Computational and Mathematical Methods最新文献

In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form Gx = b, where G is a real symmetric positive semidefinite n × n matrix. The bounds are given in terms of the condition number of G, which is the ratio κ = α/β, where α is the largest eigenvalue of G and β is the smallest nonzero eigenvalue of G. Let H denote the related iteration matrix. Then, since G has a zero eigenvalue, the spectral radius of H equals 1, and the rate of convergence is determined by the size of η, the largest eigenvalue of H whose modulus differs from 1. The bound has the form |η|² ≤ 1 − 1/(κc), where c = 2 + log₂n. The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.

A mathematical model to describe the dynamic of a multiserotype infectious disease at the population level is studied. Applied to dengue fever epidemiology, we analyse a mathematical model with time delay to describe the cross-immunity protection period, including a key parameter for the antibody-dependent enhancement (ADE) effect, the well-known features of dengue fever infection. Numerical experiments are performed to show the stability of the coexistence equilibrium, which is completely determined by the basic reproduction number and by the invasion reproduction number, as well as the bifurcation structures for different scenarios of dengue fever transmission in a population. The model shows a rich dynamical behavior, from fixed points and periodic oscillations up to chaotic behaviour with complex attractors.

Mathematical models play an important role in epidemiology. The inclusion of a spatial component in epidemiological models is especially important to understand and address many relevant ecological and public health questions, e.g., when wanting to differentiate transmission patterns across geographical regions or when considering spatially heterogeneous intervention measures. However, the introduction of spatial effects can have significant consequences on the observed model dynamics and hence must be carefully analyzed and interpreted. Cellular automata epidemiological models typically rely on simplified computational grids but can provide valuable insight into the spatial dynamics of transmission within a population by suitably accounting for the connections between individuals in the considered community. In this paper, we describe a stochastic cellular automata disease model based on an extension of the traditional Susceptible-Infected-Recovered (SIR) compartmentalization of the population, namely, the Susceptible-Hospitalized-Asymptomatic-Recovered (SHAR) formulation, in which infected individuals either present a severe form of the disease, thus requiring hospitalization, or belong to the so-called mild/asymptomatic class. The critical transmission threshold is derived analytically in the nonspatial SHAR formulation, and this generalizes previously obtained theoretical results for the SIR model. We present simulation results discussing the effect of key model parameters and of spatial correlations on model outputs and propose an algorithm for tracking the evolution of infection clusters within the considered population. Focusing on the role of import and criticality on the overall dynamics, we conclude that the current spatial setting increases the critical transmission threshold in comparison to the nonspatial model.

A cell-centered finite volume semi-Lagrangian method is presented for the numerical solution of two-dimensional coupled Burgers’ problems on unstructured triangular meshes. The method combines a modified method of characteristics for the time integration and a cell-centered finite volume for the space discretization. The new method belongs to fractional-step algorithms for which the convection and the viscous parts in the coupled Burgers’ problems are treated separately. The crucial step of interpolation in the convection step is performed using two local procedures accounting for the element where the departure point is located. The resulting semidiscretized system is then solved using a third-order explicit Runge-Kutta scheme. In contrast to the Eulerian-based methods, we apply the new method for each time step along the characteristic curves instead of the time direction. The performance of the current method is verified using different examples for coupled Burgers’ problems with known analytical solutions. We also apply the method for simulation of an example of coupled Burgers’ flows in a complex geometry. In these test problems, the new cell-centered finite volume semi-Lagrangian method demonstrates its ability to accurately resolve the two-dimensional coupled Burgers’ problems.

The main goal of this paper is to define a new one-parametric family of symmetric temporal transformations with respect to the ellipse. This new family contains as a particular case the eccentric anomaly, the regularized length of arc, and the elliptic anomaly. This family is a particular case of the biparametric family of anomalies introduced by the authors in 2016. The biparametric family comprises the most common anomalies used in the study of the two-body problem. Two approaches of this work have been taken. The first one involves the study of the analytical properties of the symmetric family of anomalies. The second approach explores the improvement of the numerical integration methods when the natural time is replaced by an anomaly of this family.

