Search Results (190)

In this research work, we proposed and studied a fractional-order model for Human African Trypanosomiasis (HAT) disease transmission, incorporating three control strategies: health education campaigns, prevention measures, and use of insecticides. The theoretical analysis of the model was presented, including the computation of disease-free equilibrium and basic reproduction number. We performed the stability analysis of the model and the results showed that the disease-free equilibrium point was locally asymptotically stable whenever ${\mathcal{R}}_0<1$ and unstable when ${\mathcal{R}}_0>1$. Furthermore, we performed parameter estimation of the model using HAT-reported cases in Tanzania. The results showed that fractional-order model had a better fit to the real data compared to the classical integer-order model. Sensitivity analysis of the basic reproduction number was performed using computed partial rank correlation coefficients to assess the effects of parameters on HAT transmission. Additionally, we performed numerical simulations of the model to assess the impact of memory effects on the spread of HAT. Overall, we observed that the order of derivatives significantly influences the dynamics of HAT transmission in the population. Moreover, we simulated the model to assess the effectiveness of proposed control strategies. We observed that the use of insecticides and prevention measures have the potential to significantly reduce the spread of HAT within the population. Full article

This study investigates Weibull distribution modeling for data under grouped observations. Two data grouping methods (equal-width and optimal) were compared for estimating parameters of the Weibull distribution using maximum likelihood estimation (MLE) in each case. Methodologically, our contribution is twofold: First, we derive the correct Fisher information matrix for grouped data in the two-parameter Weibull and use it to compute optimal interval boundaries. Second, we derive maximum likelihood estimators for data grouped under these optimal intervals. The fit of the assumed distributions was evaluated using chi-squared goodness-of-fit tests. We also calculated Asymptotic Relative Efficiency (ARE) to compare the precision of parameter estimates across different grouping approaches. Optimal boundaries yielded systematically higher ARE than equal-width grouping in 100% of comparisons for the shape parameter $c$. Gains for the scale parameter $b$ were smaller and occurred in about 62% of cases. Optimal grouping also produced generally higher chi-squared $\left({\chi }^2\right)$ goodness-of-fit p-values than equal-width grouping, indicating a better fit. From a symmetry standpoint, the Weibull distribution is inherently asymmetric, with the degree of asymmetry governed by the shape parameter $c$. We show that the choice of grouping affects the estimate of $c$ and, thus, the inferred skewness, further explaining why optimally designed intervals yield both higher precision and a more faithful representation of failure behavior. Full article

This study introduces an inverted version of the three-parameter Hjorth lifespan model, characterized by one scale parameter and two shape parameters, referred to as the inverted Hjorth (IH) distribution. This asymmetric distribution can fit various positively skewed datasets more accurately than several existing models in the literature, as it can accommodate data exhibiting an inverted (upside-down) bathtub-shaped hazard rate. We derive key properties of the model, including quantiles, moments, reliability measures, stress–strength reliability, and order statistics. Point estimation of the IH model parameters is performed using maximum likelihood and Bayesian approaches. Moreover, for interval estimation, two types of asymptotic confidence intervals and two types of Bayesian credible intervals are obtained using the same estimation methodologies. As an extension to a complete sampling plan, Type-II censoring is employed to examine the impact of data incompleteness on IH parameter estimation. Monte Carlo simulation results indicate that Bayesian point and credible estimates outperform those obtained via classical estimation methods across several precision metrics, including mean squared error, average absolute bias, average interval length, and coverage probability. To further assess its performance, two real datasets are analyzed: one from the environmental domain (minimum monthly water flows of the Piracicaba River) and another from the pharmacological domain (plasma indomethacin concentrations). The superiority and flexibility of the inverted Hjorth model are evaluated and compared with several competing models. The results confirm that the IH distribution provides a better fit than several existing lifetime models—such as the inverted Gompertz, inverted log-logistic, inverted Lomax, and inverted Nadarajah–Haghighi distributions—making it a valuable tool for reliability and survival data analysis. Full article

