Pre-exposure prophylaxis (PrEP) is an effective strategy for HIV prevention, offering individual protection and broader public health benefits. Enrollment in PrEP programs not only provides access to HIV prevention but also serves as a strategic entry point for the diagnosis of other sexually transmitted infections (STIs). Because participation requires regular HIV testing and routine STI screening (e.g., for syphilis), PrEP implementation facilitates early detection and treatment of coexisting infections, strengthening integrated sexual health surveillance and control efforts. We developed a mathematical model capturing syphilis dynamics, incorporating PrEP as a mechanism for diagnosis and treatment engagement. The model considers coinfection via high-risk sexual contact, partial protection of PrEP (HIV but not syphilis), and diagnostic pathways linked to PrEP program entry. Independent analysis of syphilis (without PrEP) established population persistence, basic reproduction numbers, and stability of disease-free and endemic equilibria. Integrating PrEP, we derived conditions under which PrEP-related parameters—particularly diagnostic access—positively influence syphilis transmission dynamics. Sensitivity analysis showed that higher PrEP adherence reduces reproduction numbers for syphilis and coinfection. Computational simulations using literature-based parameters confirmed these findings: increased PrEP use and lower discontinuation rates decreased new infections and improved treatment outcomes. These results highlight the role of PrEP in improving the detection and treatment of syphilis and HIV–syphilis coinfection.

Traffic congestion presents a critical challenge in contemporary urban environments, necessitating the development of effective traffic management systems. Microscopic traffic flow models, which offer detailed insights into individual vehicle dynamics such as car-following and lane-changing behaviors, are pivotal in addressing these challenges. However, a comprehensive review synthesizing the advancements and research trends in this field has been lacking. This paper presents a systematic review of major microscopic traffic flow research from 1950 to 2023. Our extensive search across multiple academic databases identifies significant methodologies and model equations, highlighting notable advancements in the field. The presentation reveals critical trends, including the integration of connected and autonomous vehicles, the application of machine learning techniques, and the increasing reliance on real-time data for traffic management. This paper provides a foundation for future research directions and contributes to the ongoing development of more efficient and sustainable urban traffic management strategies.

In this article, we present a new asymmetric distribution, the Topp-Leone modified Weighted Rayleigh (TLMWR) distribution, which extends the well-known Topp-Leone distribution. We derive several of its properties, including the probability density function, cumulative distribution function, survival function, failure (hazard) rate, moments, generating functions, quantile function, and order statistics. The model parameters are estimated by the method of maximum likelihood, and a simulation study is conducted to examine the finite-sample behavior of the estimators. We summarize key characteristics of the data using graphical displays and diagnostic procedures, including normality assessments and model-selection criteria. These analyses are performed on real-world data to assess the level and direction of skewness and kurtosis. The proposed distribution is then evaluated with a real-life dataset, and its performance is compared with existing and newly proposed distributions. The results support the validity of the proposed model and highlight its effectiveness relative to existing alternatives.

With rapid economic development and urbanization, many cities, particularly in China, face serious PM\(_{2.5}\) pollution issues. In this study, the city of Hefei is selected as the research area to investigate the factors influencing PM\(_{2.5}\) concentrations. Data on electricity consumption of major PM\(_{2.5}\)-emitting industries, meteorological factors (temperature, wind speed, wind direction, relative humidity), and atmospheric pollutant concentrations (NO\(_{2}\),SO\(_{2}\),O\(_{3}\),CO) are utilized to explore PM\(_{2.5}\) concentrations in Hefei from 2020 to 2021 using a generalized additive model (GAM). The aims are to identify the main influencing factors and potential control pathways for particulate matter pollution. Results reveal that CO accounts for 69% of the variation in PM\(_{2.5}\) mass concentration, suggesting it as the dominant factor in Hefei in 2020. Additionally, the major PM\(_{2.5}\)-emitting industries contribute to a 16% change in PM\(_{2.5}\) mass concentration, with a significant impact from smelting industries, which exhibit an increase in electricity consumption associated with an increase in PM\(_{2.5}\) mass concentration. Model fitting indicates that a 50% reduction in electricity consumption within the iron and steel making industries can lead to a 37% decrease in PM\(_{2.5}\) mass concentration compared to pre-reduction levels. Moreover, targeted control measures in winter result in higher reductions in PM\(_{2.5}\) pollution within a 40% reduction compared to consistent emission reductions throughout the year. These findings highlight the effectiveness of more focused control strategies based on localized circumstances. Implementing measures to restrict electricity use by key industries during high pollution seasons and in cities with high pollution levels can effectively address local PM\(_{2.5}\) pollution concerns.