In December 2019, a severe respiratory syndrome (COVID-19) caused by a new coronavirus (SARS-CoV-2) was identified in China and spread rapidly around the globe. COVID-19 was declared a pandemic by the World Health Organization (WHO) in March 2020. With eventually substantial global underestimation, more than 225 million cases were confirmed by the end of August 2021, counting more than 4.5 million deaths. COVID-19 symptoms range from mild (or no symptoms) to severe illness, with disease severity and death occurring according to a hierarchy of risks, with age and preexisting health conditions enhancing the risks of disease severity manifestation. In this paper, a mathematical model for COVID-19 transmission is proposed and analyzed. The model stratifies the studied population into two groups, older and younger. Applied to the COVID-19 outbreaks in Spain and in Italy, we find the disease-free equilibrium and the basic reproduction number for each case study. A sensitivity analysis to identify the key parameters which influence the basic reproduction number, and hence regulate the transmission dynamics of COVID-19, is also performed. Finally, the model is extended to its stochastic counterpart to encapsulate the variation or uncertainty found in the transmissibility of the disease. We observe the variability of the infectious population finding its distribution at a given time, demonstrating that for small populations, stochasticity will play an important role.

The log-logistic distribution is widely used in different fields of study such as survival analysis, hydrology, insurance, and economics. Recently, Ahsanullah and Alzaatreh studied the best linear unbiased estimators for the location and the scale parameters of the three-parameter log-logistic model. The same authors also propose a shift-invariant Hill estimator for the unknown shape parameter. In this work, we propose a new estimation method for the shape parameter. We derive its nondegenerate asymptotic behaviour and analyse its finite sample performance through a Monte Carlo simulation study. To have precise estimates, we present a method for selecting the threshold. To illustrate the improvement achieved, efficiency comparisons are also provided.

In this paper, a computational procedure for solving singularly perturbed nonlinear delay differentiation equations (SPNDDEs) is proposed. Initially, the SPNDDE is reduced into a series of singularly perturbed linear delay differential equations (SPLDDEs) using the quasilinearization technique. A trigonometric spline approach is suggested to solve the sequence of SPLDDEs. Convergence of the method is addressed. The efficiency and applicability of the proposed method are demonstrated by the numerical examples.

The notion of a coherent system allows us to formalize how the random lifetime of the system is connected to the random lifetimes of its components. These connections are also generators of new pliant distributions, being those of various mixes of minimum and maximum of random variables. In this paper, a new four-parameter lifetime probability distribution is introduced by using the notion of a coherent system. Its structural properties are assessed and evaluated, including the analytical study of its main functions, stochastic dominance results, moments, and moment generating function. The proposed distribution, in particular, is proving to be efficient at fitting data with slight negative skewness and platykurtic as well as leptokurtic nature. This is illustrated by the analysis of three relevant real-life data sets, two in reliability and another in production, exhibiting the significance of the introduced model in comparison to various well-known models in statistical literature.

Deep brain stimulation (DBS) can alleviate the movement disorders like Parkinson’s disease (PD). Indeed, it is known that aberrant beta (13-30 Hz) oscillations and the loss of dopaminergic neurons in the basal ganglia-thalamus (BGTH) and cortex characterize the akinesia symptoms of PD. However, the relevant biophysical mechanism behind this process still remains unclear. Based on the prior striatal inhibitory model, we propose an extended BGTH model incorporating medium spine neurons (MSNs) and fast-spiking interneurons (FSIs) along with the effect of DBS. We are focusing in this paper on an open-loop DBS mode, where the stimulation parameters stay constant independent of variations in the disease state, and modifications of parameters rely mainly on trial and error of medical experts. Additionally, we propose a novel combined model of the cerebellar-basal-ganglia thalamocortical network, MSNs, and FSIs and show new results that indicate that Parkinsonian oscillations in the beta-band frequency range emerge from the dynamics of such a network. Our model predicts that DBS can be used to suppress beta oscillations in globus pallidus pars interna (GPi) neurons. This research will help our better understanding of the changes in the brain activity caused by DBS, providing new insight for studying PD in the future.