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abstract: There have been many studies on the dynamics of infectious diseases considering the age structure of the population. This study analyzes the dynamics when the population is stratified by size. This kind of models are useful in the spread of a disease in fisheries where size matters, for microorganism populations or even human diseases that are driven by weight. A simple size structured SIR model is introduced for which a threshold condition, R0, equilibria and stability are established in special cases. Hethcote's approach is used to derive, from first principles, a parallel ODE size-structure system involving n-size classes.The specific case of n = 2 is partially analyzed. Constant effort harvesting is added to this model with the purpose of exploring the role of controls and harvesting. Different harvesting policies are proposed and analyzed through simulations. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2012

In this thesis, we will study basic mathematical epidemic models SIR, SIS, and SEIR. Then we will construct a modified model as a combination of SIR and SIS models. First, we will find the explicit solutions for the SIS model. and show no exact solution for the SIR model. Also, find the parametric solution for the SIR model and find a numerical solution by using Euler’s method. Then we find an approximate explicit form of the epidemic curve. Also, we will study parametric influences on the SIR and SIS models. Finally, we will suggest some recommendations to decrease the epidemic’s spread.

Βασικό χαρακτηριστικό της σύγχρονης εποχής αποτελεί η ραγδαία αύξηση του Διαδικτύου τόσο σε επίπεδο χρηστών όσο και σε επίπεδο παρεχόμενων υπηρεσιών. Συνεπώς, είναι επιτακτική η ανάγκη της προστασίας των δικτυακών και υπολογιστικών συστημάτων από διάφορες απειλές οι οποίες μπορούν να τα καταστήσουν τρωτά. Για την πλήρη προστασία όμως αυτών των συστημάτων, απαιτείται πρώτα η κατανόηση του είδους, της ταυτότητας και του τρόπου διάδοσης της απειλής. Ιδιαίτερη χρήσιμη έχει αποδειχθεί η ανάπτυξη και αναζήτηση αξιόπιστων μοντέλων ικανών να περιγράψουν αρκετά αποτελεσματικά τον τρόπο διάδοσης μιας απειλής. Η αναζήτηση τέτοιων μοντέλων αποτελεί πλέον ένα σημαντικό τομέα έρευνας στην ακαδημαϊκή και όχι μόνο κοινότητα. Σκοπός της παρούσας διπλωματικής εργασίας είναι η προσομοίωση και μελέτη των βασικών επιδημιολογικών μοντέλων SI, SIR, SIS και SIRS. Τα μοντέλα αυτά είναι εμπνευσμένα από την επιστήμη της Βιολογίας, και πλέον τη σημερινή εποχή χρησιμοποιούνται ευρέως για τη μοντελοποίηση της διάδοσης αρκετών απειλών στα δίκτυα υπολογιστών, όπως πχ. οι ιοί και τα σκουλήκια (viruses and worms). Η εργασία αποτελείται από πέντε κεφάλαια. Στο πρώτο κεφάλαιο, γίνεται και η παρουσίαση των ασυρμάτων δικτύων αισθητήρων περιγράφοντας επίσης τόσο τη δομή όσο και τα βασικά χαρακτηριστικά αυτών. Στο δεύτερο κεφάλαιο γίνεται μια παρουσίαση των βασικών ειδών του κακόβουλου λογισμικού που μπορούν να πλήξουν ένα υπολογιστικό σύστημα. Γίνεται επίσης αναφορά στα χαρακτηριστικά των κακόβουλων λογισμικών τα οποία επηρεάζουν την εξάπλωσή του. Το τρίτο κεφάλαιο επιχειρεί να εισάγει την έννοια της επιδημιολογίας στα συστήματα υπολογιστών με την ανάλυση κυρίως των ιδιαιτεροτήτων οι οποίες την χαρακτηρίζουν. Επίσης αυτό το κεφάλαιο παρουσιάζει κάποια βασικά επιδημιολογικά μοντέλα κάνοντας μια αναφορά τόσο στα βασικά χαρακτηριστικά αυτών, όσο επίσης και στον τρόπο λειτουργίας τους. Το τέταρτο κεφάλαιο το οποίο είναι και το πιο σημαντικό της εργασίας αυτής, αφιερώνεται στην παρουσίαση του εργαλείου OPNET Modeler που χρησιμοποιήσαμε και στην εκτενή περιγραφή της προσομοίωσης των μοντέλων SI, SIS, SIR και SIRS που διεξήγαμε για ένα ασύρματο δίκτυο αισθητήρων. Γίνεται παρουσίαση της λειτουργίας του δικτύου με ταυτόχρονη επεξήγηση του κώδικα που αναπτύχθηκε. Επιπλέον παρουσιάζονται και αναλύονται τα αποτελέσματα της προσομοίωσης ενώ παράλληλα περιγράφονται και τα συμπεράσματα στα οποία μας οδήγησε η εν λόγω προσομοίωση. Τέλος, στο πέμπτο κεφάλαιο, γίνεται μια αναφορά σε κάποια βασικά συμπεράσματα στα οποία οδηγηθήκαμε, ενώ περιγράφονται