During infectious disease outbreaks, media dissemination, and time delays critically influence the transmission dynamics. To capture the coupled interactions among media influence, time delay, and hospitalization resources, we propose a delayed susceptible–exposed–infected–hospitalized–recovered–media (SEIHRM) epidemic model incorporating media feedback and systematically investigate its dynamic behavior both theoretically and numerically. The model is shown to be well-posed in terms of the existence, uniqueness, positivity, and boundedness of solutions. The basic reproduction number R₀ is derived, thereby determining the stability of the disease-free equilibria and serving as a threshold for the existence of endemic equilibria. Bifurcation analysis with the time delay τ as the bifurcation parameter reveals that once τ exceeds a critical threshold τ₀, a Hopf bifurcation emerges, giving rise to sustained periodic oscillations and thereby recurrent epidemic outbreaks. Numerical simulations substantiate the theoretical results and reveal the distribution of stable equilibria, periodic oscillations, and unstable dynamics across parameter regimes. The results demonstrate that the coupling between media dissemination and time delay can drive complex oscillatory dynamics via delayed feedback, providing a mechanistic explanation for both endemic persistence and recurrent epidemic outbreaks. This study offers a novel theoretical perspective on information-feedback-driven epidemic dynamics and provides a quantitative basis for delay-based targeted intervention strategies.

Deploying machine learning models for credit decisions under regulatory mandates for explainability and reliability requires evaluation frameworks beyond predictive accuracy. Yet, benchmarking studies in credit risk still focus narrowly on the area under the receiver operating characteristic curve (AUC), neglecting probability calibration, uncertainty decomposition, and interpretability. We address this gap by benchmarking ten models, spanning deterministic and probabilistic families, across four evaluation dimensions on a proprietary dataset of 50,819 customers from a Norwegian financial institution. Each customer is represented by 124 consecutive daily account balance observations and a binary 12-month default flag. Across 180 configurations combining five feature engineering modules with four resampling strategies, variational inference-based Bayesian models (Bayesian neural network, Bayesian logistic regression, convolutional-Bayesian neural network) exhibit posterior collapse on this imbalanced tabular dataset, producing near-uniform predicted probabilities and near-zero epistemic uncertainty, particularly under oversampling. Natural gradient boosting (NGB) emerges as the most effective probabilistic model, matching random forest and XGBoost in test AUC (0.80) while providing informative aleatoric uncertainty estimates and economically interpretable SHAP attributions. We further show that feature engineering and resampling effects are non-additive and model-specific, indicating both as first-order design variables in credit risk pipelines. These findings extend the literature to probabilistic evaluation dimensions and provide practitioners with guidance for model selection in regulatory-compliant credit decision systems under similar data conditions. Note that our performance figures represent exploratory upper-bound benchmarks rather than strictly unbiased out-of-sample estimates.

In this article, a new fractional-order chemostat model incorporating delay is formulated. Making use of the Lipschitz condition, we confirm the existence and uniqueness of the solution to the formulated model. By defining a reasonable function, we prove that the solution of the proposed model is bounded. The criterion of the stability and Hopf bifurcation emerging from the developed model is established by taking advantage of the stability theory and the Hopf bifurcation theorem of fractional-order differential systems. By contriving two distinctive controllers, we can effectually adjust the stability region and the delay length of the onset of Hopf bifurcation of the developed model. Finally, computer simulation experiments are provided to support the acquired conclusions. The derived results play an important role in balancing the concentrations of the substrate and the cell mass.

This paper is concerned with an alternating direction implicit (ADI) general finite difference method (GFDM) for solving three-dimensional nonlocal diffusion equations with anisotropic diffusion coefficients. The Grünwald-Letnikov (GL) approximation is adopted to discretize the Caputo fractional derivatives on uniform grids. This spatial direction is handled by using the general difference method. A fully discrete GL-ADI general difference scheme is strictly constructed. The existence, uniqueness, stability, and convergence of numerical solution are proved. In addition, numerical experiments are carried out to confirm the validity of the theoretical analysis.

