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This article proposes a weighted-variational model as an approximated surrogate model to lessen numerical complexity and lower the execution times of brittle fracture simulations. Consequently, Monte Carlo studies of brittle fractures become possible when energy release rates are modelled as a random field. In the weighed-variational model, we propose applying a Gaussian random field with a Mat\'ern covariance function to simulate a non-homogeneous energy release rate ($G_c$) of a material. Numerical solutions to the weighed-variational model, along with the more standard but computationally demanding hybrid phase-field models, are obtained using the FEniCS open-source software. The results have indicated that the weighted-variational model is a competitive surrogate model of the hybrid phase-field method to mimic brittle fractures in real structures. This method reduces execution times by 90\%. We conducted a similar study and compared our results with an actual brittle fracture laboratory experiment. We present an example where a Monte Carlo study is carried out, modeling $G_c$ as a Gaussian Process, obtaining a distribution of possible fractures, and load-displacement curves.

We introduce a Bayesian sequential data assimilation method for COVID-19 forecasting. It is assumed that suitable transmission, epidemic and observation models are available and previously validated and the transmission and epidemic models are coded into a dynamical system. The observation model depends on the dynamical system state variables and parameters, and is cast as a likelihood function. We elicit prior distributions of the effective population size, the dynamical system initial conditions and infectious contact rate, and use Markov Chain Monte Carlo sampling to make inference and prediction of quantities of interest (QoI) at the onset of the epidemic outbreak. The forecast is sequentially updated over a sliding window of epidemic records as new data becomes available. Prior distributions for the state variables at the new forecasting time are assembled using the dynamical system, calibrated for the previous forecast. Moreover, changes in the contact rate and effective population size are naturally introduced through auto-regressive models on the corresponding parameters. We show our forecasting method's performance using a SEIR type model and COVID-19 data from several Mexican localities., Comment: 22 pages, 19 figures

In this paper, we propose a model of the dynamics of the blood glucose level during an Oral Glucose Tolerance Test (OGTT). This dynamic includes the action of insulin and glucagon in the glucose homeostasis process as the reaction of an oral stimulus. We propose a Bayesian approach in the inference of five parameters related to insulin secretion, glucagon secretion, gastrointestinal emptying, and basal glucose level. Two insulin indicators related to the glucose level in blood and in the gastrointestinal tract allow us to suggest a classification for patients with impaired insulin sensitivity.

The effective reproduction number $R_t$ measures an infectious disease's transmissibility as the number of secondary infections in one reproduction time in a population having both susceptible and non-susceptible hosts. Current approaches do not quantify the uncertainty correctly in estimating $R_t$, as expected by the observed variability in contagion patterns. We elaborate on the Bayesian estimation of $R_t$ by improving on the Poisson sampling model of Cori et al. (2013). By adding an autoregressive latent process, we build a Dynamic Linear Model on the log of observed $R_t$s, resulting in a filtering type Bayesian inference. We use a conjugate analysis, and all calculations are explicit. Results show an improved uncertainty quantification on the estimation of $R_t$'s, with a reliable method that could safely be used by non-experts and within other forecasting systems. We illustrate our approach with recent data from the current COVID19 epidemic in Mexico.

We present a compartmental SEIRD model aimed at forecasting hospital occupancy in metropolitan areas during the current COVID-19 outbreak. The model features asymptomatic and symptomatic infections with detailed hospital dynamics. We model explicitly branching probabilities and non exponential residence times in each latent and infected compartments. Using both hospital admittance confirmed cases and deaths we infer the contact rate and the initial conditions of the dynamical system, considering break points to model lockdown interventions. Our Bayesian approach allows us to produce timely probabilistic forecasts of hospital demand. The model has been used by the federal government of Mexico to assist public policy, and has been applied for the analysis of more than 70 metropolitan areas and the 32 states in the country.

We elaborate on results obtained in \cite{christen2018} for controlling the numerical posterior error for Bayesian UQ problems, now considering forward maps arising from the solution of a semilinear evolution partial differential equation. Results in \cite{christen2018} demand an estimate for the absolute global error (AGE) of the numeric forward map. Our contribution is a numerical method for computing the AGE for semilinear evolution PDEs and shows the potential applicability of \cite{christen2018} in this important wide range family of PDEs. Numerical examples are given to illustrate the efficiency of the proposed method, obtaining numerical posterior distributions for unknown parameters that are nearly identical to the corresponding theoretical posterior, by keeping their Bayes factor close to 1., Comment: 30 pages, 12 figures

In this paper we introduce a method to estimate a pressure dependent thermal conductivity coefficient arising in a heat diffusion model with applications in food technology. To address the known smoothing effect of the direct problem, we model the uncertainty of the conductivity coefficient as a hierarchical Gaussian Markov random field (GMRF) restricted to uniqueness conditions for the solution of the inverse problem established in Fraguela et al. Furthermore, we propose a Single Variable Exchange Metropolis-Hastings algorithm to sample the corresponding conditional probability distribution of the conductivity coefficient given observations of the temperature. Sensitivity analysis of the direct problem suggests that large integration times are necessary to identify the thermal conductivity coefficient. Numerical evidence indicates that a signal to noise ratio of roughly 1000 suffices to reliably retrieve the thermal conductivity coefficient.

