Statistical Analysis of an SEIR Epidemic Model

This thesis was focussed on statistical analysis methods and proposes the use of Bayesian inference to extract information contained in experimental data by estimating Ebola model parameters. The model is a system of differential equations expressing the behavior and dynamics of Ebola. Two sets of data (onset and death data) were both used to estimate parameters, which has not been done by previous researchers in (Chowell, 2004). To be able to use both data, a new version of the model has been built. Model parameters have been estimated and then used to calculate the basic reproduction number and to study the disease-free equilibrium. Estimates of the parameters were useful to determine how well the model fits the data and how good estimates were, in terms of the information they provided about the possible relationship between variables. The solution showed that Ebola model fits the observed onset data at 98.95% and the observed death data at 93.6%. Since Bayesian inference can not be performed analytically, the Markov chain Monte Carlo approach has been used to generate samples from the posterior distribution over parameters. Samples have been used to check the accuracy of the model and other characteristics of the target posteriors.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.

Optimización de la producción de la fase de avance en el túnel de Seiró (Ourense)

Este proyecto desarrolla una metodología de toma de datos y análisis para la optimización de la producción de la fase de avance de construcción del túnel de Seiró. Este túnel forma parte de las obras de construcción de la línea ferroviaria de alta velocidad corredor norte-noroeste a su paso por la provincia de Ourense. Mediante el análisis de diversos parámetros del diseño de la voladura, en la construcción de un túnel de 73 m2 de sección, en roca granítica, se pretende optimizar el tiempo de ciclo de excavación. Se estudia la variación de la fragmentación, el contorno y el avance obtenido y la influencia sobre estos parámetros del esquema de tiro y el tipo de cuele empleados y sus repercusiones en el tiempo de ciclo. El resultado del análisis concluye que el esquema de tiro óptimo, entre los usados, está formado por 112 barrenos de 51 mm de diámetro, con un cuele de 4 barrenos vacíos, que da lugar a mejores resultados de avance, contorno y fragmentación sin afectar negativamente al tiempo de ciclo. ABSTRACT This project develops a methodology for excavation cycle optimization by collecting and analyzing operational data for Seiró Tunnel construction site. This tunnel is included in construction project of high speed railway, located in N-NW direction, in Ourense province. Analyzing several blasting design parameters from the construction of a 73 m2 tunnel, in granitic rock, allows the optimization of cycle time. Variations in fragmentation, contour and advance length and influence about those parameters from blasting design und cut type used and its impact in the cycle time. Analysis results show up the best blasting design, among used, is formed of 112 holes 51 mm diameter, and a parallel cut with 4 empty holes. This configuration achieves the best fragmentation, contour and advance length without negative effects in time cycle

Stochastic effects in a seasonally forced epidemic model

The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity, is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effects such as switching between coexisting attractors induced by stochasticity often mentioned in the literature as playing an important role in the dynamics of childhood infectious diseases are also investigated. The main conclusion is that stochastic amplification, rather than these effects, is the key ingredient to understand the observed incidence patterns.

Numerical Simulation of Stochastic Di erential Equations

Stochastic differential equation (SDE) is a differential equation in which some of the terms and its solution are stochastic processes. SDEs play a central role in modeling physical systems like finance, Biology, Engineering, to mention some. In modeling process, the computation of the trajectories (sample paths) of solutions to SDEs is very important. However, the exact solution to a SDE is generally difficult to obtain due to non-differentiability character of realizations of the Brownian motion. There exist approximation methods of solutions of SDE. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial, Biology, physical, environmental systems. This Masters' thesis is an introduction and survey of numerical solution methods for stochastic differential equations. Standard numerical methods, local linearization methods and filtering methods are well described. We compute the root mean square errors for each method from which we propose a better numerical scheme. Stochastic differential equations can be formulated from a given ordinary differential equations. In this thesis, we describe two kind of formulations: parametric and non-parametric techniques. The formulation is based on epidemiological SEIR model. This methods have a tendency of increasing parameters in the constructed SDEs, hence, it requires more data. We compare the two techniques numerically.

Modelo de transmissão da dengue com competição larval uniforme

A dengue representa um sério problema de saúde pública no Brasil, que apresenta condições climáticas favoráveis ao desenvolvimento e proliferação do Aedes aegypti, vetor transmissor da doeça. O mosquito Aedes aegypti passa por diferentes estágios de desenvolvi- mento com características distintas, logo, para uma descrição mais aproximada da história de vida desta espécie e, consequentemente, do comportamento da epidemia, considera-se neste trabalho um modelo com distribuição etária, fundamental para a determinação das propriedades de estabilidade de populações que têm fases distintas de desenvolvimento. O modelo com distribuição etária para a população de mosquitos é descrito por um conjunto de equações diferenciais ordinárias com retardo de difícil análise e implementação. Inicialmente, apresenta-se o modelo SEIR com dinâmica vital, onde a população de mosquitos estabiliza rapidamente e, após, incorpora-se a ele um modelo com competição larval uniforme para população de insetos, com distribuição etária, o que provoca um período de instabilidade nas populações de larvas e adultos, estágios de desenvolvimento considerados para o vetor. A análise realizada investiga as consequências que este período de instabilidade provoca na evolução da epidemia.

