Despite the array of medical advances of our modern day society, infectious diseases still plague millions of people worldwide. Malaria, in particular, causes substantial suffering and death throughout both developed and developing countries. Aside from the socioeconomic challenges presented by the disease's prevalence in impoverished nations, one of the major difficulties scientists have encountered while attempting to eradicate the disease is the parasite's ability to become resistant to new drugs and methods of treatment. In an effort to better understand the dynamics of malaria, we analyze a mathematical model that accounts for both the treatment aspect as well as the drug resistance that accompanies it. Simulations demonstrating the effects of treatment rates and the level of resistance are studied and discussed in hopes of shedding additional light on the characteristics of this devastating epidemic.

Η ραγδαία ανάπτυξη του Διαδικτύου τόσο σε επίπεδο πλήθους χρηστών όσο και σε επίπεδο παρεχόμενων υπηρεσιών αποτελεί χαρακτηριστικό της σύγχρονης κοινωνίας. Η ανάγκη λοιπόν προστασίας των υπολογιστικών και δικτυακών συστημάτων από απειλές που μπορούν να τα καταστήσουν τρωτά είναι επιτακτική!Η προστασία των υπολογιστικών συστημάτων και δικτύων απαιτεί σε προηγούμενο στάδιο την κατανόηση και ανάλυση του είδους, της ταυτότητας και του τρόπου διάδοσης της απειλής. Η ανάπτυξη και η αναζήτηση αξιόπιστων μοντέλων ικανών να περιγράψουν τον τρόπο διάδοσης μίας απειλής αποδεικνύεται ιδιαίτερα χρήσιμη και ωφέλιμη με πολλαπλούς τρόπους , όπως το να προβλέψει μελλοντικές απειλές ή να αναπτύξει νέες μεθόδους αναχαίτισης. Η αναζήτηση μοντέλων αποτελεί πλέον ένα σημαντικό τομέα έρευνας στην ακαδημαϊκή και όχι μόνο κοινότητα. Σκοπός της παρούσας εργασίας είναι η παρουσίαση κάποιων βασικών επιδημιολογικών μοντέλων και η προσομοίωση του επισημιολογικού μοντέλου SI. Τα μοντέλα αυτά χρησιμοποιούνται σήμερα ευρέως για τη μοντελοποίηση της διάδοσης αρκετών απειλών στα δίκτυα υπολογιστών, όπως είναι για παράδειγμα οι ιοί και τα σκουλήκια ( viruses and worms). Τα επιδημιολογικά αυτά μοντέλα είναι εμπνευσμένα από την επιστήμη της βιολογίας, όπου περιγράφουν την εξάπλωση μολυσματικών ασθενειών σε ανθρώπινους πληθυσμούς. Παρουσιάζοντας πιο αναλύτικά τη δομή της παρούσας εργασίας περιγράφουμε συνοπτικά το περιεχόμενο των πέντε κεφαλαίων που ακολουθούν: Στο πρώτο κεφάλαιο γίνεται μια σύντομη ιστορική αναδρομή στον πρόγωνο του Διαδικτύου, το APRAnet. Παρουσιάζονται κάποιες βασικές έννοιες των δικτύων καθώς και τα ευρέως γνωστά μοντέλα δόμησης του Διαδικτύου, το μοντέλο αναφοράς OSI και το μοντέλο TCP/IP. Ακολουθεί μία σύντομη ιστορική αναδρομή στο κακόβουλου λογισμικού και συνοπτική αναφορά στους βασικούς τύπους κακόβουλου λογισμικού. Παρουσιάζοναι τα χαρακρηριστικά του κακόβουλου λογισμικού που επηρεάζουν την εξάπλωση του, καθώς και κάποιες τεχνικές περιορισμού της. Το τρίτο κεφάλαιο επιχειρεί να παρουσιάσει κάποιες βασικές τοπολογίες σύνθετων δικτύων που συναντάμε σήμερα και κάποια βασικά μεγέθη που χαρακτηρίζουν τα δίκτυα αυτά. Η γνώση αυτή, που αφορά στην τοπολογία και στα χαρακτηριστικά μεγέθη των σύνθετων δικτύων, είναι απαραίτητη για τη μελέτη της διάδοσης μολύνσεων στα δίκτυα. Το τέταρτο κεφάλαιο αφιερώνεται στην παρουσίαση κάποιων βασικών επιδημιολογικών μοντέλων. Στο πέμπτο κεφάλαιο παρουσιάζεται η προσομοίωση του βασικού επιδημιολογικού μοντέλου SI (Susceptible-Infectious) συναρτήση διαφόρων τοπολογιών σύνθετων δικτύων, διαφορετικού πλήθους κόμβων και μεταβλητού αριθμού συνδέσεων ανά κόμβο ή μεταβλητή πιθανότητα σύνδεσης κόμβων. Το μέγεθος το οποίο μετράται είναι μία σταθερά «T», η οποία είναι ενδεικτική για να αποφανθούμε αν μια τοπολογία σύνθετου δικτύου είναι «ανθεκτικη» ή όχι έναντι μολύνσεων… Η σταθερά αυτή περιγράφει τα βήματα που απαιτούνται, ώστε μία μόλυνση η οποία ακολουθεί το επιδημιολογικό μοντέλο διάδοσης SI, να εξελιχθεί σε επιδημία έχοντας μολύνει όλους τους κόμβους του δικτύου….Η σταθερά αυτή μπορεί να ερμηνευτεί ως μία σταθερά χρόνου. Η σταθερά αυτή αποτελεί ένα μέτρο μέτρησης των μονάδων χρόνου που απαιτούνται για να μολυνθεί ένα ολόκληρο το δίκτυο και ένα μέσο ώστε να αποκτήσουμε μία σαφή εικόνα για το ποιές τοπολογίες είναι ανθεκτικές σε μολύνσεις και ποιες όχι. / -

