Advances on Uncertainty Quantification: Theory and Modelling

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 12230

Special Issue Editors


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Guest Editor
Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Universitat Politècnica de València, 46022 Valencia, Spain
Interests: differential equations with randomness; mathematical modelling
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Guest Editor
Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Polytechnic University of Valencia, 46022 Valencia, Spain
Interests: uncertainty quantification; mathematical epidemiology

Special Issue Information

Dear Colleagues,

Numerous physical, chemical, and biological phenomena, which are very important for scientistic and technological progress, have been traditionally formulated through mathematical models based on ordinary or partial differential equations, where the data (initial/boundary conditions, source term, and/or coefficients) are expressed by means of numerical values or deterministic functions. Nevertheless, scientists really fix these data from measurements, which are always subject to error. How satisfactory the results obtained from the model will be depends on the quality of these measurements (which can frequently take a lot of time and incur high costs). In addition to measurement errors, we must consider the random character of complex external factors that can affect the system, such as pressure, temperature, and humidity in Meteorology; the composition of the land in Seismology; investor tendency and economical policy of countries and companies in Finance; the environmental and genetical factors in Epidemiology; etc. These circumstances make it more advisable to consider the data as random magnitudes; if what is to be measured is a magnitude functionally independent of others values, it would be better to consider it as a random variable; when a dependency exists with respect to other magnitudes, such as time, space, etc., it would be more advisable to interpret it not as a function but as a stochastic process. The consideration of these facts leads to the reformulation of traditional deterministic models, which, in order to improve them, must be replaced by random models. Thus, this is the main purpose of this Special Issue: to gather contributions addressing new analytic and numerical methods and their applications to nontrivial problems to solve, simulate, and approximate random and stochastic equations in a wide sense (algebraic, difference, differential, integral, etc.). Applications of these equations in the settting of applications are particularly welcome.

Prof. Dr. Juan Carlos Cortés López
Prof. Dr. Rafael Villanueva Micó
Guest Editors

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Keywords

  • Random and stochastic equations (algebraic, difference, differential, integral, etc.)
  • Random numerics
  • Optimization under uncertainty techniques and modeling
  • Parameters estimation techniques for random and stochastic equations
  • Nonlinear systems with random excitation and perturbation and linearization techniques
  • Control theory with randomness
  • Spectral expansions for random and stochastic equations
  • Random fields and applications
  • Approximation of probability densities for random and stochastic equations
  • Efficient simulation techniques for random and stochastic equations
  • Models with uncertainty in biology, physics, chemistry, economics, and engineering

Published Papers (5 papers)

