yes / Peer-to-Peer (P2P) network is increasingly becoming the most important means of trading content throughout the last years due to the constant evolvement of the cyber world. This popularity made the P2P network susceptible to the spread of malware. The detection of the cause of malware propagation is now critical to the survival of P2P networks. This paper offers a review of the current relevant mathematical propagation models that have been proposed to date to predict the propagation behavior of a malware in a P2P network. We analyzed the models proposed by researchers and experts in the field by evaluating their limitations and a possible alternative for improving the analysis of the expected behavior of a malware spread.

A teoria de redes complexas se consolidou nos últimos anos, graças ao seu potencial como ferramenta versátil no estudo de diversos sistemas discretos. É possível enumerar aplicações em áreas tão distintas como engenharia, sociologia, computação, linguística e biologia. Tem merecido atenção, por exemplo, o estudo da organização estrutural do cérebro, tanto em nível microscópico (em nível de neurônios) como regional (regiões corticais). Acredita-se que tal organização visa otimizar a dinâmica, favorecendo processos como sincronização e processamento paralelo. Estrutura e funcionamento, portanto, estão relacionados. Tal relação é abordada pela teoria de redes complexas nos mais diversos sistemas, sendo possivelmente seu principal objeto de estudo. Neste trabalho exploramos as relações entre aspectos estruturais de redes neuronais e corticais e a atividade nas mesmas. Especificamente, estudamos como a interconectividade entre o córtex e o tálamo pode interferir em estados de ativação do último, considerando-se o sistema tálamo-cortical do gato bem como alguns modelos para geração de rede encontrados na literatura. Também abordamos a relação entre a morfologia individual de neurônios e a conectividade em redes neuronais, e consequentemente o impacto da forma neuronal em dinâmicas atuando sobre tais redes e a eficiência das mesmas no transporte de informação. Como tal eficiência pode ter como consequência a facilitação de processos maléficos às redes, como por exemplo, ataques causados por vírus neurotrópicos, também exploramos possíveis correlações entre características individuais dos elementos que formam as redes complexas e danos causados por processos infecciosos iniciados nos mesmos. / Complex network theory has been consolidated along the last years, owing to its potential as a versatile framework for the study of diverse discrete systems. It is possible to enumerate applications in fields as distinct as Engineering, Sociology, Computing, Linguistics and Biology, to name a few. For instance, the study of the structural organization of the brain at the microscopic level (neurons), as well as at regional level (cortical areas), has deserved attention. It is believed that such organization aims at optimizing the dynamics, supporting processes like synchronization and parallel processing. Structure and functioning are thus interrelated. Such relation has been addressed by complex network theory in diverse systems, possibly being its main subject. In this thesis we explore the relations between structural aspects and the activity in cortical and neuronal networks. Specifically, we study how the interconnectivity between the cortex and thalamus can interfere in activation states of the latter, taking into consideration the thalamocortical system of the cat, along with networks generated through models found in literature. We also address the relation between the individual morphology of the neurons and the connectivity in neuronal networks, and consequently the effect of the neuronal shape on dynamic processes actuating over such networks and on their efficiency on information transport. As such efficiency can consequently facilitate prejudicial processes on the networks, e.g. attacks promoted by neurotropic viruses, we also explore possible correlations between individual characteristics of the elements forming such systems and the damage caused by infectious processes started at these elements.

A teoria de redes complexas se consolidou nos últimos anos, graças ao seu potencial como ferramenta versátil no estudo de diversos sistemas discretos. É possível enumerar aplicações em áreas tão distintas como engenharia, sociologia, computação, linguística e biologia. Tem merecido atenção, por exemplo, o estudo da organização estrutural do cérebro, tanto em nível microscópico (em nível de neurônios) como regional (regiões corticais). Acredita-se que tal organização visa otimizar a dinâmica, favorecendo processos como sincronização e processamento paralelo. Estrutura e funcionamento, portanto, estão relacionados. Tal relação é abordada pela teoria de redes complexas nos mais diversos sistemas, sendo possivelmente seu principal objeto de estudo. Neste trabalho exploramos as relações entre aspectos estruturais de redes neuronais e corticais e a atividade nas mesmas. Especificamente, estudamos como a interconectividade entre o córtex e o tálamo pode interferir em estados de ativação do último, considerando-se o sistema tálamo-cortical do gato bem como alguns modelos para geração de rede encontrados na literatura. Também abordamos a relação entre a morfologia individual de neurônios e a conectividade em redes neuronais, e consequentemente o impacto da forma neuronal em dinâmicas atuando sobre tais redes e a eficiência das mesmas no transporte de informação. Como tal eficiência pode ter como consequência a facilitação de processos maléficos às redes, como por exemplo, ataques causados por vírus neurotrópicos, também exploramos possíveis correlações entre características individuais dos elementos que formam as redes complexas e danos causados por processos infecciosos iniciados nos mesmos. / Complex network theory has been consolidated along the last years, owing to its potential as a versatile framework for the study of diverse discrete systems. It is possible to enumerate applications in fields as distinct as Engineering, Sociology, Computing, Linguistics and Biology, to name a few. For instance, the study of the structural organization of the brain at the microscopic level (neurons), as well as at regional level (cortical areas), has deserved attention. It is believed that such organization aims at optimizing the dynamics, supporting processes like synchronization and parallel processing. Structure and functioning are thus interrelated. Such relation has been addressed by complex network theory in diverse systems, possibly being its main subject. In this thesis we explore the relations between structural aspects and the activity in cortical and neuronal networks. Specifically, we study how the interconnectivity between the cortex and thalamus can interfere in activation states of the latter, taking into consideration the thalamocortical system of the cat, along with networks generated through models found in literature. We also address the relation between the individual morphology of the neurons and the connectivity in neuronal networks, and consequently the effect of the neuronal shape on dynamic processes actuating over such networks and on their efficiency on information transport. As such efficiency can consequently facilitate prejudicial processes on the networks, e.g. attacks promoted by neurotropic viruses, we also explore possible correlations between individual characteristics of the elements forming such systems and the damage caused by infectious processes started at these elements.

