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The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature on ODEs which, in particular, allows for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework. Although we focus on examples of Medical Statistics, the proposed framework is applicable in any context where the interest lies in estimating and interpreting the dynamics of the hazard function.
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We present a forecasting model aim to predict hospital occupancy in metropolitan areas during the current COVID-19 pandemic. Our SEIRD type 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 breakpoints to model lockdown interventions and the increase in effective population size due to lockdown relaxation. The latter features let us model lockdown-induced 2nd waves. Our Bayesian approach allows us to produce timely probabilistic forecasts of hospital demand. We have applied the model to analyze more than 70 metropolitan areas and 32 states in Mexico.
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COVID-19/epidemiologia , Teorema de Bayes , Cidades/epidemiologia , Controle de Doenças Transmissíveis , Previsões , Hospitalização , Hospitais , Humanos , México/epidemiologia , Densidade Demográfica , SARS-CoV-2/isolamento & purificaçãoRESUMO
Degradation tests are especially difficult to conduct for items with high reliability. Test costs, caused mainly by prolonged item duration and item destruction costs, establish the necessity of sequential degradation test designs. We propose a methodology that sequentially selects the optimal observation times to measure the degradation, using a convenient rule that maximizes the inference precision and minimizes test costs. In particular our objective is to estimate a quantile of the time to failure distribution, where the degradation process is modelled as a linear model using Bayesian inference. The proposed sequential analysis is based on an index that measures the expected discrepancy between the estimated quantile and its corresponding prediction, using Monte Carlo methods. The procedure was successfully implemented for simulated and real data.
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Teorema de Bayes , Modelos Lineares , Método de Monte Carlo , Humanos , Reprodutibilidade dos TestesRESUMO
Epidemic outbreak detection is an important problem in public health and the development of reliable methods for outbreak detection remains an active research area. In this paper we introduce a Bayesian method to detect outbreaks of influenza-like illness from surveillance data. The rationale is that, during the early phase of the outbreak, surveillance data changes from autoregressive dynamics to a regime of exponential growth. Our method uses Bayesian model selection and Bayesian regression to identify the breakpoint. No free parameters need to be tuned. However, historical information regarding influenza-like illnesses needs to be incorporated into the model. In order to show and discuss the performance of our method we analyze synthetic, seasonal, and pandemic outbreak data.
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Surtos de Doenças/estatística & dados numéricos , Influenza Humana/epidemiologia , Teorema de Bayes , Humanos , Modelos Estatísticos , Densidade Demográfica , São Francisco/epidemiologia , Espanha , Estatística como AssuntoRESUMO
In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of 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 using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.
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Epidemias , Modelos Biológicos , Número Básico de Reprodução , Teorema de Bayes , Simulação por Computador , Dengue/epidemiologia , Vírus da Dengue/crescimento & desenvolvimento , Métodos Epidemiológicos , Humanos , Modelos EstatísticosRESUMO
The diversity of sites and the distribution of species are fundamental pieces in the analysis of biogeographic and macroecological questions. A link between these two variables is the correlation between the species diversity of sites and the mean range size of species occurring there. Alternatively, one could correlate the range sizes of species and the mean species diversity within those ranges. Here we show that both approaches are mirror images of the same patterns, reflecting fundamental mathematical and biological relationships. We develop a theory and analyze data for North American mammals to interpret range-diversity plots in which the species diversity of sites and the geographic range of species can be depicted simultaneously. We show that such plots contain much more information than traditional correlative approaches do, and we demonstrate that the positions of points in the plots are determined to a large extent by the average, minimum, and maximum values of range and diversity but that the dispersion of points depends on the association among species and the similitude among sites. These generalizations can be applied to biogeographic studies of diversity and distribution and in the identification of hotspots of diversity and endemism.