The rapid proliferation of online rumors undermines social stability, demanding dynamic analyses that transcend traditional steady-state assumptions. This study proposes a delayed Unknowns Neutrals Forwarders Refuters (UNFR) model that introduces discrete time delays into R → U, N → R, and N → F transitions to account for memory decay and decision-making hesitation. Employing characteristic equation analysis and Hopf bifurcation theory, alongside center manifold and normal form techniques, we examine the system’s stability dynamics. We demonstrate that equilibria remain locally asymptotically stable below a critical delay; however, exceeding this threshold induces a Hopf bifurcation, giving rise to sustained periodic oscillations. Numerical simulations corroborate this transition from stability to periodicity, offering a theoretical basis for understanding the cyclical resurgence of rumors and informing effective, time-sensitive intervention strategies.

Media coverage plays a crucial role in the spread of infectious diseases. To investigate the influence of media intervention on epidemic transmission, we adopted a media-aware susceptible–infected–recovered (SIR) model based on media-intervention epidemic studies. Our primary focus was the systematic optimization of Physics-Informed Neural Network (PINN) architectures. The dynamic behavior of the epidemic system was investigated through equilibrium, stability, and bifurcation analyses. The theoretical analysis characterized the threshold dynamics of the model through the basic reproduction number R₀, while numerical simulations further indicated that media intervention can reduce the infection peak and delay its arrival. Based on this epidemic framework, PINNs were employed for numerical simulation, and three algorithm optimization approaches were systematically investigated: (1) Replacing Euler-generated synthetic reference trajectories with RK4-generated reference trajectories to examine the influence of numerical discretization accuracy on PINN training and evaluation; (2) increasing the network depth from three to five layers to enhance the approximation of nonlinear epidemic dynamics; and (3) introducing skip-connection architectures, including Residual, Highway, and an exploratory residual-integrated Highway (ResHighway) network. Comparative results showed that the Residual network achieved the best overall accuracy and stability. Although the proposed ResHighway architecture yielded marginal accuracy gains compared with the Highway network, it underperformed the lightweight Residual network. This observation implies that overcomplicating network architectures may not bring consistent performance improvements to PINNs applied in epidemic dynamic modeling.

This paper proposes a symbolic computational method using neural networks to solve Boussinesq equations. By combining activation functions such as sine, hyperbolic tangent, hyperbolic secant, and exponential, we constructed a single-hidden-layer model [3-3-1] and a double-hidden- layer model [3-3-2-1] to solve the (2+1)-dimensional and (1+1)-dimensional Boussinesq equations, respectively. This yielded multiple sets of novel, exact analytical solutions, which were visualized using 3D plots, contour plots, heatmaps, and evolution plots. To address the issues of insufficient direct neural network feature representation and poor generalization ability, this paper introduces a feature- enhanced neural network method. Model [(2-5)-3-1] was constructed for the (1+1)-dimensional equation, expanding the number of input neurons from 2 to 5; Model [(3-9)-3-1] was constructed for the (2+1)-dimensional equation, expanding the number of input neurons from 3 to 9. Both models retain 3 hidden-layer neurons and optimize the network structure based on the equation’s dimension. By reconstructing the physical, spatiotemporal, and nonlinear characteristics of the equations and coupling them with inherent constraints, this method offers the advantages of richer solution results, higher accuracy, and significantly improved model generalization. It does not require bilinear transformations, features a simple structure, and has low computational cost, providing a new, viable solution for solving strongly nonlinear water wave equations.

This work investigates the exponential stability of a perturbed non-linear (or linear) wave equation in a sub-critical regime with respect to both the damping and perturbation terms. Using an energy-based analysis, we first establish the decay of the total energy over a finite time interval T > 0. When the damping region satisfies the geometric control condition (GCC), and the perturbations contribute to the dissipation mechanism, we prove that the energy of the system decays exponentially by combining the multiplier method with a unique continuation principle. In addition to the theoretical analysis, we present numerical simulations for one- and two-dimensional damped wave equations with localized internal damping. Finite difference schemes are employed to approximate the dynamics and quantify the dissipation of energy, and the results confirm the theoretical predictions: Introducing a localized damping region induces strong stabilization, and the total discrete energy decays exponentially over time.

In recent years, the study of partial differential equations (PDEs) and their systems in graph settings has emerged as a prominent research area, driven by their significant applications in image processing and data clustering. Motivated by these developments, we aim to extend certain existence results for biharmonic-Kirchhoff systems and biharmonic systems from the Euclidean setting (continuous models) to their counterparts in locally finite graphs (discrete models). Our primary focus is on establishing the existence of infinitely many solutions for a biharmonic-Kirchhoff system defined on a locally finite graph by employing the symmetric mountain pass theorem as our key analytical tool. Under the assumption that the nonlinear term is super-quartic, we prove that the system admits infinitely many solutions. Additionally, we present analogous results for the biharmonic system. A notable observation from our study is that the results obtained in the locally finite graph setting surpass those in the Euclidean setting. This improvement is fundamentally attributed to the more favorable embedding properties inherent to the locally finite graph framework.