We consider the lossless compression bound of any individual data sequence. Conceptually, its Kolmogorov complexity is such a bound yet uncomputable. According to Shannon’s source coding theorem, the average compression bound is $nH$, where n is the number of words and H is the entropy of an oracle probability distribution characterizing the data source. The quantity $nH\left({\widehat{\theta }}_n\right)$ obtained by plugging in the maximum likelihood estimate is an underestimate of the bound. Shtarkov showed that the normalized maximum likelihood (NML) distribution is optimal in a minimax sense for any parametric family. Fitting a data sequence—without any a priori distributional assumption—by a relevant exponential family, we apply the local asymptotic normality to show that the NML code length is $nH\left({\widehat{\theta }}_n\right)+{\displaystyle \frac{d}{2}}log{\displaystyle \frac{n}{2\pi }}+log{\int }_{\Theta }{\left|I\left(\theta \right)\right|}^{1/2}d\theta +o\left(1\right)$, where d is dictionary size, $\left|I\right(\theta \left)\right|$ is the determinant of the Fisher information matrix, and $\Theta$ is the parameter space. We demonstrate that sequentially predicting the optimal code length for the next word via a Bayesian mechanism leads to the mixture code whose length is given by $nH\left({\widehat{\theta }}_n\right)+{\displaystyle \frac{d}{2}}log{\displaystyle \frac{n}{2\pi }}+log{\displaystyle \frac{|I\left({\widehat{\theta }}_n\right){|}^{1/2}}{w\left({\widehat{\theta }}_n\right)}}+o\left(1\right)$, where $w\left(\theta \right)$ is a prior. The asymptotics apply to not only discrete symbols but also continuous data if the code length for the former is replaced by the description length for the latter. The analytical result is exemplified by calculating compression bounds of protein-encoding DNA sequences under different parsing models. Typically, compression is maximized when parsing aligns with amino acid codons, while pseudo-random sequences remain incompressible, as predicted by Kolmogorov complexity. Notably, the empirical bound becomes more accurate as the dictionary size increases. Full article

Type 2 diabetes (T2D) is a chronic metabolic disorder requiring effective management to avoid complications. Metformin is a first-line drug agent and is routinely prescribed for the control of glycemia, but its underlying dynamics are complicated and not fully quantified. This paper formulates a control-oriented and interpretable mathematical model that integrates metformin dynamics into a classic beta-cell–insulin–glucose (BIG) regulation system. The paper’s applicability to theoretical and clinical settings is enhanced by rigorous mathematical analysis, which guarantees the model is globally bounded, well-posed, and biologically meaningful. One of the key features of the study is its global stability analysis using Lyapunov functions, which demonstrates the asymptotic stability of critical equilibrium points under realistic physiological constraints. These findings support the predictive reliability of the model in explaining long-term glycemic regulation. Bifurcation analysis also clarifies the dynamic interplay between glucose production and utilization by identifying parameter thresholds that signify transitions between homeostasis and pathological states. Residual analysis, which detects Gaussian-distributed errors, underlines the robustness of the fitting process and suggests possible refinements by including temporal effects. Sensitivity analysis highlights the predominant effect of the initial dose of metformin on long-term glucose regulation and provides practical guidance for optimizing individual treatment. Furthermore, changing the two considered metformin parameters from their optimal values—altering the dose by ±50% and the decay rate by ±20%—demonstrates the flexibility of the model in simulating glycemic responses, confirming its adaptability and its potential for optimizing personalized treatment strategies. Full article