Malaria is a preventable and treatable blood-borne disease whose complications can be fatal. Although many interventions exist in order to reduce the impacts of malaria, the optimal method of distributing these interventions in a geographical area with limited resources must be determined. This thesis refines a model that uses an integer linear program and a compartmental model of epidemiology called an SIR model of ordinary differential equations. The objective of the model is to find an intervention strategy over multiple time steps and multiple geographic regions that minimizes the number of days people spend infected with malaria. In this paper, we refine the resolution of the model and conduct sensitivity analysis on its parameter values.

Epidemic models help researchers understand and predict the nature of a potential epidemic. This study analyzes and improves network evolution technology that evolves contact networks so that simulated epidemics on the network mimic a specified epidemic pattern. The evolutionary algorithm incorporates the novel recentering-restarting algorithm, which is adopted into the optimizer to allow for efficient search of the space of networks. It also implements the toggle-delete representation which allows for broader search of solution space. Then, a diffusion character based method is used for analyzing the contact networks. A comparison of simulated epidemics that result from changing patient zero for a single contact network is performed. It is found that the location of patient zero is important for the behaviour of an epidemic. The social fabric representation is invented and then tested for parameter choices. The response to vaccination strategies (including ring vaccination) is then tested by incorporating them into the epidemic simulations. / Ontario Graduate Scholarship (OGS), Natural Sciences and Engineering Research Council of Canada (NSERC)