Un modello matematico in epidemologia

Fornisce una breve trattazione di due tipi di modelli matematici applicabili nel campo dell'epidemologia, prendendo spunto da un articolo del biologo matematico A. Korobeinikov, "Lyapunov function and global properties for SEIR and SEIS epidemic models".

Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa

The 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. To better understand the spread of infection in the affected countries, it is crucial to know the number of secondary cases generated by an infected index case in the absence and presence of control measures, i.e., the basic and effective reproduction number. In this study, I describe the EBOV epidemic using an SEIR (susceptible-exposed-infectious-recovered) model and fit the model to the most recent reported data of infected cases and deaths in Guinea, Sierra Leone and Liberia. The maximum likelihood estimates of the basic reproduction number are 1.51 (95% confidence interval [CI]: 1.50-1.52) for Guinea, 2.53 (95% CI: 2.41-2.67) for Sierra Leone and 1.59 (95% CI: 1.57-1.60) for Liberia. The model indicates that in Guinea and Sierra Leone the effective reproduction number might have dropped to around unity by the end of May and July 2014, respectively. In Liberia, however, the model estimates no decline in the effective reproduction number by end-August 2014. This suggests that control efforts in Liberia need to be improved substantially in order to stop the current outbreak.

(Table T1) Magnetic properties of MORB from ODP Leg 187 sites

The core samples of mid-ocean-ridge basalts (including Indian and Pacific type) recovered from the Southeast Indian Ridge (SEIR) area near the Australian Antarctic Discordance during Ocean Drilling Program Leg 187 were studied using rock magnetism, mineralogy, and petrography methods. On the basis of thermomagnetic analyses and low-temperature magnetometry, the dominant magnetic carrier in most of the basalt samples (pillow basalts) is characterized as titanomaghemite, which presumably formed by low-temperature oxidation of primary titanomagnetite. Some samples from unaltered massive basalts contain nearly unoxidized titanomagnetite as the main magnetic mineral. A metadiabase sample showing greenschist facies metamorphism contains magnetic minerals dominated by magnetite. The pillow basalts contain titanomaghemite ranging from stable single-domain to pseudosingle-domain (PSD) grains, and the majority are characterized by a single stable component of remanence. The massive basalts show hysteresis features of larger PSD grains and contain a very low coercivity remanence. The values of natural remanent magnetization (NRM) of the samples in this SEIR area are on the same order as those of other oceanic ridge basalts. They show a general decreasing trend of NRM with increasing crust age. However, the values of NRM show no correlation either with the tectonic zonations (Zone A vs. Zone B) or with the mantle provinces (Pacific vs. Indian types).

Geochemistry of Ninetyeast Ridge basaltic glasses

The Ninetyeast Ridge (NER), a north-south striking, 5,000 km long, 77 to 43 Ma chain of basaltic submarine volcanoes in the eastern Indian Ocean formed as a hotspot track created by rapid northward migration of the Indian Plate over the Kerguelen hotspot. Based on the major and trace element contents of unaltered basaltic glasses from six locations along the NER, we show that the NER was constructed by basaltic magma derived from at least three geochemically distinct mantle sources: (1) a source enriched in highly incompatible elements relative to primitive mantle like the source of the 29-24 Ma flood basalts in the Kerguelen Archipelago; (2) an incompatible element-depleted source similar to the source of Mid-Ocean Ridge Basalt (MORB) erupted along the currently active Southeast Indian Ridge (SEIR); and (3) an incompatible element-depleted source that is compositionally and mineralogically distinct from the source of SEIR MORB. Specifically, this depleted mantle source was garnet-bearing and had higher Y/Dy and Nb/Zr, but lower Zr/Sm, than the SEIR MORB source. We infer that this third source formed as a garnet-bearing residue created during a previous melting event, perhaps an initial partial melting of the mantle hotspot. Subsequently, this residue partially melted over a large pressure range, from slightly over 3 GPa to less than 1 GPa, and to a high extent (~ 30%) thereby creating relatively high SiO2 and FeO contents in some NER basalts relative to SEIR MORB.

The Southeast Indian Ridge between 88°E and 118°E: Gravity anomalies and crustal accretion at intermediate spreading rates