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Research

20 pages, 332 KiB  
Article
Likelihood Function through the Delta Approximation in Mixed SDE Models
by Nelson T. Jamba, Gonçalo Jacinto, Patrícia A. Filipe and Carlos A. Braumann
Mathematics 2022, 10(3), 385; https://doi.org/10.3390/math10030385 - 27 Jan 2022
Cited by 3 | Viewed by 2645
Abstract
Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle [...] Read more.
Stochastic differential equations (SDE) appropriately describe a variety of phenomena occurring in random environments, such as the growth dynamics of individual animals. Using appropriate weight transformations and a variant of the Ornstein–Uhlenbeck model, one obtains a general model for the evolution of cattle weight. The model parameters are α, the average transformed weight at maturity, β, a growth parameter, and σ, a measure of environmental fluctuations intensity. We briefly review our previous work on estimation and prediction issues for this model and some generalizations, considering fixed parameters. In order to incorporate individual characteristics of the animals, we now consider that the parameters α and β are Gaussian random variables varying from animal to animal, which results in SDE mixed models. We estimate parameters by maximum likelihood, but, since a closed-form expression for the likelihood function is usually not possible, we approximate it using our proposed delta approximation method. Using simulated data, we estimate the model parameters and compare them with existing methodologies, showing that the proposed method is a good alternative. It also overcomes the existing methodologies requirement of having all animals weighed at the same ages; thus, we apply it to real data, where such a requirement fails. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
21 pages, 512 KiB  
Article
Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months
by José M. Garrido, David Martínez-Rodríguez, Fernando Rodríguez-Serrano, Sorina-M. Sferle and Rafael-J. Villanueva
Mathematics 2021, 9(10), 1132; https://doi.org/10.3390/math9101132 - 17 May 2021
Cited by 2 | Viewed by 2326
Abstract
Mathematical models have been remarkable tools for knowing in advance the appropriate time to enforce population restrictions and distribute hospital resources. Here, we present a mathematical Susceptible-Exposed-Infectious-Recovered (SEIR) model to study the transmission dynamics of COVID-19 in Granada, Spain, taking into account the [...] Read more.
Mathematical models have been remarkable tools for knowing in advance the appropriate time to enforce population restrictions and distribute hospital resources. Here, we present a mathematical Susceptible-Exposed-Infectious-Recovered (SEIR) model to study the transmission dynamics of COVID-19 in Granada, Spain, taking into account the uncertainty of the phenomenon. In the model, the patients moving throughout the hospital’s departments (intra-hospitalary circuit) are considered in order to help to optimize the use of a hospital’s resources in the future. Two main seasons, September–April (autumn-winter) and May–August (summer), where the hospital pressure is significantly different, have been included. The model is calibrated and validated with data obtained from the hospitals in Granada. Possible future scenarios have been simulated. The model is able to capture the history of the pandemic in Granada. It provides predictions about the intra-hospitalary COVID-19 circuit over time and shows that the number of infected is expected to decline continuously from May without an increase next autumn–winter if population measures continue to be satisfied. The model strongly suggests that the number of infected cases will reduce rapidly with aggressive vaccination policies. The proposed study is being used in Granada to design public health policies and perform wise re-distribution of hospital resources in advance. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
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25 pages, 683 KiB  
Article
Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization
by Anirban Mondal and Jia Wei
Mathematics 2021, 9(9), 1067; https://doi.org/10.3390/math9091067 - 10 May 2021
Cited by 2 | Viewed by 1391
Abstract
In this article, we study uncertainty quantification for flows in heterogeneous porous media. We use a Bayesian approach where the solution to the inverse problem is given by the posterior distribution of the permeability field given the flow and transport data. Permeability fields [...] Read more.
In this article, we study uncertainty quantification for flows in heterogeneous porous media. We use a Bayesian approach where the solution to the inverse problem is given by the posterior distribution of the permeability field given the flow and transport data. Permeability fields within facies are assumed to be described by two-point correlation functions, while interfaces that separate facies are represented via smooth pseudo-velocity fields in a level set formulation to get reduced dimensional parameterization. The permeability fields within facies and pseudo-velocity fields representing interfaces can be described using Karhunen–Loève (K-L) expansion, where one can select dominant modes. We study the error of posterior distributions introduced in such truncations by estimating the difference in the expectation of a function with respect to full and truncated posteriors. The theoretical result shows that this error can be bounded by the tail of K-L eigenvalues with constants independent of the dimension of discretization. This result guarantees the feasibility of such truncations with respect to posterior distributions. To speed up Bayesian computations, we use an efficient two-stage Markov chain Monte Carlo (MCMC) method that utilizes mixed multiscale finite element method (MsFEM) to screen the proposals. The numerical results show the validity of the proposed parameterization to channel geometry and error estimations. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
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13 pages, 843 KiB  
Article
Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations
by Leonid Shaikhet
Mathematics 2020, 8(8), 1302; https://doi.org/10.3390/math8081302 - 6 Aug 2020
Cited by 11 | Viewed by 2189
Abstract
So called SIR epidemic model with distributed delay and stochastic perturbations is considered. It is shown, that the known sufficient conditions of stability in probability of the equilibria of this model, formulated immediately in the terms of the system parameters, can be improved [...] Read more.
So called SIR epidemic model with distributed delay and stochastic perturbations is considered. It is shown, that the known sufficient conditions of stability in probability of the equilibria of this model, formulated immediately in the terms of the system parameters, can be improved by virtue of the method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). It is also shown, that stability can be investigated immediately via numerical simulation of a solution of the considered model. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
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19 pages, 1609 KiB  
Article
Probabilistic Study of the Effect of Anti-Epileptic Drugs Under Uncertainty: Cost-Effectiveness Analysis
by Isabel Barrachina-Martínez, Ana Navarro-Quiles, Marta Ramos, José-Vicente Romero, María-Dolores Roselló and David Vivas-Consuelo
Mathematics 2020, 8(7), 1120; https://doi.org/10.3390/math8071120 - 9 Jul 2020
Cited by 1 | Viewed by 2510
Abstract
Epilepsy is one of the most ancient diseases. Despite the efforts of scientists and doctors to improve the quality of live of epileptic patients, the disease is still a mystery in many senses. Anti-epileptic drugs are fundamental to reduce epileptic seizures but it [...] Read more.
Epilepsy is one of the most ancient diseases. Despite the efforts of scientists and doctors to improve the quality of live of epileptic patients, the disease is still a mystery in many senses. Anti-epileptic drugs are fundamental to reduce epileptic seizures but it have some adverse effects, which influence the quality of life outcomes of the patients. In this paper, we study the effectiveness of anti-epileptic drugs taking into account the inherent uncertainty. We establish a model, which allows to represent the natural history of epilepsy, using Markov chains. After randomizing the mathematical model, we compute the first probability density function of the solution stochastic process applying the random variable transformation technique. We also take advantage of this method to determine the distribution of some key quantities in medical decision, such as the time until a certain proportion of the population remains in each state and the incremental cost-effectiveness ratio. The study is completed computing all these quantities using data available in the literature. In addition, regarding the incremental cost-effectiveness ratio, different third generation anti-epileptic treatments are compared with the Brivaracetam, a new third generation anti-epileptic drug. Full article
(This article belongs to the Special Issue Advances on Uncertainty Quantification: Theory and Modelling)
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