Hydrological time series frequently exhibit periodic trends with variables such as rainfall, runoff, and evaporation rates often following annual cycles. Seasonal variations further contribute to the complexity of these data sets. A critical aspect of analyzing such phenomena is estimating realistic return intervals, making the precise determination of these values essential. Given this importance, selecting an appropriate probability distribution is paramount. To address this need, we introduce a flexible probability model specifically designed to capture periodicity in hydrological data. We thoroughly examine its fundamental mathematical and statistical properties, including the asymptotic behavior of the probability density function (PDF) and hazard rate function (HRF), to enhance predictive accuracy. Our analysis reveals that the PDF exhibits polynomial decay as $x\to \infty$, ensuring heavy-tailed behavior suitable for extreme events. The HRF demonstrates decreasing or non-monotonic trends, reflecting variable failure risks over time. Additionally, we conduct a simulation study to evaluate the performance of the estimation method. Based on these results, we refine return period estimates, providing more reliable and robust hydrological assessments. This approach ensures that the model not only fits observed data but also captures the underlying dynamics of hydrological extremes. Full article

We introduce a modern methodology for constructing global analytical approximations of special functions over their entire domains. By integrating the traditional method of matching asymptotic expansions—enhanced with Padé approximants—with differential evolution optimization, a modern machine learning technique, we achieve high-accuracy approximations using elegantly simple expressions. This method transforms non-elementary functions, which lack closed-form expressions and are often defined by integrals or infinite series, into simple analytical forms. This transformation enables deeper qualitative analysis and offers an efficient alternative to existing computational techniques. We demonstrate the effectiveness of our method by deriving an analytical expression for the Fermi gas pressure that has not been previously reported. Additionally, we apply our approach to the one-loop correction in thermal field theory, the synchrotron functions, common Fermi–Dirac integrals, and the error function, showcasing superior range and accuracy over prior studies. Full article

This paper introduces the record-based transmuted Rayleigh distribution of order 3 (rbt-R), a three-parameter extension of the classical Rayleigh model designed to address data characterized by high skewness and heavy tails. While traditional generalizations of the Rayleigh distribution enhance model flexibility, they often lack sufficient adaptability to capture the complexity of empirical distributions encountered in applied statistics. The rbt-R model incorporates two additional shape parameters, a and b, enabling it to represent a wider range of distributional shapes. Parameter estimation for the rbt-R model is performed using the maximum likelihood method. Simulation studies are conducted to evaluate the asymptotic properties of the estimators, including bias and mean squared error. The performance of the rbt-R model is assessed through empirical applications to four datasets: nicotine yields and carbon monoxide emissions from cigarette data, as well as breaking stress measurements from carbon-fiber materials. Model fit is evaluated using standard goodness-of-fit criteria, including AIC, AIC_c, BIC, and the Kolmogorov–Smirnov statistic. In all cases, the rbt-R model demonstrates a superior fit compared to existing Rayleigh-based models, indicating its effectiveness in modeling highly skewed and heavy-tailed data. Full article

In this study, we introduce a novel distribution related to the gamma distribution, referred to as the generalized incomplete gamma distribution. This new family is defined through a stochastic representation involving a linear transformation of a random variable following a distribution derived from the upper incomplete gamma function. As a result, the proposed distribution exhibits a probability density function that effectively captures data exhibiting asymmetry and both mild and high levels of kurtosis, providing greater flexibility compared to the conventional gamma distribution. We analyze the probability density function and explore fundamental properties, including moments, skewness, and kurtosis coefficients. Parameter estimation is conducted via the maximum likelihood method, and a Monte Carlo simulation study is performed to assess the asymptotic properties of the maximum likelihood estimators. To illustrate the applicability of the proposed distribution, we present two case studies involving real-world datasets related to mineral concentration and the length of odontoblasts in guinea pigs, demonstrating that the proposed distribution provides a superior fit compared to the gamma, inverse Gaussian, and slash-type distributions. Full article

