt the given market conditions. We find out that the parameters from technical models are more volatile than their counterpart form market neutral strategies and calibration must be done in a high-frequency sampling manner to constantly track the currently market situation. As a whole, the goal of this project is to provide an overview of a quantitative approach to investment reviewing basic strategies and illustrating them by means of a back-testing with real financial market data. The sources of the data used in this project are Bloomberg for intraday time series and Yahoo! for daily prices. All numeric computations and graphics used and shown in this project were implemented in MATLAB^R scratch from scratch as a part of this thesis. No other mathematical or statistical software was used.

Epidemic models with an infected-infectious period

The introduction of an infective-infectious period on the geographic spread of epidemics is considered in two different models. The classical evolution equations arising in the literature are generalized and the existence of epidemic wave fronts is revised. The asymptotic speed is obtained and improves previous results for the Black Death plague

Estimation of electronic coupling in π -stacked donor-bridge-acceptor systems: correction of the two-state model

Comparison of donor-acceptor electronic couplings calculated within two-state and three-state models suggests that the two-state treatment can provide unreliable estimates of Vda because of neglecting the multistate effects. We show that in most cases accurate values of the electronic coupling in a π stack, where donor and acceptor are separated by a bridging unit, can be obtained as Ṽ da = (E2 - E1) μ12 Rda + (2 E3 - E1 - E2) 2 μ13 μ23 Rda2, where E1, E2, and E3 are adiabatic energies of the ground, charge-transfer, and bridge states, respectively, μij is the transition dipole moments between the states i and j, and Rda is the distance between the planes of donor and acceptor. In this expression based on the generalized Mulliken-Hush approach, the first term corresponds to the coupling derived within a two-state model, whereas the second term is the superexchange correction accounting for the bridge effect. The formula is extended to bridges consisting of several subunits. The influence of the donor-acceptor energy mismatch on the excess charge distribution, adiabatic dipole and transition moments, and electronic couplings is examined. A diagnostic is developed to determine whether the two-state approach can be applied. Based on numerical results, we showed that the superexchange correction considerably improves estimates of the donor-acceptor coupling derived within a two-state approach. In most cases when the two-state scheme fails, the formula gives reliable results which are in good agreement (within 5%) with the data of the three-state generalized Mulliken-Hush model

Exactly solvable phase oscillator models with syncrhonization dynamics

Populations of phase oscillators interacting globally through a general coupling function f(x) have been considered. We analyze the conditions required to ensure the existence of a Lyapunov functional giving close expressions for it in terms of a generating function. We have also proposed a family of exactly solvable models with singular couplings showing that it is possible to map the synchronization phenomenon into other physical problems. In particular, the stationary solutions of the least singular coupling considered, f(x) = sgn(x), have been found analytically in terms of elliptic functions. This last case is one of the few nontrivial models for synchronization dynamics which can be analytically solved.

Dynamic models of viral replication and latency.

PURPOSE OF REVIEW: HIV targets primary CD4(+) T cells. The virus depends on the physiological state of its target cells for efficient replication, and, in turn, viral infection perturbs the cellular state significantly. Identifying the virus-host interactions that drive these dynamic changes is important for a better understanding of viral pathogenesis and persistence. The present review focuses on experimental and computational approaches to study the dynamics of viral replication and latency. RECENT FINDINGS: It was recently shown that only a fraction of the inducible latently infected reservoirs are successfully induced upon stimulation in ex-vivo models while additional rounds of stimulation make allowance for reactivation of more latently infected cells. This highlights the potential role of treatment duration and timing as important factors for successful reactivation of latently infected cells. The dynamics of HIV productive infection and latency have been investigated using transcriptome and proteome data. The cellular activation state has shown to be a major determinant of viral reactivation success. Mathematical models of latency have been used to explore the dynamics of the latent viral reservoir decay. SUMMARY: Timing is an important component of biological interactions. Temporal analyses covering aspects of viral life cycle are essential for gathering a comprehensive picture of HIV interaction with the host cell and untangling the complexity of latency. Understanding the dynamic changes tipping the balance between success and failure of HIV particle production might be key to eradicate the viral reservoir.

Application of theoretical models to financial innovation

The present paper is aimed at providing a general strategic overview of the existing theoretical models that have applications in the field of financial innovation. Whereas most financialdevelopments have relied upon traditional economic tools, a new stream of research is defining a novel paradigm in which mathematical models from diverse scientific disciplines are being applied to conceptualize and explain economic and financial behavior. Indeed, terms such as ‘econophysics’ or ‘quantum finance’ have recently appeared to embrace efforts in this direction. As a first contact with such research, the project will present a brief description of some of the main theoretical models that have applications in finance and economics, and will try to present, if possible, potential new applications to particular areas in financial analysis, or new applicable models. As a result, emphasiswill be put on the implications of this research for the financial sector and its future dynamics.