Although slow spreading ridges characterized by a deep axial valley and fast spreading ridges characterized by an axial bathymetric high have been extensively studied, the transition between these two modes of axial morphology is not well understood. We conducted a geophysical-survey of the intermediate spreading rate Southeast Indian Ridge between 88 degrees E and 118 degrees E, a 2300-km-long section of the ridge located between the Amsterdam hot spot and the Australian-Antarctic Discordance where satellite gravity data suggest that the Southeast Indian Ridge (SEIR) undergoes a change from an axial high in the west to an axial valley in the east. A basic change in axial morphology is found near 103 degrees 30'E in the shipboard data; the axis to the west is marked by an axial high, while a valley is found to the east. Although a well-developed axial high, characteristic of the East Pacific Rise (EPR), is occasionally present, the more common observation is a rifted high that is lower and pervasively faulted, sometimes with significant (> 50 m throw) faults within a kilometer of the axis. A shallow axial valley (< 700 m deep) is observed from 104 degrees E to 114 degrees E with a sudden change to a deep (>1200 m deep) valley across a transform at 114 degrees E. The changes in axial morphology along the SEIR are accompanied by a 500 m increase in near-axis ridge flank depth from 2800 m near 88 degrees E to 3300 m near 114 degrees E and by a 50 mGal increase in the regional level of mantle Bouguer gravity anomalies over the same distance, The regional changes in depth and mantle Bouguer anomaly (MBA) gravity can be both explained by a 1.7-2.4 km change in crustal thickness or by a mantle temperature change of 50 degrees C-90 degrees C. In reality, melt supply (crustal thickness) and mantle temperature are linked, so that changes in both may occur simultaneously and these estimates serve as upper bounds. The along-axis MBA gradient is not uniform. Pronounced steps in the regional level of the MBA gravity occur at 103 degrees 30'E-104 degrees E and at 114 degrees E-116 degrees E and correspond to the changes in the nature of the axial morphology and in the amplitude of abyssal hill morphology suggesting that the different forms of morphology do not grade into each other but rather represent distinctly different forms of axial (s)tructure and tectonics with a sharp transition between them. The change from an axial high to an axial valley requires a threshold effect in which the strength of the lithosphere changes quickly. The presence or absence of a quasi-steady state magma chamber may provide such a mechanism. The different forms of axial morphology are also associated with different intrasegment MBA gravity patterns. Segments with an axial high have an MBA low located at a depth minimum near the center of the segment, At EPR-like segments, the MBA low is about 10 mGal with along-axis gradients of 0.15-0.25 mGal/km, similar to those observed at the EPR, Rifted highs have a shallower low and lower gradients suggesting an attenuated composite magma chamber and a reduced and perhaps episodic melt supply. Segments with a shallow axial valley have very flat along-axis MBA profiles with little correspondence between axial depth and axial MBA gravity.

The Southeast Indian Ridge between 88°E and 118°E: Variations in crustal accretion at constant spreading rate

The temperature of the mantle and the rate of melt production are parameters which play important roles in controlling the style of crustal accretion along mid-ocean ridges. To investigate the variability in crustal accretion that develops in response to variations in mantle temperature, we have conducted a geophysical investigation of the Southeast Indian Ridge (SEIR) between the Amsterdam hotspot and the Australian-Antarctic Discordance (88 degrees E-118 degrees E). The spreading center deepens by 2100 m from west to east within the study area. Despite a uniform, intermediate spreading rate (69-75 mm yr-l), the SEIR exhibits the range in axial morphology displayed by the East Pacific Rise and the Mid-Atlantic Ridge (MAR) and usually associated with variations in spreading rate. The spreading center is characterized by an axial high west of 102 degrees 45'E, whereas an axial valley is prevalent east of this longitude. Both the deepening of the ridge axis and the general evolution of axial morphology from an axial high to a rift valley are not uniform. A region of intermediate morphology separates axial highs and MAR-like rift valleys. Local transitions in axial morphology occur in three areas along the ridge axis. The increase in axial depth toward the Australian-Antarctic Discordance may be explained by the thinning of the oceanic crust by similar to 4 km and the change in axial topography. The long-wavelength changes observed along the SEIR can be attributed to a gradient in mantle temperature between regions influenced by the Amsterdam and Kerguelen hot spots and the Australian-Antarctic Discordance. However, local processes, perhaps associated with an heterogeneous mantle or along-axis asthenospheric flow, may give rise to local transitions in axial topography and depth anomalies.

Modelli matematici applicati alle epidemie

Il mio elaborato di tesi è basato sull'analisi generale dei modelli matematici alla base delle epidemie e lo studio dei punti di equilibrio dei sistemi che descrivono tali modelli. Il primo capitolo fornisce una descrizione degli elementi di base per lo studio che verrà poi affrontato nei capitoli successivi; nel capitolo in questione ci si riduce a introdurre nomenclatura, parametri e assunzioni di base. Nei capitoli secondo e terzo vengono esposti due tipi di modelli SIR diversi (epidemico e endemico), modelli alla base dell'epidemiologia che spiegano, sotto determinate ipotesi, il comportamento degli individui all'esplodere di una epidemia e il loro cambiamento di compartimento (suscettibile, infetto o rimosso) nel tempo. Di essi viene costruito il sistema di equazioni differenziali e analizzato il comportamento asintotico, per poi studiare i punti di equilibrio e la loro stabilità. Nel capitolo quarto viene analizzato allo stesso modo il modello SEIR, interessante perché fornisce una visione leggermente più sofisticata e vicina alla realtà, poiché tra i compartimenti possibili si suppone esistere anche un compartimento degli esposti, cioè di coloro che sono stati infettati ma che ancora non sono in grado di trasmettere la malattia. L'appendice finale illustra l'apparato matematico utile allo studio di questi modelli: le Equazioni Differenziali infatti sono uno strumento necessario per la descrizione dei fenomeni naturali e per lo studio della stabilità dei punti di equilibrio.