The applications of differential equations are many. However, many differential equations modelling real-world scenarios are very complex and it can be of great difficulty to find an exact solution if one even exists. Thus, it is of importance to be able to approximate solutions of differential equations. Here, a method using neural networks is explored and its performance is compared to that of a numerical method. To illustrate the method, two first order, two second order and two first order systems of ordinary differential equations are explored. The systems are the Lotka-Volterra system and the SEIR (Susceptible, Exposed, Infected, Removed) epidemiological model. The first four examples have exact solutions to compare to and the observations are then used as a basis when discussing the results of the systems. The results of the thesis show that while the neural network method takes longer to deliver an approximation, it continuously gives better approximations than the implicit Euler method used for comparison. The main contribution of this thesis is the comparison done of the performances of the neural network method and the implicit Euler method. / Det finns många användningsområden för differentialekvationer. Däremot är många differentialekvationer som modellerar verkligheten komplexa och det kan vara svårt, om inte omöjligt, att hitta en exakt lösning. På grund av detta är det viktigt att ha metoder som kan approximera lösningar till differentialekvationer. Därför undersöks här en metod som använder sig av neurala nätverk. Dess resultat blir sedan jämförda med en numerisk metod. För att illustrera metoden presenteras två ekvationer av första ordningen, två ekvationer av andra ordningen och två system av differentialekvationer. Systemen som undersöks är Lotka-Volterra ekvationerna samt SEIR (Susceptible, Exposed, Infected, Removed) modellen. De första fyra exemplen som undersöks har exakta lösningar att jämföra med och dessa observationer används sedan vid diskussionerna gällande systemen. Resultaten visar att medan metoden som använder neurala nätverkar tar längre tid att exekvera, så ger metoden bättre approximationer än den implicita Euler metoden som användes som jämförelse. Det huvudsakliga bidraget med det här examensarbetet är jämförelsen av hur de två metoderna presterar.

Millions of users world-wide are sharing content using Peer-to-Peer (P2P) networks, such as Skype and Bit Torrent. While such new innovations undoubtedly bring benefits, there are nevertheless some associated threats. One of the main hazards is that P2P worms can penetrate the network, even from a single node and then spread rapidly. Understanding the propagation process of such worms has always been a challenge for researchers. Different techniques, such as simulations and analytical models, have been adopted in the literature. While simulations provide results for specific input parameter values, analytical models are rather more general and potentially cover the whole spectrum of given parameter values. Many attempts have been made to model the worm propagation process in P2P networks. However, the reported analytical models to-date have failed to cover the whole spectrum of all relevant parameters and have therefore resulted in high false-positives. This consequently affects the immunization and mitigation strategies that are adopted to cope with an outbreak of worms. The first key contribution of this thesis is the development of a susceptible, exposed, infectious, and Recovered (SEIR) analytical model for the worm propagation process in a P2P network, taking into account different factors such as the configuration diversity of nodes, user behaviour and the infection time-lag. These factors have not been considered in an integrated form previously and have been either ignored or partially addressed in state-of-the-art analytical models. Our proposed SEIR analytical model holistically integrates, for the first time, these key factors in order to capture a more realistic representation of the whole worm propagation process. The second key contribution is the extension of the proposed SEIR model to the mobile M-SEIR model by investigating and incorporating the role of node mobility, the size of the worm and the bandwidth of wireless links in the worm propagation process in mobile P2P networks. The model was designed to be flexible and applicable to both wired and wireless nodes. The third contribution is the exploitation of a promising modelling paradigm, Agent-based Modelling (ABM), in the P2P worm modelling context. Specifically, to exploit the synergies between ABM and P2P, an integrated ABM-Based worm propagation model has been built and trialled in this research for the first time. The introduced model combines the implementation of common, complex P2P protocols, such as Gnutella and GIA, along with the aforementioned analytical models. Moreover, a comparative evaluation between ABM and conventional modelling tools has been carried out, to demonstrate the key benefits of ease of real-time analysis and visualisation. As a fourth contribution, the research was further extended by utilizing the proposed SEIR model to examine and evaluate a real-world data set on one of the most recent worms, namely, the Conficker worm. Verification of the model was achieved using ABM and conventional tools and by then comparing the results on the same data set with those derived from developed benchmark models. Finally, the research concludes that the worm propagation process is to a great extent affected by different factors such as configuration diversity, user-behaviour, the infection time lag and the mobility of nodes. It was found that the infection propagation values derived from state-of-the-art mathematical models are hypothetical and do not actually reflect real-world values. In summary, our comparative research study has shown that infection propagation can be reduced due to the natural immunity against worms that can be provided by a holistic exploitation of the range of factors proposed in this work.

