Mathematical modeling is a broad field that greatly impacts and contributes to collaborative research investigations. The models Improve research on the fundamental the quantitative attitude and dynamics of diseases that are infectious that harm humans, including COVID-19, human immunodeficiency virus (HIV), and hepatitis B virus. This paper
introduces a framework that supports the analysis of COVID-19 using an epidemic model that incorporates vaccination and treatment. The framework enables the examination of both non-pharmaceutical interventions and pharmaceutical interventions. The preservation of the basic reproduction number ensures the standardization of the stability of disease-free (DFE) and endemic equilibria. The local stability of the endemic and disease-free equilibria is proven using the Routh-Hurwitz criterion. The demonstration of disease-free and endemic equilibria convergence and divergence is demonstrated through the utilization of Standard and non-standard finite differences (SFD and NSFD, respectively) techniques.
It might be argued that SFD schemes, specifically Runge-Kutta order four (RK-4) and Euler schemes, demonstrate convergence at smaller step sizes. However, the NSFD scheme is intended to enhance understanding of the dynamic behavior of the continuous model. Empirical evidence demonstrates that the NSFD technique converges, irrespective of the chosen step size. The latter refers to a powerful, effective, and dependable technique
that provides a clear representation of the continuous model. Numerical simulations are employed to validate all the data, enhancing our understanding of the causes of the illness. The theoretical and quantitative results of its study can serve a valuable for mechanism tracking the transmission as to COVID-19.
Raed Hameed Mahdi, Shah Zeb, Ayesha Kamran et al.· Punjab University journal of...· 0 citations
This study presents an adaptive modified Runge-Kutta compact scheme for the numerical simulation of unsteady k − ω turbulent nanofluid flow over a heated moving surface under local thermal non-equilibrium conditions. The surface-interfacial model incorporates mixed convection, viscous dissipation, turbulence transport, and separate energy equations for the base fluid and nanoparticle phases, with the effective thermal conductivity described by Xue’s formulation. The proposed time-integration method is explicit and combined with a compact finite-difference discretization that provides fourth-order spatial accuracy. The temporal coefficients are selected to achieve second-order accuracy, and the method is further enhanced through adaptive time stepping based on local error control. Stability analysis for the scalar convection–diffusion problem and conditional convergence analysis for the corresponding system formulation are also established. Numerical comparisons show that the proposed adaptive scheme yields lower error than existing adaptive Euler- and Runge-Kutta-based schemes. The computed results further demonstrate that thermal buoyancy increases the mean velocity, whereas larger Prandtl numbers reduce the thermal boundary-layer thickness of the fluid and nanoparticle phases. In addition, stronger interphase coupling modifies the two-temperature fields in a manner consistent with local thermal nonequilibrium. A machine-learning model is also employed to predict eddy viscosity, and its reliability is confirmed through profile comparisons, contour analyses, sensitivity assessments, and Taylor diagram evaluations. Overall, the proposed framework provides an accurate and efficient computational tool for surface-associated turbulent nanofluid transport with interfacial thermal nonequilibrium.
M. Arif, Yasir Nawaz· Advances in Differential Equ...· 0 citations
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