abstract: In contemporary society, sustainability and public well-being have been pressing challenges. Some of the important questions are:how can sustainable practices, such as reducing carbon emission, be encouraged? , How can a healthy lifestyle be maintained?Even though individuals are interested, they are unable to adopt these behaviors due to resource constraints. Developing a framework to enable cooperative behavior adoption and to sustain it for a long period of time is a major challenge. As a part of developing this framework, I am focusing on methods to understand behavior diffusion over time. Facilitating behavior diffusion with resource constraints in a large population is qualitatively different from promoting cooperation in small groups. Previous work in social sciences has derived conditions for sustainable cooperative behavior in small homogeneous groups. However, how groups of individuals having resource constraint co-operate over extended periods of time is not well understood, and is the focus of my thesis. I develop models to analyze behavior diffusion over time through the lens of epidemic models with the condition that individuals have resource constraint. I introduce an epidemic model SVRS ( Susceptible-Volatile-Recovered-Susceptible) to accommodate multiple behavior adoption. I investigate the longitudinal effects of behavior diffusion by varying different properties of an individual such as resources,threshold and cost of behavior adoption. I also consider how behavior adoption of an individual varies with her knowledge of global adoption. I evaluate my models on several synthetic topologies like complete regular graph, preferential attachment and small-world and make some interesting observations. Periodic injection of early adopters can help in boosting the spread of behaviors and sustain it for a longer period of time. Also, behavior propagation for the classical epidemic model SIRS (Susceptible-Infected-Recovered-Susceptible) does not continue for an infinite period of time as per conventional wisdom. One interesting future direction is to investigate how behavior adoption is affected when number of individuals in a network changes. The affects on behavior adoption when availability of behavior changes with time can also be examined. / Dissertation/Thesis / M.S. Computer Science 2013

Associamos independentemente a cada vértice v de un grafo infinito G um raio de infecção aleatório R_v e definimos um modelo de percolação sujeito às seguintes regras: (1) no tempo zero só a raiz é declarada infectada, (2) um vértice é declarado infectado em um instante t, t>0, se está a uma distância no maximo R_v de algum vértice v previamente infectado, e (3) vértices infectados permanecem infectados para sempre. Dizemos que há sobrevivência em uma realização particular do modelo se o número final de vértices infectados é infinito. Neste trabalho damos condições suficientes sobre o grafo G para a transição de fase deste modelo, estabelecendo limitantes não triviais para o parâmetro crítico quando os raios R_v têm distribuição geometrica de parâmetro 1-p. Além disto, restringindo nosso estudo para o caso das árvores esfericamente simétricas, obtemos um melhor limitante superior para este parâmetro. Finalmente, concluímos que o parâmetro crítico para o modelo nas árvores homogêneas de grau d+1 se comporta assintoticamente como 1/(2d). / We assign independently to each vertex v of an infinite graph G, a random radius of infection R_v and define a percolation model subject to the following rules: (1) at time zero, only the root is declared infected, (2) a vertex is declared infected at time t, t>0, if it is at distance at most R_v of some vertex v previously infected, and (3) infected vertices stay infected forever. We say that there is survival in a particular realization of the model if the final number of infected vertices is infinite. In this work, we give sufficient conditions on the graph G for the phase transition of this model, by stating non-trivial bounds for the critical parameter when the radii have geometrical distribution with parameter 1-p. In addition, restricting our study to the case of the spherically symmetric trees, we obtain an improved upper bound for this critical parameter. Finally, we conclude that the critical parameter for the model on homogeneous trees of degree (d+1) behaves asymptotically as 1/(2d).