The last two decades have seen several large-scale epidemics of international impact, including human, animal and plant epidemics. Policy makers face health challenges that require epidemic predictions based on limited information. There is therefore a pressing need to construct models that allow us to frame all available information to predict an emerging outbreak and to control it in a timely manner. The aim of this thesis is to develop an early-warning modelling approach that can predict emerging disease outbreaks. Based on Bayesian techniques ideally suited to combine information from different sources into a single modelling and estimation framework, I developed a suite of approaches to epidemiological data that can deal with data from different sources and of varying quality. The SEIR model, particle filter algorithm and a number of influenza-related datasets were utilised to examine various models and methodologies to predict influenza outbreaks. The data included a combination of consultations and diagnosed influenza-like illness (ILI) cases for five influenza seasons. I showed that for the pandemic season, different proxies lead to similar behaviour of the effective reproduction number. For influenza datasets, there exists a strong relationship between consultations and diagnosed datasets, especially when considering time-dependent models. Individual parameters for different influenza seasons provided similar values, thereby offering an opportunity to utilise such information in future outbreaks. Moreover, my findings showed that when the temperature drops below 14°C, this triggers the first substantial rise in the number of ILI cases, highlighting that temperature data is an important signal to trigger the start of the influenza epidemic. Further probing was carried out among Maltese citizens and estimates on the under-reporting rate of the seasonal influenza were established. Based on these findings, a new epidemiological model and framework were developed, providing accurate real-time forecasts with a clear early warning signal to the influenza outbreak. This research utilised a combination of novel data sources to predict influenza outbreaks. Such information is beneficial for health authorities to plan health strategies and control epidemics.

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.

No description available.

Dans ce mémoire, nous considérons un modèle épidémiologique multigroupe dans une population hétérogène, pour décrire la situation de l’épidémie de la COVID-19 dans une résidence pour personnes âgées. L’hétérogénéité liée ici à l’âge reflète une transmission élevée dûe à des interactions accrues, et un taux de mortalité plus élevé chez les personnes âgées. Du point de vue mathématique, nous obtenons un modèle SEIR multigroupe d’équations intégro-différentielles dans lequel nous considérons une distribution générale de la période infectieuse. Nous utilisons la méthode des fonctions de Lyapunov et une approche de la théorie des graphes pour déterminer le rôle du nombre de reproduction de base \(\mathcal{R}_0\) : l’état d’équilibre sans maladie est globalement asymptotiquement stable et l’épidémie s’éteint dans les deux groupes lorsque \(\mathcal{R}_0 \leq 1\), par contre elle persiste et l’état d’équilibre endémique est globalement asymptotiquement stable lorsque \(\mathcal{R}_0>1\). Les simulations numériques illustrent l’impact des stratégies de contrôle de la santé publique. / In this thesis, we consider a multiple group epidemiological model in a heterogeneous population to describe COVID-19 outbreaks in an elderly residential population. Age-based heterogeneity reflects higher transmission with enhanced interactions, and higher fatality rates in the elderly. Mathematically, we analyse a SEIR model in the form of a system of integro-differential equations with general distribution function for the infectious period. Lyapunov functions and graph-theoretical methods are employed to establish the role played by the basic reproduction ratio \(\mathcal{R}_0\) : global asymptotic stability of the disease-free equilibrium and no sustained outbreak when \(\mathcal{R}_0 \leq 1\), as opposed to persistent outbreak and globally asymptotic endemic equilibrium when \(\mathcal{R}_0>1\). Numerical simulations are presented to illustrate public health control strategies.