This study investigates the effects of warming water on growth in seven demersal fish species including Atlantic cod (Gadus morhua), American plaice (Hippoglossoides platessoides), Greenland halibut (Reinhardtius hippoglossoides), roughhead grenadier (Macrourus berglax) and three species of redfish (Sebastes spp.) in the Northwest Atlantic and compares the changes in growth across species. Length-at-age data were collected from EU bottom trawl surveys from 1993 to 2018, and bottom temperature data were obtained from the Copernicus Marine Service. Generalised additive mixed models (GAMMs) were used to describe the temperature effects on growth. The analysis was carried out separately for males and females. Both sexes of all species except American plaice showed significant temperature effects on growth. To obtain the growth parameters, von Bertalanffy growth functions (VBGFs) were fitted to the predictions from best-fit GAMMs for all species and both sexes under five different bottom temperature scenarios (3, 3.5, 4, 4.5 and 5 °C). The predictions from all best-fit GAMMs were broadly similar in form to the fitted von Bertalanffy growth functions (R² > 90%). Increased bottom temperature generally resulted in a decrease in the asymptotic length (L_∞) and an increase in the growth rate (k). The species with the most dramatic increase in k over the temperature range of 3 °C to 5 °C was Atlantic cod, for which k increased from 0.05 to 0.13 year⁻¹ in females and from 0.08 to 0.14 year⁻¹ in males. The maximum length (L_max), predicted by the VBGF at maximum age generally declined from 3 °C to 5 °C. The species with the most pronounced decline in L_max was beaked redfish (S. mentella). An increase in the proportion of smaller individuals could impact population productivity and result in lower biomass available to fisheries. Uneven changes in fish growth in the warming ocean could also have wider ecological implications and alter the trophic landscape. Full article

Diphtheria is a severely infectious and deadly bacterial disease with Corynebacterium diphtheriae as the causative agent. Since the COVID-19 pandemic, contagious diseases such as diphtheria have re-emerged due to disruptions in routine childhood immunization programs worldwide. Nigeria is witnessing a significant increase in diphtheria outbreaks likely due to an inadequate health care system and insufficient public enlightenment campaign. This paper presents a mathematical epidemic diphtheria model in Nigeria, which includes a public enlightenment campaign to assess its positive impact on the prevalence of the disease. The mathematical analysis of the model reveals two equilibrium points: the diphtheria infection-free equilibrium and the endemic equilibrium. These equilibrium points are shown to be stable globally asymptotically if ${\mathcal{R}}_c<1$ and ${\mathcal{R}}_c>1$, respectively. The model was fit using the confirmed diphtheria cases data of Kano State from January to December 2023. Sensitivity analysis indicates that the transmission rate and recovery rate of asymptomatic peopleare crucial parameters to be considered in developing effective strategies for diphtheria control and prevention. This analysis also reveals that the implementation of a high-level public enlightenment campaign and its high efficacy effectively reduce the prevalence of diphtheria. Finally, numerical simulations show that combining the public enlightenment campaign and isolating infected individuals is the best strategy to contain the spread of diphtheria. Full article

To address a key issue in functional time series analysis on testing the randomness of an observed series, we propose an IID test for functional time series by generalizing the Brock–Dechert–Scheinkman (BDS) test, which is commonly used for testing nonlinear independence. Similarly to the BDS test, the proposed functional BDS test can be used to evaluate the suitability of prediction models as a model specification test and to detect nonlinear structures as a nonlinearity test. We establish asymptotic results for the test statistic of the proposed test in a generic separate Hilbert space and show that it enjoys the same asymptotic properties as those for the univariate case. To address the practical issue of selecting hyperparameters, we provide the recommended range of the hyperparameters. Using empirical data on the VIX index, empirical studies are conducted that feature the applications of the proposed test to evaluate the adequacy of the fAR$\left(1\right)$ and fGARCH$(1,1)$ models in fitting the daily curves of cumulative intraday returns (CIDR) of the index. The results reveal that the proposed test remedies some shortcomings of the existing independence test. Specifically, the proposed test can detect nonlinear temporal structures, while the existing test can only detect linear structures. Full article