Practical algorithms for a family of waterfilling solutions

Many engineering problems that can be formulatedas constrained optimization problems result in solutionsgiven by a waterfilling structure; the classical example is thecapacity-achieving solution for a frequency-selective channel.For simple waterfilling solutions with a single waterlevel and asingle constraint (typically, a power constraint), some algorithmshave been proposed in the literature to compute the solutionsnumerically. However, some other optimization problems result insignificantly more complicated waterfilling solutions that includemultiple waterlevels and multiple constraints. For such cases, itmay still be possible to obtain practical algorithms to evaluate thesolutions numerically but only after a painstaking inspection ofthe specific waterfilling structure. In addition, a unified view ofthe different types of waterfilling solutions and the correspondingpractical algorithms is missing.The purpose of this paper is twofold. On the one hand, itoverviews the waterfilling results existing in the literature from aunified viewpoint. On the other hand, it bridges the gap betweena wide family of waterfilling solutions and their efficient implementationin practice; to be more precise, it provides a practicalalgorithm to evaluate numerically a general waterfilling solution,which includes the currently existing waterfilling solutions andothers that may possibly appear in future problems.

Monte Carlo Methods in Parameter Estimation of Nonlinear Models

Yksi keskeisimmistä tehtävistä matemaattisten mallien tilastollisessa analyysissä on mallien tuntemattomien parametrien estimointi. Tässä diplomityössä ollaan kiinnostuneita tuntemattomien parametrien jakaumista ja niiden muodostamiseen sopivista numeerisista menetelmistä, etenkin tapauksissa, joissa malli on epälineaarinen parametrien suhteen. Erilaisten numeeristen menetelmien osalta pääpaino on Markovin ketju Monte Carlo -menetelmissä (MCMC). Nämä laskentaintensiiviset menetelmät ovat viime aikoina kasvattaneet suosiotaan lähinnä kasvaneen laskentatehon vuoksi. Sekä Markovin ketjujen että Monte Carlo -simuloinnin teoriaa on esitelty työssä siinä määrin, että menetelmien toimivuus saadaan perusteltua. Viime aikoina kehitetyistä menetelmistä tarkastellaan etenkin adaptiivisia MCMC menetelmiä. Työn lähestymistapa on käytännönläheinen ja erilaisia MCMC -menetelmien toteutukseen liittyviä asioita korostetaan. Työn empiirisessä osuudessa tarkastellaan viiden esimerkkimallin tuntemattomien parametrien jakaumaa käyttäen hyväksi teoriaosassa esitettyjä menetelmiä. Mallit kuvaavat kemiallisia reaktioita ja kuvataan tavallisina differentiaaliyhtälöryhminä. Mallit on kerätty kemisteiltä Lappeenrannan teknillisestä yliopistosta ja Åbo Akademista, Turusta.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background: Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization. Results: Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity. Conclusions: Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.

A measure of stability as a criterion for the verification and analysis of simulation models

The aim of this study is to define a new statistic, PVL, based on the relative distance between the likelihood associated with the simulation replications and the likelihood of the conceptual model. Our results coming from several simulation experiments of a clinical trial show that the PVL statistic range can be a good measure of stability to establish when a computational model verifies the underlying conceptual model. PVL improves also the analysis of simulation replications because only one statistic is associated with all the simulation replications. As well it presents several verification scenarios, obtained by altering the simulation model, that show the usefulness of PVL. Further simulation experiments suggest that a 0 to 20 % range may define adequate limits for the verification problem, if considered from the viewpoint of an equivalence test.

A comparison of electricity spot prices simulation using ARMA-GARCH and mean-reverting models

The aim of this work is to compare two families of mathematical models for their respective capability to capture the statistical properties of real electricity spot market time series. The first model family is ARMA-GARCH models and the second model family is mean-reverting Ornstein-Uhlenbeck models. These two models have been applied to two price series of Nordic Nord Pool spot market for electricity namely to the System prices and to the DenmarkW prices. The parameters of both models were calibrated from the real time series. After carrying out simulation with optimal models from both families we conclude that neither ARMA-GARCH models, nor conventional mean-reverting Ornstein-Uhlenbeck models, even when calibrated optimally with real electricity spot market price or return series, capture the statistical characteristics of the real series. But in the case of less spiky behavior (System prices), the mean-reverting Ornstein-Uhlenbeck model could be seen to partially succeeded in this task.