Newcastle disease continues to have a significant economic impact on farmers and food security. Thisstudy develops and analyzes a deterministic mathematical model to investigate the effect of vaccinationon the transmission dynamics of Newcastle disease in caged chicken populations. The model is basedon the Susceptible Exposed-Infected Vaccinated Revaccinated Recovered Susceptible (SEIVVrRS) framework,adapted to capture the unique characteristics of Newcastle disease transmission. The disease free equilibriumof the model was computed, and the basic reproduction number for Newcastle disease was calculated usingthe next generation matrix method. Both analytical results and numerical simulations show that frequentvaccinations increase the number of susceptible chickens by reducing the at-risk chicken population. Additionally, re-vaccination significantly enhances immunity, resulting in a higher number of recovered chickens.Sensitivity analysis indicates that the recruitment rate of chickens, the effective contact rate between susceptible and infectious chickens, and the natural death rate of chickens are the most sensitive parameters fortargeting in disease control strategies. Therefore, the findings from this study can support farmers and foodsecurity practitioners in decision making regarding Newcastle disease control strategies and emphasize theircrucial role in poultry disease management

This study places a significant emphasis on assessing the efficiency of numerical methods, specifically in the context of solving linear equations of the form Ax = b, where A is a square matrix, x is a solvent vector, and b is a column vector representing real-world phenomena. The investigation compares the effectiveness of the Refined Successive Over Relaxation (RSOR) method to the standard Successive Over-relaxation (SOR) method. The core evaluation criteria encompass computational time (in seconds), convergence behavior, and the number of iterations necessary to approximate the solvents of five distinct real-world phenomena: Model Problem 1 (MP1) involving an Electrical Circuit, Model Problem 2 (MP2) focusing on Beam Deflection, Model Problem 3 (MP3) addressing Damped Vibrations of a Stretching Spring, Model Problem 4 (MP4) dealing with Linear Springs and Masses, and Model Problem 5 (MP5) focusing on Temperature Distribution on Heated Plate. The RSOR method generally outperforms the SOR method, particularly with a constant relaxation parameter (ω) in the range 1.0 < ω < 1.2. The RSOR method is favored for its robustness and efficiency with less need for fine-tuning ω, whereas the SOR method can achieve superior performance if the optimal ω is found, although this often requires time-consuming trial and error. Despite the potential for better performance with an optimal ω, the RSOR method’s consistent results make it the more practical choice in many cases. The study also explores the stability of the systems of linear equations arising from these phenomena by calculating their condition numbers (K(A)). More interestingly, the results reveal that all systems MP1 to MP4 exhibit instability when subjected to even modest perturbations, shedding light on potential challenges in their solvents. This research not only underscores the advantages of the RSOR method but also emphasizes the importance of understanding the stability of numerical solvents in the context of real-world problems. Additionally, the results for MP5 demonstrates that tiny changes to the original matrix’s coefficients have no effect on the desired solvent because the perturbed matrix’s condition number is the same as the original matrix’s, making the problem well-structured. The problem becomes ill-conditioned if there is an increase or decrement to the matrix’s coefficients that is bigger than 10⁻⁵. In summary, Sparse systems are sensitive to perturbations, resulting in instability. If the tolerance |k(A⁰_i) − k(A_i)| > 10⁻⁵ for all positive integers i, then the problem becomes poorly structured.

This study incorporates thermal energy, temperature‐dependent properties, thermal‐dynamic irreversibility in MHD fluid flow embedded in a porous vertical medium driven by Newtonian cooling. The study considers an incompressible, electrically conducting fluid flowing laminarly in a vertical channel bounded by two infinite parallel plates. The governing equations are developed and discretized using Crank‐Nicolson finite difference scheme (CN‐FDM) where the discretized equations are simulated using MATLAB software. The findings reveal that rising magnetic field by 2% lowers both velocity and temperature up to 0.2% with intensified entropy reduction near the walls. Amplifying permeability by 4% intensifies fluid motion while Bejan number falls by approximately 7%, stabilizing unrecovered fluid friction. Escalating the radioactive parameter by 6% improves the temperature profile while lowering Bejan number. Rising pressure gradient deteriorates the Bejan number by approximately 9% caused by stabilized fluid friction across the channel. Moreover, the temperature profile and Bejan number fall by 12% due to an increase in Prandtl number by 5%. Fluid motion deteriorates by 4% while entropy production at the central region of the channel is reduced by 13% with the increase in Eckert number. Escalating heat generation by 8% amplifies the temperature profile by 16% and reduces fluid resistance by 23%, with intensified entropy production near the walls. Fine‐tuning buoyancy forces may lower entropy production by approximately 30%. Velocity and temperature reach a steady state with buoyancy, while time increases, wall shear and heat transfer before stabilization. The findings reveal that variations of these physical quantities have a great influence to coefficient of skin friction and Nusselt number on both walls. These results inform the MHD and thermal‐fluid systems designers to ensure fine‐tuning of magnetic field parameter, Darcy number, Eckert number, internal heat generation, Prandtl number, and buoyance effects that enhance effective thermodynamic in these systems.

We introduce a Susceptible-Corrupted-Paying Well-Recovered (SCPwR) model specifically designed to analyze corruption dynamics within the public sector. This model is demonstrated to be well posed both epidemiologically and mathematically. Our results show that all model solutions remain positive and bounded given initial conditions within a meaningful set. We investigate the existence of unique corruption-free and endemic equilibrium points and calculate the basic reproduction number. The local and global stability of these equilibrium points is then analyzed. Our findings indicate that the system has a locally asymptotically stable corruption-free equilibrium when R_e < 1 and unstable when R_e > 1 . Additionally, the corruption endemic equilibrium point E^* is globally asymptotically stable only if dL/dt< 0 . Numerical implementation of the model suggests that corruption will persist in public sectors if civil servants are not adequately compensated.