Both extreme-excess modeling and extreme-value analysis of precipitation events frequently utilize the Generalized Pareto (GP) distribution to model peaks above a selected threshold. However, selecting an appropriate threshold remains a complex and challenging task, which has discouraged many practitioners from employing Pareto or Pareto–Poisson distributions for extreme-value analysis. Recent analyses of threshold selection methods proposed in the technical literature, particularly when applied to rainfall records with high quantization levels, have shown that nonparametric methods are often unreliable. Additionally, methods relying on the asymptotic properties of the GP distribution tend to produce unrealistically high threshold estimates. In contrast, graphical methods and goodness-of-fit (GoF) metrics that account for the pre-asymptotic behavior of the GP distribution have demonstrated better performance. Despite these improvements, there remains no automatic and statistically robust methodology for threshold selection. This study develops an automatic, statistically sound procedure for optimal threshold selection, leveraging weighted mean square errors and internally studentized residuals. The proposed method outperforms existing approaches in terms of accuracy, as demonstrated through numerical experiments and its application to real-world data from the NOAA NCDC Daily Rainfall Database. Results indicate that the method not only improves threshold estimation precision but also enhances the reliability of extreme-value analysis for precipitation records, making it a valuable tool for hydrological applications. The findings emphasize the practical implications of the method for analyzing extreme rainfall events and its potential for broader climatological studies. Full article

The purpose of this study was to compare heart rate recovery (HRR) after a maximal treadmill (MAX-TM) and three-minute all-out (3MT) test between firefighters (FF) and a control (CON) group. Nine male CON and nine male FF participants completed height (m), weight (kg), body fat percent (BF%), normalized handgrip (GRIP_NORM, kg/kg), and MAX-TM with direct gas analysis to capture aerobic capacity (VO_2PEAK, mL/kg/min). A shuttle-sprint 3MT was used to measure critical velocity (CV, m/s) and D′ (m). Non-linear models determined HR decay (HRR_τ), HR asymptote (HR_∞), and HR amplitude (HR_amp). Two-way GROUP (FF vs. CON) by TEST (MAX-TM vs. 3MT) repeated measures ANOVAs indicated a significant TEST (F = 7.004, p = 0.018) effect on HR_amp. When divided by VO_2PEAK classification (FITNESS), a significant TEST effect was observed (F = 7.661, p = 0.014) on HR_amp. VO_2PEAK was significantly related to CV (r = 0.583, p = 0.011), GRIP_NORM (r = 0.668, p = 0.002), and BF% (r = −0.890, p < 0.001). Complete autonomic nervous system recovery may depend on the intensity of task demands and cardiorespiratory fitness. Full article

Capturing the factors influencing yellow fever (YF) outbreaks is essential for effective public health interventions, especially in regions like Nigeria, where the disease poses significant health risks. This study explores the synergistic effects of active case detection (ACD) and early hospitalization on controlling YF transmission dynamics. We develop a dynamic model that integrates vaccination, active case detection, and hospitalization to enhance our understanding of disease spread and inform prevention strategies. Our methodology encompasses mechanistic dynamic modeling, optimal control analysis, parameter estimation, model fitting, and sensitivity analyses to study YF transmission dynamics, ensuring the robustness of control measures. We employ advanced mathematical techniques, including next-generation matrix methods, to accurately compute the reproduction number and assess outbreak transmissibility. Rigorous qualitative analysis of the model reveals two equilibria: disease-free and endemic, demonstrating global asymptotic stability and its impact on overall YF transmission dynamics, significantly affecting control and prevention mechanisms. Furthermore, through sensitivity analysis, we identify crucial parameters of the model that require urgent attention for more effective YF control. Moreover, our results highlight the critical roles of ACD and early hospitalization in reducing YF transmission. These insights provide a foundation for informed decision making and resource allocation in epidemic control efforts, ultimately contributing to the enhancement of public health strategies aimed at mitigating the impact of YF outbreaks. Full article