Computational and Mathematical Methods
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 Journal profile

Computational and Mathematical Methods is an interdisciplinary journal dedicated to publishing the world's top research in the expanding area of computational mathematics, science and engineering. 

 Editor spotlight

Chief Editor, Professor Jesús Vigo Aguiar, is based at University of Salamanca, Spain. His core expertise is in mathematical applications. 

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Research Article

Modeling Dengue Immune Responses Mediated by Antibodies: Insights on the Biological Parameters to Describe Dengue Infections

Dengue fever is a viral mosquito-borne disease, a significant global health concern, with more than one third of the world population at risk of acquiring the disease. Caused by 4 antigenically distinct but related virus serotypes, named DENV-1, DENV-2, DENV-3, and DENV-4, infection by one serotype confers lifelong immunity to that serotype and a short period of temporary cross immunity to other related serotypes. Severe dengue is epidemiologically associated with a secondary infection caused by a heterologous serotype via the so-called antibody-dependent enhancement (ADE), an immunological process enhancing a new infection. Within-host dengue modeling is restricted to a small number of studies so far. With many open questions, the understanding of immunopathogenesis of severe disease during recurrent infections is important to evaluate the impact of newly licensed vaccines. In this paper, we revisit the modeling framework proposed by Sebayang et al. and perform a detailed sensitivity analysis of the well-known biological parameters and its possible combinations to understand the existing data sets. Using numerical simulations, we investigate features of viral replication, antibody production, and infection clearance over time for three possible scenarios: primary infection, secondary infection caused by homologous serotype, and secondary infection caused by heterologous serotype. Besides, describing well the infection dynamics as reported in the immunology literature, our results provide information on parameter combinations to best describe the differences on the immunological dynamics of secondary infections with homologous and heterologous viruses. The results presented here will be used as baseline to investigate a more complex within-host dengue model.

Research Article

Box-Cox Transformations and Bias Reduction in Extreme Value Theory

The Box-Cox transformations are used to make the data more suitable for statistical analysis. We know from the literature that this transformation of the data can increase the rate of convergence of the tail of the distribution to the generalized extreme value distribution, and as a byproduct, the bias of the estimation procedure is reduced. The reduction of bias of the Hill estimator has been widely addressed in the literature of extreme value theory. Several techniques have been used to achieve such reduction of bias, either by removing the main component of the bias of the Hill estimator of the extreme value index (EVI) or by constructing new estimators based on generalized means or norms that generalize the Hill estimator. We are going to study the Box-Cox Hill estimator introduced by Teugels and Vanroelen, in 2004, proving the consistency and asymptotic normality of the estimator and addressing the choice and estimation of the power and shift parameters of the Box-Cox transformation for the EVI estimation. The performance of the estimators under study will be illustrated for finite samples through small-scale Monte Carlo simulation studies.

Research Article

Seasonally Forced SIR Systems Applied to Respiratory Infectious Diseases, Bifurcations, and Chaos

Summary. We investigate models to describe respiratory diseases with fast mutating virus pathogens such that after some years the aquired resistance is lost and hosts can be infected with new variants of the pathogen. Such models were initially suggested for respiartory diseases like influenza, showing complex dynamics in reasonable parameter regions when comparing to historic empirical influenza like illness data, e.g., from Ille de France. The seasonal forcing typical for respiratory diseases gives rise to the different rich dynamical scenarios with even small parameter changes. Especially the seasonality of the infection leads for small values already to period doubling bifurcations into chaos, besides additional coexisting attractors. Such models could in the future also play a role in understanding the presently experienced COVID-19 pandemic, under emerging new variants and with only limited vaccine efficacies against newly upcoming variants. From first period doubling bifurcations, we can eventually infer at which close by parameter regions complex dynamics including deterministic chaos can arise.

Research Article

The Rate of Convergence of the SOR Method in the Positive Semidefinite Case

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 , where is a real symmetric positive semidefinite matrix. The bounds are given in terms of the condition number of , which is the ratio , where is the largest eigenvalue of and is the smallest nonzero eigenvalue of . Let denote the related iteration matrix. Then, since has a zero eigenvalue, the spectral radius of equals 1, and the rate of convergence is determined by the size of , the largest eigenvalue of whose modulus differs from 1. The bound has the form , where The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.

Research Article

Study of a Set of Symmetric Temporal Transformations for the Study of the Orbital Motion

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.

Research Article

Deep Brain Stimulation with a Computational Model for the Cortex-Thalamus-Basal-Ganglia System and Network Dynamics of Neurological Disorders

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.

Computational and Mathematical Methods
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Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.