In this paper, we model edge traffic with a conformable fractional partial differential equation that keeps memory in time and space. The solution represents a unit‐free attack pressure, built from a z‐scored edge series, a quiet period baseline, and a partially absorbing boundary that reflects scrubbing and rate limits. Using a closed form atomic solution, we split the edge signal into a growth part and a damped part for early warning, and use a simple spatial factor to estimate how far pressure can travel inside. The result is a small, readable model that turns one edge metric into clear, tier‐specific actions during DDoS attacks.

This study assesses whether Bitcoin’s linkage with AI equities remains robust after accounting for equity risk sentiment. To this end, the study employs the multiscale quantile‐on‐quantile correlation (MSQQC) and multiscale quantile‐on‐quantile partial correlation (MSQQPC) approaches, using daily data covering 02/01/2019–16/06/2025. The results indicate that BTC–AI comovement is strongly state‐ and frequency‐dependent rather than stable across the joint distribution or across horizons. In the high‐frequency band, dependence is weak and only intermittently significant, with localised negative regions around BTC ≈ 0.20 with AI ≈ 0.30–0.50 and BTC ≈ 0.30 with AI ≈ 0.70. In the mid‐frequency band, significance concentrates in the tails, showing negative dependence under downside stress conditions such as BTC ≈ 0.10–0.30 with AI ≈ 0.10, alongside sign changes when BTC is in upper‐tail states. In the low‐frequency band, dependence becomes broadly positive and significant across most quantile combinations, with limited decoupling when AI is highly elevated (≈ 0.80–0.90) and BTC is also in upper quantiles (≈ 0.70–0.90). Importantly, conditioning on VIX and VVIX does not materially alter these patterns, suggesting that sentiment influences segments of short‐run dependence but does not overturn the longer‐run BTC–AI linkage. The study derives policy recommendations from these findings.

We introduce a novel family of trigonometric basis functions equipped with a shape parameter, analogous to Bernstein functions. These basis functions are employed to construct Bézier‐like curves, termed “trigo‐curves”, which retain the fundamental properties of classical Bézier curves while offering enhanced shape control through parameter adjustment, independent of control point positions. We establish the monotonicity‐preserving nature of these curves. Additionally, we develop a new class of linear positive operators based on these trigonometric basis functions. The operators, incorporating an auxiliary parameter, are thoroughly analyzed for their fundamental properties. We establish their convergence rate, derive a modified Voronovskaja theorem, and obtain error bounds in terms of the modulus of continuity. Furthermore, the monotonicity‐preserving properties of these operators are also investigated.

The study introduces a novel numerical framework for chaotic financial models, characterized by sensitive dynamics. Our method diverges from traditional liberalization and subdivision approaches, directly tackling chaotic systems without simplification. By integrating two linear multistep methods via a multistep inversion, we achieve seamless continuous‐to‐discrete formulation, ensuring robust convergence and precision.

Various studies have examined generalized additive models (GAMs), comparing thin plate splines (tp), P‐splines (ps), cubic regression splines (cr), and Gaussian processes (gp) for discrete choice data, function approximation, and in the presence of multicollinearity and outliers. Some studies have applied ps to models with correlated and heteroscedastic errors, while others have reviewed multiple smoothing term packages for modeling GAMs. This study seeks to examine the performance of semiparametric models in the presence of different noise and outliers within the framework of GAMs through simulation. The study adopted four GAMs, cr, ps, tp, and gp, for simulated data with different noise and outliers with varying sample sizes. According to our investigation, the cr model performs well in terms of deviance for the majority of sample sizes and all types of noise. With higher sample sizes, the ps model frequently performs well, particularly in terms of AIC and GCV under noise that is heteroscedastic and Gaussian. The gp model excels with the smallest sample size under Gaussian and lognormal noise in terms of GCV, and the tp model frequently performs best under exponential and lognormal noise for larger samples in terms of AIC and GCV. For data containing outliers, the cr and tp models are effective with smaller sample sizes, while the gp model excels with larger sample sizes based on AIC and GCV. Regarding deviance, the cr model consistently performs best across all sample sizes. Our results show that the sample size and kind of noise in the data have a significant impact on the smoothing model’s performance. No single model consistently outperforms the others for all noise types and sample sizes, suggesting that the choice of model should be based on the specific goal of a study.