Estudamos um sistema de partículas a tempo discreto cuja dinâmica é a seguinte. Considere que no instante inicial sobre cada inteiro não negativo há uma partícula, inicialmente inativa. A partícula da origem é ativada e instantaneamente ativa um conjunto aleatório contíguo de partículas que estão a sua direita. Como regra, no instante seguinte ao que foi ativada, cada partícula ativa realiza esta mesma dinâmica de modo independente de todo o resto. Dizemos que o processo sobrevive se em qualquer momento sempre há ao menos uma partícula ativa. Chamamos este processo de Firework, associando a dinâmica de ativação de uma partícula inativa a uma infecção ou explosão. Nosso interesse é estabelecer se o processo tem probabilidade positiva de sobrevivência e apresentar limites para esta probabilidade. Isto deve ser feito em função da distribuição da variável aleatória que define o raio de ação de uma partícula. Associando o processo de ativação a uma infecção, podemos pensar este modelo como um modelo epidêmico. Consideramos também algumas variações dessa dinâmica. Dentre elas, variantes com partículas distribuídas sobre a semirreta dos reais positivos (nesta vertente, existem condições para as distâncias entre partículas consecutivas) e também com as partículas distribuídas sobre vértices de árvores. Estudamos também para esses casos a transição de fase e probabilidade de sobrevivência. Nesta variante os resultados obtidos são funções da sequência de distribuições dos alcances das explosões e da estrutura dos lugares onde se localizam as partículas. Consideramos também variações do modelo onde cada partícula ao ser ativada, permanece ativa durante um tempo aleatório e nesse período emite explosões que ocorrem em instantes aleatórios. / We studied a discrete time particle system whose dynamic is as follows. Consider that at time zero, on each non-negative integer, there is a particle, initially inactive. A particle which is placed at origin is activated and instantly activates a contiguous random set of particles that is on its right. As a rule, the next moment to what it has been activated, each active particle carries the same behavior independently of the rest. We say that the process survives if the amount of particles activated along the process is infinite. We call this the Firework process, associating the activation dynamic of a particle to an infection or explosion process. Our interest is to establish whether the process has positive probability of survival and to present limits to this probability. This is done according to the distribution random variable that defines the radius of infection of each active particle, Associating the activation process to an infection, we think this model as a model epidemic. We also consider some variations of this dynamic. Among them, variants with particles distributed over the half line (there are conditions for the distances between consecutive particles) and also with particles distributed over the vertices of a tree. We studied phase transitions and the correspondent survival probability. In this variant the results depend on the sequence of probability distributions for the range of the explosions and on the particles displacement. We also consider a variation where each particle after activated, remains active during a random time period emitting explosions that occur in random moments.