La fièvre de la Vallée du Rift (FVR) est une zoonose due à un virus transmissible par une large variété de vecteurs. Au cours des quinze dernières années, elle a rendu malades des dizaines de milliers de personnes, entrainé des centaines de décès humains et provoqué la mort de plus de 100 000 ruminants domestiques en Afrique et dans la péninsule arabique. Suite à la découverte de la présence du virus à Mayotte, les autorités sanitaires, le Cirad (Centre de coopération internationale en recherche agronomique pour le développement) et les éleveurs ont créé en 2009 un système de surveillance en santé animale, le SESAM. L‘objectif de cette thèse était de faire un état des lieux de la situation épidémiologique de la FVR chez les ruminants à Mayotte en 2012-2013, d’évaluer les perspectives de persistance de la maladie et d’étudier les perceptions et les priorités des éleveurs vis-à-vis de cette maladie. En 2012, le fonctionnement du dispositif de surveillance de la FVR a été révisé avec un renouvellement partiel du groupe d’éleveurs sentinelles et un renforcement de la surveillance événementielle basée sur la recherche systématique de la FVR en cas d’avortement ou de mortalité anormale. Une baisse continue de la séroprévalence de 2010 à 2013 a été observée. En 2012-2013, le taux d’incidence est resté stable avec environ deux séroconversions pour 100 animaux-ans. Ce taux est très inférieur au taux d’incidence de 18% observé en 2011. Le virus de la FVR n’a été détecté dans aucun des prélèvements réalisés dans le cadre de la déclaration officielle des avortements chez les ruminants (n=41). Des critères d’évaluation de la performance du dispositif de surveillance ont été proposés et ont mis en évidence une amélioration de la qualité des données entre 2010 et 2013. Un modèle dynamique de type SIR a permis d’estimer le niveau de transmission hôte-vecteur attendu qui expliquerait au mieux la séroprévalence observée de 2008 à 2013 à Mayotte. La probabilité de transmission hôte-vecteur estimée par ce modèle est cinq fois plus faible qu’attendue. Dans ces conditions, la probabilité de persistance de la FVR, en l’absence de nouvelle introduction d’animaux virémiques reste inférieure à 10%, cinq ans après l’arrivée du virus. Enfin, la priorisation des problèmes sanitaires à Mayotte par une démarche d’épidémiologie participative et des réunions d’éleveurs a mis en évidence que les cinq problèmes spontanément rapportés comme les plus importants chez les bovins sont dans l’ordre le charbon symptomatique, les tiques, le syndrome « bavite-fièvre-grippe», le syndrome « dermatophilose-boutons » et la diarrhée. La FVR est absente de ce classement mais les avortements arrivent en 9ème position et à la première selon le critère de « risque ». La surveillance, et les approches interdisciplinaires de modélisation et de sciences humaines doivent être poursuivies pour évaluer plus précisément le risque de réémergence de la FVR à Mayotte et anticiper les réponses à y apporter. La surveillance doit aussi s’adapter aux attentes des éleveurs et développer l’approche syndromique ainsi que s’intégrer pleinement aux dispositifs régionaux et nationaux. / Rift Valley fever (RVF) is a vector-borne zoonosis, with a wide variety of potential competent vectors. During the last fifteen years, RVF caused tens of thousands of human cases, hundreds of human deaths and more than 100,000 domestic ruminant deaths in Africa and in the Arabic peninsula. After the first detection of RVF in Mayotte, an animal health surveillance network has been created in 2009, namely the SESAM. The aim of this PhD was to assess the epidemiological situation of Mayotte toward RVF in 2012-2013, to assess the persistence probability of RVF in the territory and to document farmers’ perceptions and health priorities. In 2012, RVF surveillance has been revised in order to partially renew the pool of the sentinel herds. The passive surveillance with systematic RVF detection assay on abortion and abnormal mortality was enforced. A continuous decrease in the seroprevalence was observed based on 2010-2013 data. During 2012-2013, a steady incidence rate of about 2 seroconversions per 100 animal-year was observed. This rate is much lower than the previous assessment of 18% during 2011. RVF virus was not detected in any of the 41 declared abortion cases. Surveillance performance criteria were discussed and assessed showing mainly an increase in data quality between 2010 and 2013. A SIR dynamic model was built based on vector knowledge and observed seroprevalence in Mayotte from 2008 to 2013. Estimated host-vector transmission rate was fivefold lower than expected according to current literature. Persistence probability, without reintroduction of viremic animals, was predicted to be 10% five years after virus introduction. Finally, health problem prioritization in Mayotte was studied through focus groups, gathering 164 farmers. The five main issues stated were: blackleg, ticks, a respiratory « fever/flu-like» syndrome, a dermatologic syndrome and diarrhoea. RVF was absent from the priority problems list in which abortions were ranked 9th. However abortions came first when « risk » criteria was taken into account, raising issues of cultural risk perceptions. Further surveillance and research using transdisciplinary approaches that mix mathematical modelling and the humanities should be continued with the view to assess more precisely reemergence probability in Mayotte and anticipate mitigating measures. Surveillance and research should also align with farmers’ expectations. The development of syndromic surveillance (abortion, death, etc.) requires strengthening surveillance network and identification data quality. Mayotte surveillance components should also be fully integrated within the regional and national schemes.

In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graph-theoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phase-lock oscillations.

This thesis is comprised of three parts. The first part describes a novel framework for computing importance measures on graph vertices. The concept of a D-spectrum is introduced, based on vertex ranks within certain chains of nested sub-graphs. We show that the D- spectrum integrates the degree distribution and coreness information of the graph as two particular such chains. We prove that these spectra are realized as fixed points of certain monotone and contractive SDSs we call t-systems. Finally, we give a vertex deletion algorithm that efficiently computes D-spectra, and we illustrate their correlation with stochastic SIR-processes on real world networks. The second part deals with the topology of the intersection nerve for a bi-secondary structure, and its singular homology. A bi-secondary structure R, is a combinatorial object that can be viewed as a collection of cycles (loops) of certain at most tetravalent planar graphs. Bi-secondary structures arise naturally in the study of RNA riboswitches - molecules that have an MFE binary structural degeneracy. We prove that this loop nerve complex has a euclidean 3-space embedding characterized solely by H2(R), its second homology group. We show that this group is the only non-trivial one in the sequence and furthermore it is free abelian. The third part further describes the features of the loop nerve. We identify certain disjoint objects in the structure of R which we call crossing components (CC). These are non-trivial connected components of a graph that captures a particular non-planar embedding of R. We show that each CC contributes a unique generator to H2(R) and thus the total number of these crossing components in fact equals the rank of the second homology group. / Doctor of Philosophy / This Thesis is divided into three parts. The first part describes a novel mathematical framework for decomposing a real world network into layers. A network is comprised of interconnected nodes and can model anything from transportation of goods to the way the internet is organized. Two key numbers describe the local and global features of a network: the number of neighbors, and the number of neighbors in a certain layer, a node has. Our work shows that there are other numbers in-between the two, that better characterize a node. We also give explicit means of computing them. Finally, we show that these numbers are connected to the way information spreads on the network, uncovering a relation between the network’s structure and dynamics on said network. The last two parts of the thesis have a common theme and study the same mathematical object. In the first part of the two, we provide a new model for the way riboswtiches organize themselves. Riboswitches, are RNA molecules within a cell, that can take two mutually opposite conformations, depending on what function they need to perform within said cell. They are important from an evolutionary standpoint and are actively studied within that context, usually being modeled as networks. Our model captures the shapes of the two possible conformations, and encodes it within a mathematical object called a topological space. Once this is done, we prove that certain numbers that are attached to all topological spaces carry specific values for riboswitches. Namely, we show that the shapes of the two possible conformations for a riboswich are always characterized by a single integer. In the last part of the Thesis we identify what exactly in the structure of riboswitches contributes to this number being large or small. We prove that the more tangled the two conformations are, the larger the number. We can thus conclude that this number is directly proportional to how complex the riboswitch is.