On the global arena, drug and substance abuse (DSA) problem has caused unprecedented suffering. The number of the youth and adult population developing addictions has been on a steady rise, according to the United Nations Office on Drugs and Crime (UNODC) report. A new mathematical model incorporating the impact of policing and rehabilitation on proliferation of DSA in the community is designed and examined in this research. This study postulates that the propagation of DSA is comparable to that of horrendous contagious disease. The effective reproduction number ( R 0 ) of the proposed deterministic model provides the substratum for stability analysis. From the analysis, the DSA‐free equilibrium (DSAFE) is corroborated to be locally asymptotically stable when R 0 < 1 and unstable for R 0 > 1. The local stability of DSA endemic equilibrium (DSAEE) is explored by employing the centre manifold technique. The study rules out the possibility of occurrence of backward bifurcation. Using Python software, various simulation scenarios were run in order to correlate the outcomes with the analytical findings. The simulation results show that increasing the rate at which light drug abusers (LDAs) and heavy drug abusers (HDAs) are identified and placed under structured treatment and therapy, as well as proactive response by intercepting and dismantling trafficking networks via efficacious law enforcement, would adversely diminish drug abuse menace not only in Kenya but in the world at large.

In this work, a compartmental model for the coinfection of bacterial meningitis and Hepatitis B was formulated and analyzed. The most sensitive parameters were identified, including the contact rates for Hepatitis B virus (HBV) and bacterial meningitis, the progression rate of bacterial meningitis, and the recruitment rate. Based on the results of the sensitivity analysis, the system was extended into an optimal control model by incorporating four intervention strategies: bacterial meningitis prevention, Hepatitis B vaccination for susceptible individuals, and treatments for individuals infected with bacterial meningitis, those with chronic Hepatitis B, and those who are coinfected. The controls are characterized using Pontryagin’s maximum principle. The optimality system is simulated using the forward–backward sweep Runge–Kutta scheme. Subsequently, the incremental cost‐effectiveness ratio (ICER) analysis method is applied to determine the most cost‐effective strategy. The results indicate that combining bacterial meningitis prevention efforts with Hepatitis B vaccination represents the most cost‐effective intervention.

Reference [7] in the reference list should be corrected to "Novikov (1965)".

A proposed nonlinear autoregressive model was employed to predict daily new COVID‐19 infections in Iraq. This model was applied to actual COVID‐19 data from 3 months in 2022 and met the stability criteria set by the proposed nonlinear regression model, based on a model developed by Japanese scientist Ozaki. The primary objective of the article was to examine the stability of this proposed nonlinear time series model, aimed at assessing the model’s stability and determining its stability conditions. To achieve this, the article applied Ozaki’s method to convert the nonlinear model, which depends on a singular point, into a linear model that fulfills the stability conditions required for the study. The approximation method helped identify the singular point, its stability conditions, and the limit cycle of the proposed model. The article also presented numerical case studies that satisfied the mathematical criteria for finding the singular point and its stability conditions. The findings indicated that such numerical case studies meeting stability conditions exist when the nonlinear function is decreasing, as illustrated in both Ozaki’s model and the proposed model. The proposed model was applied to COVID‐19 data from Iraq over a 3‐month period in 2022. This data was transformed into a stable form, and a Python program used the converted real data and proposed model to estimate the real parameters of the first‐order model, fulfilling the established stability conditions and forecasting future new cases of the disease, which was the research’s main goal. The statistical criteria used in the fifth paragraph of the research are the Akaike information criterion (AIC), Bayesian information criterion (BIC), and normalized Bayesian information criterion (NBIC).

Nosocomial infections, such as those caused by carbapenem‐resistant Klebsiella pneumoniae (CRKP) transmitted into hospitals, worsen patient outcomes, strain healthcare infrastructure, increase treatment costs, prolong hospital stays, and complicate medical treatment due to their high level of antibiotic resistance. Taking into account these detrimental aspects, this study deals with a dynamic system of nonlinear differential equations with maintenance preventive strategies, and treatment applying combination therapy has been considered to prevent CRKP transmission and reduce antibiotic resistance while accounting for two hospital populations. The analytical analysis includes positivity, boundedness, existence of a unique solution, stability at equilibrium points, reproduction number ( R 0 ), sensitivity, and convergence of state variables. And, the numerical simulations were performed using the fourth‐order Runge–Kutta method. We find that the proper implementation of hygiene practices among hospitalized patients and the consistent use of personal protective equipment (PPE) by healthcare workers play an important role in preventing CRKP transmission in hospitals. Additionally, applying combination therapy as a treatment approach for CRKP reduces antibiotic resistance. Consequently, this improves hospital safety and ensures a healthier environment by minimizing the spread of infectious agents and reducing the risks of nosocomial infections.