Estudamos, através de modelagem matemática, aspectos estacionários, espaciais e temporais relacionados à propagação e controle de doenças infecciosas de transmissão direta por contato pessoa-a-pessoa. Elaboramos modelos matemáticos determinísticos fundamentados no princípio de ação de massas em Epidemiologia, levando em consideração a simetria no número de contatos entre suscetíveis e infectados, o que nos permitiu estimar a taxa per capita de contatos potencialmente infectantes e, por conseguinte, a força de infecção e os possíveis efeitos de diferentes programas de vacinação. O desenvolvimento do modelo de estado estacionário foi feito com base em dados sorológicos de rubéola (Azevedo Neto 1992) para uma população que ainda não havia sido imunizada por meio de vacinação. Analisamos, então, o efeito de três diferentes esquemas de vacinação para a rubéola, nos seguintes intervalos de idade: de 1 a 2 anos, de 7 a 8 anos e de 14 a 15 anos. A incerteza estatística na idade média de infecção foi estimada com o auxílio do método de Monte Carlo e tal metodologia foi aplicada a dados de varicela e hepatite A. Estudamos também o aspecto espacial, com a inclusão da variável distância na formulação de um modelo SIR e análise da influência do alcance de interação entre indivíduos. E, através do estudo da força de infecção em função da idade e do tempo, pudemos analisar, de modo qualitativo, diferentes cenários na evolução temporal de uma doença infecciosa. / We have studied, based on mathematical modelling, stationary, spatial and temporal features related to the propagation and control of directly transmitted infectious diseases through person-to-person contact. We have developed deterministic mathematical models founded on the mass-action principle of Epidemiology, taking into account the symmetry of contacts among susceptible and infectious individuals. Such symmetry enabled us to estimate the potentially infective per capita contact rate and, therefore, the force of infection and the possible effects of different vaccination programmes. The steady state modelling has been based on rubella serological data of a non-immunized population (Azevedo Neto 1992) and we have analysed three different vaccination schemes against rubella in the following age intervals: from 1 to 2 years of age, from 7 to 8 years of age, and from 14 to 15 years of age. The serological data variability has been considered in the estimation of the statistical uncertainty of the average age at infection by means of the Monte Carlo method and we have applied this methodology to varicella and hepatitis A data. The spatial feature in a SIR model has been studied with the analysis of the influence of the interaction range among individuals. We have also studied the force of infection as a function of age and time and we have analysed, in a qualitative way, different situations in the time evolution of an infectious disease.

O comportamento de sistemas epidêmicos é frequentemente descrito de maneira determinística, através do emprego de equações diferenciais ordinárias. Este trabalho visa fornecer uma visão estocástica do problema, traçando um paralelo entre o encontro de indivíduos em uma população e o choque entre partículas de uma reação química. Através dessa abordagem é apresentado o algoritmo de Gillespie, que fornece uma forma simples de simular a evolução de um sistema epidêmico. Fundamentos de processos estocásticos são apresentados para fundamentar uma técnica para a estimação de parâmetros através de dados reais. Apresentamos ainda o modelo de Tau-leaping e o modelo difusivo elaborados através de equações diferenciais estocásticas que são aproximações do modelo proposto por Gillespie. A aplicação dos modelos apresentados é exemplificada através do estudo de dados reais da epidemia de dengue ocorrida no estado do Rio de Janeiro entre os anos de 2012 e 2013. / The behavior of epidemic systems is often described in a deterministic way, through the use of ordinary differential equations. This paper aims to provide a stochastic view of the problem, drawing a parallel between the encounter between individuals in a population and the clash between particles of a chemical reaction. Through this approach is presented the Gillespie algorithm, which provides a simple way to simulate the evolution of an epidemic system. Fundamentals of stochastic process theory are presented to support a technique for estimating parameters through real data. We present the model of Tau-leaping and the diffusive model elaborated by stochastic differential equations that are approximations of the model proposed by Gillespie. The application of the presented models is exemplified through the study of real data of the dengue epidemic occurred in the state of Rio de Janeiro between the years of 2012 and 2013.

Associamos independentemente a cada vértice v de un grafo infinito G um raio de infecção aleatório R_v e definimos um modelo de percolação sujeito às seguintes regras: (1) no tempo zero só a raiz é declarada infectada, (2) um vértice é declarado infectado em um instante t, t>0, se está a uma distância no maximo R_v de algum vértice v previamente infectado, e (3) vértices infectados permanecem infectados para sempre. Dizemos que há sobrevivência em uma realização particular do modelo se o número final de vértices infectados é infinito. Neste trabalho damos condições suficientes sobre o grafo G para a transição de fase deste modelo, estabelecendo limitantes não triviais para o parâmetro crítico quando os raios R_v têm distribuição geometrica de parâmetro 1-p. Além disto, restringindo nosso estudo para o caso das árvores esfericamente simétricas, obtemos um melhor limitante superior para este parâmetro. Finalmente, concluímos que o parâmetro crítico para o modelo nas árvores homogêneas de grau d+1 se comporta assintoticamente como 1/(2d). / We assign independently to each vertex v of an infinite graph G, a random radius of infection R_v and define a percolation model subject to the following rules: (1) at time zero, only the root is declared infected, (2) a vertex is declared infected at time t, t>0, if it is at distance at most R_v of some vertex v previously infected, and (3) infected vertices stay infected forever. We say that there is survival in a particular realization of the model if the final number of infected vertices is infinite. In this work, we give sufficient conditions on the graph G for the phase transition of this model, by stating non-trivial bounds for the critical parameter when the radii have geometrical distribution with parameter 1-p. In addition, restricting our study to the case of the spherically symmetric trees, we obtain an improved upper bound for this critical parameter. Finally, we conclude that the critical parameter for the model on homogeneous trees of degree (d+1) behaves asymptotically as 1/(2d).