και πεδία πάνω στη μελέτη της διάδοσης ενός κακόβουλου λογισμικού σε ένα υπολογιστικό δίκτυο, τα οποία μπορούν να μελετηθούν εκτενέστερα μελλοντικά. / A basic characteristic of contemporary days is the boom of the Internet either in terms of users or in terms of services rendered. Therefore, there is an imperative need to protect the network and computational systems from various threats which can render them vulnerable. However, for the full protection of these systems, it is required in the first place to get to know the type, the identity and the propagation mode of the threat. Of significant use has proved to be the development and the pursuit of models capable of describing quite effectively the way a threat is spread. The pursuit of such models constitutes nowadays a significant sector of research, including, but not limited to the academic community. The intention of the present diploma thesis is the simulation and study of the basic epidemic models SI, SIR, SIS and SIRS. These models are inspired from the science of Biology, and they are widely used nowadays for the modeling of the spread of various threats in computer networks such as viruses and worms. This dissertation consists of five chapters. In the first chapter, there is taking place the presentation of wireless sensor networks and there is also a description of their structure and their basic characteristics. In the second chapter there is a presentation of the basic types of malicious software that can hit a computational system. There is also reference to the characteristics of malicious software that affect their propagation. The third chapter attempts to introduce the concept on epidemiology in computer systems, analyzing mainly the particularities characterizing her. In addition, this chapter presents some basic epidemic models, referring both to their basic characteristics and their mode of operation. The fourth chapter, which is also the most significant one of the present thesis, is dedicated to the presentation of the tool OPNET Modeler that we used too in the thorough description of the simulation of the models SI, SIR, SIS and SIRS that we carried out for a wireless sensor network. It is taking place the presentation of the network’s operation mode with a simultaneous explanation of the code that was developed. Moreover, there are presented and analyzed the results of the simulation when at the same time are also described the conclusions that were derived from the present simulation. Finally, in the fifth chapter, there is a reference to some basic conclusions in which we were led, where there are also described fields concerning the study of malicious software propagation in a computational network, which can studied further in the future.

The problem of changing scale in models of a system is relevant in many different fields. In this thesis we investigate the problem in models of biological systems, particularly infectious disease spread and population dynamics. We investigate this problem using the process algebra \emph{Weighted Synchronous Calculus of Communicating Systems} (WSCCS). In WSCCS we can describe the different types of individual in a population and study the population by placing many of these individuals in parallel. We present an algorithm that allows us to rigorously derive mean field equations (MFE) describing the average change in the population. The algorithm takes into account the Markov chain semantics of WSCCS such that as the system being considered becomes larger, the approximation offered by the MFE tends towards the mean of the Markov chain. The traditional approach to developing population level equations of a system involves making assumptions about the behaviour of the entire population. Our approach means that the population level dynamics explained by the MFE are a direct consequence of the behaviour of individuals, which is more readily observed and measured than the behaviour of the population. In this way we develop MFE models of several different systems and compare the equations obtained to the traditional mathematical models of the system.