In recent decades, a wide range of Hardy-Hilbert-type integral inequalities have been established. This article focuses on a one-parameter result introduced by Waadallah Tawfeeq Sulaiman in 2010, which has a unique structure: the double integral involves a power-sum of the variables, as well as a technical power-minimum. The sharp constant factor is also elegantly expressed in terms of the beta function. However, the parameter involved is subject to restrictions on its values. In this article, we refine the inequality by removing this restriction and addressing a theoretical gap in the original proof to yield a sharper result. We provide a thorough, step-by-step proof and demonstrate how this new result can be used to derive additional variants and extensions.

We study the Abel-type family \(y’=C\,y^r(1-y)^s\) under a parity-driven mapping of \((r,s)\), which yields symmetric dynamics for odd \(k\) and asymmetric, potentially stiff dynamics for even \(k\). We correct the normalization by peaking at the true maximizer \(y^\star=r/(r+s)\) and provide the analytic Jacobian \(g'(y)\) for implicit solvers. A matched-accuracy benchmarking protocol sweeps rtol/atol and reports global errors against ultra-tight references (separable/explicit for odd \(k\), Radau for even \(k\)), alongside wall time, \(nfev\), \(njev\), linear-solve counts, rejected steps, and step-size histories. Stiffness is quantified through the proxy \(\tau(t)=1/\lvert g'(y(t))\rvert\) and correlated with step-size adaptation; trajectories are constrained to \(y\in[0,1]\) via terminal events. Across tolerances, DOP853 and LSODA are strong all-rounders in non-stiff regimes, while Radau/BDF dominate when asymmetry and proximity to multiple roots induce stiffness; observed orders align with nominal ones under matched error. The study clarifies how parity and nonlinearity govern solver efficiency for polynomial nonlinearities and provides full environment details and code for reproducibility.

Let \(G\) be a graph of order \(n\) and size \(m\), with adjacency matrix eigenvalues \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_n\). The energy of \(G\), denoted by \(\mathcal{E}(G)\), is defined as the sum of the absolute values of its eigenvalues. A classical upper bound on the energy, originally established by McClelland [1], states that \(\mathcal{E}(G) \leq \sqrt{2mn}\,.\) In this paper, we refine the spectral analysis of graph energy by deriving an exact analytical expression relating \(\mathcal{E}(G)\) to the variance of the vector of absolute eigenvalues \(x = (|\lambda_1|, |\lambda_2|, \dots, |\lambda_n|)\,.\) Specifically, we prove that \(\mathcal{E}(G) = \sqrt{2mn – n^2 \operatorname{Var}(x)},\) providing a more precise and quantitative spectral characterization of graph energy. As an application, this identity allows us to derive improved lower bounds for \(\mathcal{E}(G)\), thereby strengthening and generalizing previously known inequalities. Furthermore we conjecture that for any non-singular graph \(G\) of order \(n\), \(\mathcal{E}(G) \geq 2 \sqrt{\langle d \rangle (n-1)},\) where \(\langle d \rangle = 2m/n\) is the average vertex degree of \(G\). Equality holds if and only if \(G \cong K_n\).

First, this paper provides some approximation and estimation type results for some moments of the Gauss function, motivated by the fact that the moments of even orders \(n=2l,\ l\in \mathbb{N}\mathrm{=}\mathrm{\{}0,1,\dots \}\) of the function \(exp\left(-t^2\right)\) on bounded intervals . Second, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subseteq \left[0,{1}/{\sqrt{2}}\right]\) is studied and solved. Here the point is using Jensen inequality. Third, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \(\left[0,b\right]\subset \left[0,+\infty \right)\) is deduced, via elements of complex analysis (Vitali’s theorem). The convergence holds uniformly on compact subsets of the complex plane. Fourth, the asymptotic behavior of the sequence of all moments on \(\left[0,1\right],\ \)as \(n\to \infty ,\) for an arbitrary function \(f\in C\left(\left[0,1\right]\right)\) is determined precisely, by means of Korovkin’s approximation theorem. Consequently, a similar result for complex analytic functions is deduced, using Vitali’s theorem. This is the fifth aim of the paper.