Estudamos um sistema de partículas a tempo discreto cuja dinâmica é a seguinte. Considere que no instante inicial sobre cada inteiro não negativo há uma partícula, inicialmente inativa. A partícula da origem é ativada e instantaneamente ativa um conjunto aleatório contíguo de partículas que estão a sua direita. Como regra, no instante seguinte ao que foi ativada, cada partícula ativa realiza esta mesma dinâmica de modo independente de todo o resto. Dizemos que o processo sobrevive se em qualquer momento sempre há ao menos uma partícula ativa. Chamamos este processo de Firework, associando a dinâmica de ativação de uma partícula inativa a uma infecção ou explosão. Nosso interesse é estabelecer se o processo tem probabilidade positiva de sobrevivência e apresentar limites para esta probabilidade. Isto deve ser feito em função da distribuição da variável aleatória que define o raio de ação de uma partícula. Associando o processo de ativação a uma infecção, podemos pensar este modelo como um modelo epidêmico. Consideramos também algumas variações dessa dinâmica. Dentre elas, variantes com partículas distribuídas sobre a semirreta dos reais positivos (nesta vertente, existem condições para as distâncias entre partículas consecutivas) e também com as partículas distribuídas sobre vértices de árvores. Estudamos também para esses casos a transição de fase e probabilidade de sobrevivência. Nesta variante os resultados obtidos são funções da sequência de distribuições dos alcances das explosões e da estrutura dos lugares onde se localizam as partículas. Consideramos também variações do modelo onde cada partícula ao ser ativada, permanece ativa durante um tempo aleatório e nesse período emite explosões que ocorrem em instantes aleatórios. / We studied a discrete time particle system whose dynamic is as follows. Consider that at time zero, on each non-negative integer, there is a particle, initially inactive. A particle which is placed at origin is activated and instantly activates a contiguous random set of particles that is on its right. As a rule, the next moment to what it has been activated, each active particle carries the same behavior independently of the rest. We say that the process survives if the amount of particles activated along the process is infinite. We call this the Firework process, associating the activation dynamic of a particle to an infection or explosion process. Our interest is to establish whether the process has positive probability of survival and to present limits to this probability. This is done according to the distribution random variable that defines the radius of infection of each active particle, Associating the activation process to an infection, we think this model as a model epidemic. We also consider some variations of this dynamic. Among them, variants with particles distributed over the half line (there are conditions for the distances between consecutive particles) and also with particles distributed over the vertices of a tree. We studied phase transitions and the correspondent survival probability. In this variant the results depend on the sequence of probability distributions for the range of the explosions and on the particles displacement. We also consider a variation where each particle after activated, remains active during a random time period emitting explosions that occur in random moments.

abstract: Using a simple $SI$ infection model, I uncover the overall dynamics of the system and how they depend on the incidence function. I consider both an epidemic and endemic perspective of the model, but in both cases, three classes of incidence functions are identified. In the epidemic form, power incidences, where the infective portion $I^p$ has $p\in(0,1)$, cause unconditional host extinction, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The case of non-extinction in upper density-dependent incidences extends to the case where a latent period is included. Using data from experiments with rhanavirus and salamanders, maximum likelihood estimates are applied to the data. With these estimates, I generate the corrected Akaike information criteria, which reward a low likelihood and punish the use of more parameters. This generates the Akaike weight, which is used to fit parameters to the data, and determine which incidence functions fit the data the best. From an endemic perspective, I observe that power incidences cause initial condition dependent host extinction for some parameter constellations and global stability for others, homogeneous incidences have host extinction for certain parameter constellations and host survival for others, and upper density-dependent incidences never cause host extinction. The dynamics when the incidence function is homogeneous are deeply explored. I expand the endemic considerations in the homogeneous case by adding a predator into the model. Using persistence theory, I show the conditions for the persistence of each of the predator, prey, and parasite species. Potential dynamics of the system include parasite mediated persistence of the predator, survival of the ecosystem at high initial predator levels and ecosystem collapse at low initial predator levels, persistence of all three species, and much more. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2017