In this manuscript, we develop a fractional-order mathematical model to characterize the propagation dynamics of COVID-19 outbreaks and assess the influence of vaccination interventions. The model comprises a set of eight nonlinear fractional-order differential equations in the Caputo sense. To establish the existence and uniqueness of solutions, we employ the fixed-point technique. Furthermore, we employ the effective fractional Adams-Bashforth numerical scheme to explore both the approximate solutions and the dynamic behavior inherent to the examined model. All of the results are numerically visualized through the consideration of various fractional orders. Furthermore, the real data from three different countries are compared with the simulated results, and good agreements are obtained, revealing the effectiveness of this work.

In this article, we methodically investigate the fractional stochastic Kraenkel–Manna–Merle system (KMMS), which explains how a magnetic field propagates in a ferromagnet with zero conductivity and may shed light on a number of intriguing scientific occurrences. A suitable wave transformation is used to convert the governing equation into an ordinary differential equation (ODE). We thoroughly evaluate the innovative soliton solutions in the forms of dark, bright–dark, dark–bright, periodic, singular, hyperbolic, mixed trigonometric, and rational forms using the improved $\mathcal{F}$-expansion approach and the new extended direct algebraic method (NEDAM). Furthermore, a sensitivity analysis is carried out to investigate the impact of different factors on the behavior of the system. In order to shed light on the model's physical behavior, the study displays graphical plots of the chosen solutions using the selected methodologies. These techniques offer a strong foundation for resolving nonlinear fractional differential equations, which are crucial for simulating intricate ferromagnetistic physical processes. The resulting solutions demonstrate the fractional stochastic KMMS's complex structures and dynamic behavior.