A emergência de fenômenos coletivos e correlações de longo alcance impossibilitam a inferência de propriedades de sistemas como um todo a partir de suas partes componentes. A modelagem destes sistemas frequentemente ocorre mediante emprego de operadores de spin localizados em grafos com topologias não-triviais. Aqui, mostramos que o operador de momento angular de muitos corpos une o estudo de diversos sistemas complexos, desde a sistemas epidêmicos até cadeias magnéticas de spin. Para o modelo epidêmico SIS, determinamos a matriz de transição do processo estocástico correspondente e mostramos suas soluções para grafos regulares e aleatórios, por meio de técnicas geralmente empregadas em sistemas fortemente correlacionados. Já no modelo de Dicke, identificamos o vínculo que explica a relevância e o efeito finito de operadores anti-girantes para duas espécies atômicas confinadas numa cavidade óptica que interagem com radiação eletromagnética. Por fim, o papel do momento angular também é identificado para duas cadeias quânticas de spin 1/2 acopladas, as quais modelam nanoestruturas magnéticas heterogêneas. A estrutura de bandas é calculada, enquanto efeitos espúrios de superfície são removidos pela introdução de quasipartículas dotadas de grau de liberdade de spin adicional / The emergence of collective phenomena and long range correlations makes it impossible to infer the properties of whole systems from their components. Their modeling often occurs through the use of localized spin operators, taking place within graphs with non-trivial topologies. Here, we show that the many-body angular momentum operator connects the study of several complex systems, ranging from epidemic systems to magnetic spinchains. For the SIS epidemic model, we calculate the transition matrix of the corresponding stochastic process and show the corresponding solutions for regular and random graphs, using techniques generally employed in strongly correlated systems. For the Dicke model we identify the constraint that explains the relevance and finite size effect of anti-rotating operators, for two atomic species, confined within an optical cavity, and interacting with electromagnetic radiation. Finally, the role of angular momentum is also identified for two coupled quantum spinchains 1/2 which model heterogeneous magnetic nanostructures. The band structure is calculated, while spurious surface effects are removed due to the introduction of quasiparticles with an additional spin degree of freedom.

Individual-level models (ILMs) are models that can use the spatial-temporal nature of disease data to capture the disease dynamics. Parameter estimation is usually done via Markov chain Monte Carlo (MCMC) methods, but correlation between model parameters negatively affects MCMC mixing. Introducing a normalization constant to alleviate the correlation results in MCMC convergence over fewer iterations, however this negatively effects computation time. It is important that model fitting is done as efficiently as possible. An upper-truncated distance kernel is introduced to quicken the computation of the likelihood, but this causes a loss in goodness-of-fit. The normalization constant and upper-truncated distance kernel are evaluated as components in various ILMs via a simulation study. The normalization constant is seen not to be worthwhile, as the effect of increased computation time is not outweighed by the reduced correlation. The upper-truncated distance kernel reduces computation time but worsens model fit as the truncation distance decreases. / Studies have been funded by OMAFRA & NSERC, with computing equipment provided by CSI.

Pesquisas sobre a propagação de epidemias são uma constante devido a sua relevância para a contenção de doenças. Porém, devido aos diversos tipos de doenças existentes, a observação de um comportamento genérico e aproximado torna-se impraticável. Neste âmbito, a elaboração de modelos matemáticos epidêmicos auxiliam no fornecimento de informações que podem ser usadas por orgãos públicos para o combate de surtos epidêmicos reais. Em paralelo, por causa do grande volume de dados que são processados na execução da simulação desses modelos, o constante aumento dos recursos computacionais desenvolvidos vem em auxílio desta tarefa. O objetivo desta dissertação é estudar o comportamento da disseminação de uma epidemia simulada computacionalmente através do modelo epidêmico SIR em reticulados quadrados considerando duas propriedades: a existência de vértices vazios e a movimentação aleatória dos agentes. Essas propriedades são conhecidas por taxas de diluição e mobilidade, respectivamente. Para alcançar esse objetivo, algumas técnicas físico-estatística, tais como a análise das transições de fase e fenômenos críticos, foram aplicadas. Através destas técnicas, é possível observar a passagem do sistema da fase em que ocorre um surto epidêmico para a fase em que a epidemia é contida, bem como estudar a dinâmica do modelo quando ele está na criticidade, ou seja, no ponto de mudança de fase, conhecido por ponto crítico. Foi constatado que a taxa de diluição influencia a disseminação das epidemias porque desloca a transição de fase negativamente, reduzindo o valor crítico da imunização. Por sua vez, a taxa da movimentação dos agentes favorece o espalhamento da doença, pois a transição de fase é positivamente deslocada e seu ponto crítico, aumentado. Além disso foi observado que, apesar desse incremento, ele não é completamente restaurado devido às restrições de mobilidade dos agentes e ao alto grau de desconectividade da rede causado pelas altas taxas de diluição. Neste trabalho nós mostramos as razões deste comportamento. / Research on the spreading of epidemics are frequent because of their relevance for the containment of diseases. However, due to the variety of existing illnesses, the observation of an approximated generic behavior becomes impractical. In this context, the development of mathematical models of epidemics assists in providing information that can be used to make strategic decisions for the combat of real epidemic outbreaks. In parallel, because