Heterogeneity, influenced by diverse factors such as age, gender, immunity, behavior, and spatial distribution, plays a critical role in the dynamics of infectious disease transmission. Discrete mathematical structures, including matrices and graphs, can offer effective tools for modeling the interactions among these diverse factors, resulting heterogeneous epidemiological models. This dissertation explores analytical approaches, specifically utilizing eigenvalues and eigenvectors of discrete structures, to characterize heterogeneity within mathematical models of infectious diseases. Theoretical results, along with numerical simulations, enhance our understanding of heterogeneous epidemiological processes and their significant implications for disease control strategies. In this dissertation, we introduce a unified approach to establish the final size formula in heterogeneous epidemic models, based on a new concept of “total infectious contacts” as an eigenvector-based aggregation of disease compartments. This approach allows us to identify the peak of total infectious contacts, offering a novel method to pinpoint the turning point of a disease outbreak. Furthermore, we examine spatial heterogeneity through two distinct mathematical frameworks: the Lagrangian and Eulerian models. The Lagrangian model assesses the epidemiological consequences of spatio-temporal residence time matrices, while the Eulerian model investigates “Turing instability” as a new underlying mechanism for spatial heterogeneity observed in disease prevalence data.

Cette thèse traite de l'impact des perturbations naturelles et anthropiques sur les structures des populations et des communautés afin de discuter la problématique d'une gestion forestière qui s'inspire des - et compose avec les - perturbations naturelles. En première partie, nous étudions les effets d'un gradient de perturbations sur la diversité des communautés de plantes. A l'aide de modèles de dynamique en patchs ("patch models"), nous montrons que la relation perturbations-diversité dépend des mécanismes de compétition (hiérarchie compétitive, "founder effect"), des mécanismes de coexistence (compromis entre compétition et colonisation, niche de succession), des mécanismes affectant le processus de colonisation (limitation par le recrutement, fréquence dépendance négative des taux de colonisation et immigration) et enfin des mesures de diversité privilégiées (richesse en espèce, diversité de Shannon). L'élaboration d'hypothèses mécanistes intégrant les caractéristiques de l'agent de perturbations (dont les actions anthropiques), les descripteurs utilisés (intensité, fréquence, stochasticité etc.), les traits de vie des espèces, les mécanismes de coexistence et les facteurs biotiques et abiotiques contrôlés permettraient d'affiner les prédictions qualitatives et quantitatives sur la relation perturbations-diversité et de favoriser ainsi leurs dimensions opérationnelles. En seconde partie, nous rappelons différents concepts (équilibre, résistance, résilience, persistance, constance, permanence) permettant de caractériser le comportement des systèmes en présence de perturbations. Nous présentons une méthode d'analyse basée sur le concept de permanence qui permet d'évaluer différents scénarios écologiques ou scénarios de gestion. La permanence consiste à se focaliser sur le comportement des variables lorsque ces dernières sont proches de valeurs jugées critiques (ex. extinction d'une espèce). Nous appliquons cette méthode d'évaluation à la problématique des forêts de protection dans les Alpes du Nord. A partir d'un modèle de dynamique de pessières d'altitude, nous testons différents scénarios de gestion (prélèvements agrégés/aléatoire, différents taux de prélèvements) et déterminons leurs conséquences sur la dynamique d'indicateurs de propriétés de stabilité (diversité des hauteurs, diversité des diamètres) et d'indicateurs de fonction de protection (somme des diamètres pour les chutes de blocs, taille des trouées pour les avalanches). La possibilité d'améliorer conjointement propriétés de stabilité et efficacité de protection dépend de la nature et de l'intensité des interventions ainsi que du type de danger. Cette partie permet d'illustrer une réflexion de gestion basée sur des compromis plutôt que sur l'optimisation simultanée d'un ensemble d'indicateurs. En conclusion, nous replaçons nos résultats dans le contexte d'une gestion qui s'appuie sur la connaissance du processus de perturbation. La notion de dynamique en patchs (trouées) et la notion de permanence représentent, selon nous, des cadres conceptuels et méthodologiques intéressant pour établir un pont entre écologie des perturbations et gestion forestière. Nous ouvrons des perspectives de recherche à partir de l'exemple des sapinières-pessières dans les Alpes du Nord, exemple qui intègre à la fois la problématique de la coexistence des espèces et celle de la réalisation des fonctions de production et de protection dans des écosystèmes gérés où dominent les perturbations par le vent.