This paper investigates the soliton solution and periodic lump solution of the (3 + 1)- dimensional generalized Bogoyavlensky–Konopelchenko equation with variable coefficients. First, by introducing specific constraint conditions, a novel bilinear expression for the aforementioned equation is derived. Subsequently, with the aid of symbolic computation, the corresponding soliton and periodic lump solution are successfully constructed. Finally, three-dimensional plots, density plots, and x-axis projection diagrams are presented to analyze the interaction behaviors of the obtained solutions, which provides intuitive visual evidence for the physical characteristics of these solutions.

A novel average-valued condition for consensus of multi-agent systems (MASs) is developed via a quantized adaptive control protocol, a framework that simultaneously addresses multiple self-delays, time-varying coupling, and logarithmic quantizers. A adaptive controller is designed, where each follower applies a quantized error signal and a shared adaptive gain that is updated by the quantized sum of all followers’ squared errors with a fixed adaptation rate. By constructing a Lyapunov functional and applying a generalized Halanay inequality that exploits the time-average property of the coupling strength, we prove that the adaptive gain is nondecreasing, remains bounded, and converges to a finite limit. A gain threshold, depending explicitly on the average coupling strength and the delay-related Lipschitz constants, is established: If the gain exceeds this threshold, exponential consensus is achieved; otherwise, asymptotic consensus still holds. Numerical simulations on a leader-follower network with Lorenz-type node dynamics and two self-delays validate the theoretical results, including the effect of different adaptation rates, quantization densities, and delay configurations.

In this paper, we consider a semilinear reaction–diffusion equation with a mixed local–nonlocal spatial operator and a time integral operator. The equation involves the mixed local–nonlocal Laplacian operator $\mathcal{L}=-\Delta+(-\Delta)^{\alpha / 2}$ and the Riemann–Liouville fractional integral operator $I_{0 \mid t}^{1-\gamma}$. Our main goal is to characterize the large–time asymptotic behavior of global mild solutions.

This paper introduces and examines a flexible numerical strategy for solving the time-fractional telegraph equation. A finite difference scheme is used for temporal discretization, while spatial discretization is achieved using a discontinuous Galerkin method that incorporates generalized alternating numerical fluxes. From an implementation standpoint, these generalized alternating fluxes offer enhanced flexibility and greater adaptability compared with conventional flux selections. We prove that the resulting numerical scheme is unconditionally stable and convergent. Several numerical examples are provided to demonstrate the robustness, reliability, and accuracy of the proposed method.

This work addresses the consensus tracking problem for connected vehicle systems under time-varying communication topologies and persistent intermittent communication. A coordinated event-triggered protocol relying only on local and neighboring vehicle information is developed to enhance resilience against channel blockages while reducing bandwidth usage and energy consumption. The communication digraph switches within a finite set of directed graphs, each containing a spanning tree rooted at the lead vehicle. By constructing a piecewise Lyapunov functional and applying the average dwell-time technique, sufficient conditions in terms of linear matrix inequalities are derived to guarantee exponential stability and leader-following consensus in the tracking error dynamics. The proposed scheme is proven to exclude Zeno behavior, ensuring a positive lower bound on inter event intervals. Simulation results confirm the effectiveness and superiority of the strategy under simultaneous switching topologies and intermittent communication.

Hepatitis B virus (HBV) infection can progress from acute infection to chronic liver disease, posing a major public health concern. In this work, a mathematical model incorporating immune response and vaccination is developed using susceptible, infected, chronic, and recovered populations. Positivity and boundedness of solutions are established, and disease-free and endemic equilibria are analyzed. The basic reproduction numbers and stability conditions are derived using the next-generation matrix and Jacobian methods. Critical thresholds are obtained as $\Upsilon^{*}$ = 0.2237 for immune response and Γ^* = 0.7466 for vaccination. Numerical simulations using the fourth-order Runge–Kutta method demonstrate that enhanced immune response and vaccination can reduce HBV infection and chronic cases. The findings provide quantitative evidence on the role of immune response and vaccination in controlling HBV transmission and disease progression.