of the large volume of data which has to be processed in the simulation of these models, the increase of computational performance helps with this task. The objective of this thesis is to study the behavior of the spreading of an epidemic, by computationally simulating an SIR epidemic model on square lattices, considering two properties: the existence of empty vertices and random movement of agents. These properties are known as dilution rate and mobility rate, respectively. To achieve this goal, techniques of statistical physics, such as the analysis of phase transition and power laws, were applied. With these techniques, it is possible to observe the transition of the system from the phase in which an outbreak occurs to the phase where the epidemic is contained. Additionally, we studied the dynamics of the model when it is in criticality, that is, at the point of phase transition, known as the critical point. It was found that a higher dilution rate reduces the spreading of epidemics because it shifts the phase transition negatively, reducing the value of its critical point. On the other hand, increasing the rate of movement of the agents favors the spreading of the disease, because the phase transition is shifted positively and its critical point is increased. It was noticed that, despite of this increasing, this point is not completely restored due to restricted mobility of agents and the high degree of the network disconectivity caused by the high dilution rates. In this work we show the reasons for this behavior.

Cette thèse porte sur l'analyse mathématique de modèles d'évolution sur des réseaux. La thèse se compose de trois chapitres. Les deux premiers portent sur un modèle de propagation d'épidémies dans des réseaux et le troisième porte sur l'équation de Price, qui intervient dans la modélisation de la croissance des réseaux complexes.L'essentiel de la thèse est constituée des deux premiers chapitres, où nous proposons et analysons un modèle épidémique de type SIS avec diffusion non locale. Ce modèle est obtenu à partir d'un modèle discret, en supposant que le degré des noeuds du réseau est ici une variable continue à valeurs positives. Le réseau est ainsi modélisé par la distribution de probabilité des degrés des noeuds du réseau, où a lieu la transmission épidémique. La migration le long des arêtes du réseau correspond à une diffusion non locale. Le système d'évolution en temps pour les densités d'individus sains et infectés se ramène ainsi à un système couplé d'équations différentielles non linéaires avec des termes non locaux, qui sont des moyennes sur le réseau de ces densités. Nous analysons ce système d'évolution, en étudiant successivement le cas d'une transmission limitée (Chapitre 1) et non limitée (Chapitre 2).Nous prouvons tout d'abord rigoureusement l'existence d'une unique solution, locale ou globale, par une méthode de point fixe. Nous établissons ensuite des conditions de seuils nécessaires et suffisantes pour l'existence d'un équilibre endémique. Puis nous étudions la stabilité linéaire des équilibres sain et endémique et comparons nos résultats à ceux obtenus sur le modèle discret de départ. Dans le cas de la transmission limitée et de coefficients de diffusion égaux, on se ramène à une équation de type Fisher avec diffusion non locale, pour laquelle on établit un principe de comparaison. Ceci nous permet d'étudier le comportement asymptotique en temps pour des données initiales quelconques.Le dernier chapitre porte sur l'équation de Price, qui est un modèle de croissance des réseaux.Il se présente sous la forme d'une relation de récurrence donnant l'évolution de la distribution des degrés dans un réseau de taille croissante. Nous montrons rigoureusement la convergence de la solution du modèle de Price vers la solution stationnaire et nous montrons que celle-ci est équivalente à une loi puissance, dont nous précisons l'exposant. / This thesis is devoted to the mathematic analysis of time-dependent models on complex networks. There are three chapters. The first two chapters concern a model for the spread of epidemics on networks while the third chapter concerns Price equation, which arises as a model for the growth of complex networks.Most part of this thesis is concentrated in the first two chapters, in which we propose and analyze a SIS-type epidemic model with nonlocal diffusion. This model is derived from a discrete model, by considering here the degree as a continuous variable taking nonnegative values. Hence the network is described by the degree distribution of its nodes, where the epidemic transmission takes place. Migration occurs along the edges of the network and corresponds to nonlocal diffusion. The evolution system for the density of susceptible and infected individuals reads as a coupled system of nonlinear equations with nonlocal terms, which are given by the mean values of these densities on the network. We provide the analysis of this time-dependent system, distinguishing the cases of limited transmision (chapter 1) and illimited transmission (chapter 2).We first rigorously prove the existence of a unique solution to the system, either locally or globally in time, using a fixed point method. Next we establish necessary and sufficient threshold conditions for the existence of an endemic equilibrium. We then investigate the linear stability of both the disease-free and the endemic equilibrium and compare our results to the ones obtained for the discrete system. In the case of equal diffusivities and illimited transmission, we reduce the system to a Fisher-type equation with nonlocal diffusion, for which we prove a comparison principle. This allows us to study the large-time asymptotics of the solution for arbitrary initial data.The last chapter deals with Price equation, which is a model for the growth of networks. The model reads as a discrete recursive equation that provides the time-evolution of the probability distribution of the degrees in a growing network. We show rigorously that the solution converges to a stationary state exhibiting a power-law tail, whose exponent is explicitly given.