In this paper we demonstrate a computational method to solve the inverse scattering problem for a star-shaped, smooth, penetrable obstacle in 2D. Our method is based on classical ideas from computational geometry. First, we approximate the support of a scatterer by a point cloud. Secondly, we use the Bayesian paradigm to model the joint conditional probability distribution of the non-convex hull of the point cloud and the constant refractive index of the scatterer given near field data. Of note, we use the non-convex hull of the point cloud as spline control points to evaluate, on a finer mesh, the volume potential arising in the integral equation formulation of the direct problem. Finally, in order to sample the arising posterior distribution, we propose a probability transition kernel that commutes with affine transformations of space. Our findings indicate that our method is reliable to retrieve the support and constant refractive index of the scatterer simultaneously. Indeed, our sampling method is robust to estimate a quantity of interest such as the area of the scatterer. We conclude pointing out a series of generalizations of our method., Comment: 20 pages, figures 8

We generalize the results of \cite{Capistran2016} on expected Bayes factors (BF) to control the numerical error in the posterior distribution to an infinite dimensional setting when considering Banach functional spaces and now in a prior setting. The main result is a bound on the absolute global error to be tolerated by the Forward Map numerical solver, to keep the BF of the numerical vs. the theoretical model near to 1, now in this more general setting, possibly including a truncated, finite dimensional approximate prior measure. In so doing we found a far more general setting to define and prove existence of the infinite dimensional posterior distribution than that depicted in, for example, \cite{Stuart2010}. Discretization consistency and rates of convergence are also investigated in this general setting for the Bayesian inverse problem., Comment: 42 pages, 6 figures

In this paper, we carry out a computational study using the spectral decomposition of the fluctuations of a two-pathogen epidemic model around its deterministic attractor, i.e., steady state or limit cycle, to examine the role of partial vaccination and between-host pathogen interaction on early pathogen replacement during seasonal epidemics of influenza and respiratory syncytial virus.

Before the current pandemic, influenza and respiratory syncytial virus (RSV) were the leading etiological agents of seasonal acute respiratory infections (ARI) around the world. In this setting, medical doctors typically based the diagnosis of ARI on patients' symptoms alone and did not routinely conduct virological tests necessary to identify individual viruses, limiting the ability to study the interaction between multiple pathogens and to make public health recommendations. We consider a stochastic kinetic model (SKM) for two interacting ARI pathogens circulating in a large population and an empirically-motivated background process for infections with other pathogens causing similar symptoms. An extended marginal sampling approach, based on the linear noise approximation to the SKM, integrates multiple data sources and additional model components. We infer the parameters defining the pathogens' dynamics and interaction within a Bayesian model and explore the posterior trajectories of infections for each illness based on aggregate infection reports from six epidemic seasons collected by the state health department and a subset of virological tests from a sentinel program at a general hospital in San Luis Potos\'{i}, M\'{e}xico. We interpret the results and make recommendations for future data collection strategies., Comment: A previous version of this manuscript was mistakenly uploaded as a separate submission (arXiv:1903.10905)

In the bayesian analysis of Inverse Problems most relevant cases the forward maps (FM, or regressor function) are defined in terms of a system of (O, P)DE's with intractable solutions. These necessarily involve a numerical method to find approximate versions of such solutions and lead to a numerical/approximate posterior distribution. Recently several results have been published on the regularity conditions required on such numerical methods to ensure converge of the numerical to the theoretical posterior. However, more practical guidelines are needed to ensure a suitable working numerical posterior. ]Capistran2016] prove for ODE's that the Bayes Factor of the approximate vs the theoretical model tends to 1 in the same order as the numerical method order. In this work we generalize the latter paper in that we consider 1) also PDE's, 2) correlated observations, 3) practical guidelines in a multidimensional setting and 4) explore the use of expected Bayes Factors. This permits us to obtain bounds on the absolute global errors to be tolerated by the FM numerical solver, which we illustrate with some examples. Since the Bayes Factor is kept above 0.95 we expect that the resulting numerical posterior is basically indistinguishable from the theoretical posterior, even though we are using an approximate numerical FM. The method is illustrated with some examples using synthetic data., Comment: 3 Figures

Under the hypothesis that both influenza and respiratory syncytial virus (RSV) are the two leading causes of acute respiratory infections (ARI), in this paper we have used a standard two-pathogen epidemic model as a regressor to explain, on a yearly basis, high season ARI data in terms of the contact rates and initial conditions of the mathematical model. The rationale is that ARI high season is a transient regime of a noisy system, e.g., the system is driven away from equilibrium every year by fluctuations in variables such as humidity, temperature, viral mutations and human behavior. Using the value of the replacement number as a phenotypic trait associated to fitness, we provide evidence that influenza and RSV coexists throughout the ARI high season through superinfection.

In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step., Comment: 28 pages, 6 figures

In this paper we introduce a novel method to conduct inference with models defined through a continuous-time Markov process, and we apply these results to a classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for first and second moments for the state variables. These approximate moments are in turn matched to the moments of an inputed generic discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference to estimate epidemic parameters using informative priors. Excellent estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies., Comment: 30 pages, 5 figures, submitted for peer review

Motivated by the need, in some Bayesian likelihood free inference problems, of imputing a multivariate counting distribution based on its vector of means and variance-covariance matrix, we define a generic multivariate discrete distribution. Based on blending the Binomial, Poisson and Negative-Binomial distributions, and using a normal multivariate copula, the required distribution is defined. This distribution tends to the Multivariate Normal for large counts and has an approximate pmf version that is quite simple to evaluate., Comment: 9 pages, 1 fgure, 1 table