abstract: The increased number of novel pathogens that potentially threaten the human population has motivated the development of mathematical and computational modeling approaches for forecasting epidemic impact and understanding key environmental characteristics that influence the spread of diseases. Yet, in the case that substantial uncertainty surrounds the transmission process during a rapidly developing infectious disease outbreak, complex mechanistic models may be too difficult to be calibrated quick enough for policy makers to make informed decisions. Simple phenomenological models that rely on a small number of parameters can provide an initial platform for assessing the epidemic trajectory, estimating the reproduction number and quantifying the disease burden from the early epidemic phase. Chapter 1 provides background information and motivation for infectious disease forecasting and outlines the rest of the thesis. In chapter 2, logistic patch models are used to assess and forecast the 2013-2015 West Africa Zaire ebolavirus epidemic. In particular, this chapter is concerned with comparing and contrasting the effects that spatial heterogeneity has on the forecasting performance of the cumulative infected case counts reported during the epidemic. In chapter 3, two simple phenomenological models inspired from population biology are used to assess the Research and Policy for Infectious Disease Dynamics (RAPIDD) Ebola Challenge; a simulated epidemic that generated 4 infectious disease scenarios. Because of the nature of the synthetically generated data, model predictions are compared to exact epidemiological quantities used in the simulation. In chapter 4, these models are applied to the 1904 Plague epidemic that occurred in Bombay. This chapter provides evidence that these simple models may be applicable to infectious diseases no matter the disease transmission mechanism. Chapter 5, uses the patch models from chapter 2 to explore how migration in the 1904 Plague epidemic changes the final epidemic size. The final chapter is an interdisciplinary project concerning within-host dynamics of cereal yellow dwarf virus-RPV, a plant pathogen from a virus group that infects over 150 grass species. Motivated by environmental nutrient enrichment due to anthropological activities, mathematical models are employed to investigate the relevance of resource competition to pathogen and host dynamics. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Speech is a non-stationary signal and contains modulations in both spectral and temporal domains. Based on the type of modulations studied, most speech processing algorithms can be classified into short-time analysis algorithms, narrow-band analysis algorithms, or joint spectro-temporal analysis algorithms. While traditional methods of speech analysis study the modulation along either time (Short-time analysis algorithms) or frequency (Narrowband analysis) at a time. A new class of algorithms that work simultaneously along both temporal as well as spectral dimensions, called the spectro-temporal analysis algorithms, have become prominent over the past decade. Joint spectro-temporal analysis (also referred to as 2-D speech analysis) has shown promise in applications such as formant estimation, pitch estimation, speech recognition, etc. Over the past decade, 2-D speech analysis has been independently motivated from several directions. Broadly these motivations for 2-D speech models can be grouped into speech-production motivated, source-separation/machine- learning motivated and neurophysiology motivated. In this thesis, we develop 2-D speech model based on the speech production motivation. The overall organization of the thesis is as follows: We first develop the context of 2-D speech processing in Chapter one, we then proceed to develop a 2-D multicomponent AM-FM model for narrowband spectrogram patch of voiced speech and experiment with the perceptual significance of number of components needed to represent a spectrogram patch in Chapter two. In Chapter three we develop a demodulation algorithm called the inphase and the quadrature phase demodulation (IQ), compared to the state-of-the art sinusoidal demodulation, the AM obtained using this method is more robust to carrier estimation errors. The demodulation algorithm was verified on call voiced sentences taken from the TIMIT database. In chapter four we develop a demodulation algorithm based on Riesz transform, a natural extension of the Hilbert transform to higher dimensions, unlike the sinusoidal and the IQ demodulation techniques, Riesz-transform-based demodulation does not require explicit carrier estimation and is also robust to pitch discontinuous in patches. The algorithm was validated on all voiced sentences from the TIMIT database. Both IQ and Riesz-transform-based methods were found to give more accurate estimates of the 2-D AM (relates to vocal tract) and 2-D carrier (relates to source) compared with the sinusoidal modulation. In Chapter five we show application of the demodulated AM and carrier to pitch estimation and for creation of hybrid sounds. The hybrid sounds created were found to have better perceptual quality compared with their counterparts created using the linear prediction analysis. In Chapter six we summarize the work and present with possible directions of future research.