Louping ill virus (LIV) is a tick borne disease that causes mortality in red grouse, an economically important game bird of British uplands. The aim of this thesis is to extend previously published models of LIV , to consider the potential impact of different management strategies. In addition a new route of infection and the seasonal biology of both grouse and ticks will be explored. Grouse chicks are known to eat ticks as part of their diet in the first three weeks of life which may contribute to virus persistence if chicks consume infected ticks. This novel route of infection is incorporated in to the model which predicts that ingestion increases the range of host densities for which the virus is able to persist. The ingestion of ticks by grouse also reduces the tick population so that for low host densities the ingestion of ticks by grouse reduces the tick population so virus cannot persist. The model is adapted to take account of the seasonal biology of grouse and ticks. Although the temporal predictions of the seasonal models show some differences the addition of seasonality does not alter the model predictions of when LIV is likely to persist at different grouse and deer densities. Consequently seasonality is felt to be unimportant when considering management strategies. The treatment of sheep with acaricide in an attempt to reduce the tick population on a grouse moor is currently being trialled in Scotland. We use a model to predict the likely effect of this strategy at different deer densities. The number of ticks found attached to sheep varies so we consider the effect of tick attachment rates as well as acaricide efficacy. Although we predict that acaricide treated sheep can reduce the tick population and therefore LIV in grouse in some circumstances the treatment is less effective in the presence of deer. Consequently we use a model to make theoretical predictions of the effectiveness of acaricide treated deer as a control strategy for reducing LIV in red grouse. The effect of culling deer on LIV in grouse is also modelled and contrasted with the effect of acaricide use. It is predicted that acaricide treatment of deer could be highly effective, particularly if the deer density is first reduced by culling. Finally we considered the direct treatment of red grouse with acaricide. Female grouse can be given an acaricidal leg band which protects her directly and indirectly protects her chicks as they acquire some acaricide whilst brooding. Trials have suggested this can reduce tick burdens for individuals. We use the model to determine the potential effect that treating individual broods may have on the whole grouse population. The model predictions suggest that unless acaricide efficacy on chicks is high and long lasting treating individual broods is unlikely to reduce LIV in the whole population but will still provide some benefit for the individuals. The effectiveness of treatment is reduced by higher deer densities. The success of the management strategies considered in this thesis appear to be restricted by the presence of deer. It may therefore be that a combination of treatments including the treatment of deer may be of the greatest benefit to the grouse population.

Classical Continuous System Simulation (CSS) is restricted to modelling continuous flows, and therefore, cannot correctly realise a conceptual model with discrete objects. The development of Full Potential CSS solves this problem by (1) handling discrete quantities as discrete and continuous matter as continuous, (2) preserving the sojourn time distribution of a stage, (3) implementing attributes correctly, and (4) describing different types of uncertainties in a proper way. In order to apply Full Potential CSS a new software, StochSD, has been developed. This thesis evaluates StochSD's ability to model Full Potential CSS, where the points 1-4 above are included. As a test model a well-defined conceptual epidemic model, which includes all aspects of Full Potential CSS, was chosen. The study was performed by starting with a classical SIR model and then stepwise add the different aspects of the Conceptual Model. The effects of each step were demonstrated in terms of size and duration of the epidemic. Finally, the conceptual model was also realised as an Agent Based Model (ABM). The results from 10 000 replications each of the CSS and ABM models were compared and no statistical differences could be confirmed. The conclusion is that StochSD passed the evaluation.