This paper developed a novel nonlinear Susceptible–Vaccinated–Exposed–Infectious–Treated–Recovered–Susceptible (SVEITRS) compartmental model to investigate the transmission dynamics of tuberculosis (TB) in Algeria over the period between 1990–2024, explicitly accounting for partial Bacillus Calmette-Guérin (BCG) vaccine efficacy, endogenous reactivation of latent infection, and exogenous reinfection. The basic reproduction number ${\mathcal{R}}_0$ was derived using the next-generation matrix approach, and rigorous analytical results were established for the local and global stability of both the disease-free and endemic equilibrium points. Model parameters were estimated by fitting the model to Algerian TB surveillance data, achieving a coefficient of determination of $R^2=0.84$, which indicates a strong agreement between the model predictions and reported epidemiological trends. A global sensitivity analysis based on partial rank correlation coefficients (PRCCs) reveals that demographic recruitment and transmission-related parameters exert the greatest influence on TB dynamics, whereas intervention-related parameters such as vaccination and treatment rates have a comparatively weaker impact. These findings indicate that vaccination alone is insufficient to eliminate TB and highlight the need for integrated control strategies that combine sustained vaccination coverage, early case detection, and effective management of latent infections. The proposed modeling framework provides a quantitative basis for evaluating TB control policies in high-burden settings such as Algeria.

Bipolar fuzzy sets (BPFs) provide a suitable framework for knowledge representation if some data contains imprecise and ambiguous information. In this manuscript, the lower and upper bounds of the Seidel Laplacian energy of a bipolar fuzzy graph were examined with suitable illustrative examples. Moreover, the energy of a bipolar fuzzy graph, the Laplacian energy of a bipolar fuzzy graph, and the Seidel Laplacian energy of a bipolar fuzzy graph were examined. Furthermore, to address complex multi-criteria decision-making (MCDM) problems involving uncertainty and bipolar information, we proposed novel score functions: The score function, improved score function, and double improved score function. These functions were demonstrated through examples to effectively handle ambiguity and duality in decision-makers' inputs represented via bipolar fuzzy sets.

This research extensively investigates the solitary wave solutions of the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. Three mathematical methods called the simple equation method, the modified extended auxiliary equation mapping method, and the exponential-expansion method are gradually employed to derive the analytical solutions. Moreover, Mathematica 13.0 software is used to perform the analytical computations and graphical simulations. The explored outcomes have significant applications in the realm of magnetic fields. After the careful selection of parametric values under constrained conditions, some solutions are plotted in 2-dimensional and 3-dimensional spaces to understand the physical phenomena of the concerned model. Importantly, our findings affirm that the employed methods not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. Hence, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies.

In this paper, we analyze and provide innovative soliton solutions for a (2+1)-dimensional generalized Korteweg-de Vries (gKdV) problem. We obtain phase shifts and dispersion relations by using the generalized Arnous technique and the Riccati equation approach, thus allowing different soliton solutions to be developed. Several precise solutions with special structural properties, including kink and solitary soliton solutions, are included in our study. This detailed examination demonstrates the complex behavior of the model and its capability to explain a large scale of nonlinear wave occurrences in many physical settings. Thus, in scientific domains such as fluid mechanics, plasma physics, and wave propagation in media ranging from ocean surfaces to optical fibers, our results are crucial to comprehend the principles behind the production and propagation of many complicated phenomena. Finally, we provide 2D and 3D graphs for various solutions that have been obtained using Maple.