Pesquisas sobre a propagação de epidemias são uma constante devido a sua relevância para a contenção de doenças. Porém, devido aos diversos tipos de doenças existentes, a observação de um comportamento genérico e aproximado torna-se impraticável. Neste âmbito, a elaboração de modelos matemáticos epidêmicos auxiliam no fornecimento de informações que podem ser usadas por orgãos públicos para o combate de surtos epidêmicos reais. Em paralelo, por causa do grande volume de dados que são processados na execução da simulação desses modelos, o constante aumento dos recursos computacionais desenvolvidos vem em auxílio desta tarefa. O objetivo desta dissertação é estudar o comportamento da disseminação de uma epidemia simulada computacionalmente através do modelo epidêmico SIR em reticulados quadrados considerando duas propriedades: a existência de vértices vazios e a movimentação aleatória dos agentes. Essas propriedades são conhecidas por taxas de diluição e mobilidade, respectivamente. Para alcançar esse objetivo, algumas técnicas físico-estatística, tais como a análise das transições de fase e fenômenos críticos, foram aplicadas. Através destas técnicas, é possível observar a passagem do sistema da fase em que ocorre um surto epidêmico para a fase em que a epidemia é contida, bem como estudar a dinâmica do modelo quando ele está na criticidade, ou seja, no ponto de mudança de fase, conhecido por ponto crítico. Foi constatado que a taxa de diluição influencia a disseminação das epidemias porque desloca a transição de fase negativamente, reduzindo o valor crítico da imunização. Por sua vez, a taxa da movimentação dos agentes favorece o espalhamento da doença, pois a transição de fase é positivamente deslocada e seu ponto crítico, aumentado. Além disso foi observado que, apesar desse incremento, ele não é completamente restaurado devido às restrições de mobilidade dos agentes e ao alto grau de desconectividade da rede causado pelas altas taxas de diluição. Neste trabalho nós mostramos as razões deste comportamento. / Research on the spreading of epidemics are frequent because of their relevance for the containment of diseases. However, due to the variety of existing illnesses, the observation of an approximated generic behavior becomes impractical. In this context, the development of mathematical models of epidemics assists in providing information that can be used to make strategic decisions for the combat of real epidemic outbreaks. In parallel, because of the large volume of data which has to be processed in the simulation of these models, the increase of computational performance helps with this task. The objective of this thesis is to study the behavior of the spreading of an epidemic, by computationally simulating an SIR epidemic model on square lattices, considering two properties: the existence of empty vertices and random movement of agents. These properties are known as dilution rate and mobility rate, respectively. To achieve this goal, techniques of statistical physics, such as the analysis of phase transition and power laws, were applied. With these techniques, it is possible to observe the transition of the system from the phase in which an outbreak occurs to the phase where the epidemic is contained. Additionally, we studied the dynamics of the model when it is in criticality, that is, at the point of phase transition, known as the critical point. It was found that a higher dilution rate reduces the spreading of epidemics because it shifts the phase transition negatively, reducing the value of its critical point. On the other hand, increasing the rate of movement of the agents favors the spreading of the disease, because the phase transition is shifted positively and its critical point is increased. It was noticed that, despite of this increasing, this point is not completely restored due to restricted mobility of agents and the high degree of the network disconectivity caused by the high dilution rates. In this work we show the reasons for this behavior.