Stochastic differential equation models are important and provide more valuable outputs to examine the dynamics of SARS-CoV-2 virus transmission than traditional models. SARS-CoV-2 virus transmission is a contagious respiratory disease that produces asymptomatically and symptomatically infected individuals who are susceptible to multiple infections. This work was purposed to introduce an epidemiological model to represent the temporal dynamics of SARS-CoV-2 virus transmission through the use of stochastic differential equations. First, we formulated the model and derived the well-posedness to show that the proposed epidemiological problem is biologically and mathematically feasible. We then calculated the stochastic reproductive parameters for the proposed stochastic epidemiological model and analyzed the model extinction and persistence. Using the stochastic reproductive parameters, we derived the condition for disease extinction and persistence. Applying these conditions, we have performed large-scale numerical simulations to visualize the asymptotic analysis of the model and show the effectiveness of the results derived.

The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ( $\mathcal{A}\mathcal{B}\mathcal{C}$) derivative. A systematic and convergent technique known as the Laplace Adomian decomposition method (LADM) is applied to extract a semi-analytical solution for the considered equation. The notion of fixed point theory is used for the derivation of the results related to the existence of at least one and unique solution of the mKdV equation involving under $\mathcal{A}\mathcal{B}\mathcal{C}$-derivative. The theorems of fixed point theory are also used to derive results regarding to the convergence and Picard's X-stability of the proposed computational method. A proper investigation is conducted through graphical representation of the achieved solution to determine that the $\mathcal{A}\mathcal{B}\mathcal{C}$ operator produces better dynamics of the obtained analytic soliton solution. Finally, 2D and 3D graphs are used to compare the exact solution and approximate solution. Also, a comparison between the exact solution, solution under Caputo-Fabrizio, and solution under the $\mathcal{A}\mathcal{B}\mathcal{C}$ operator of the proposed equation is provided through graphs, which reflect that $\mathcal{A}\mathcal{B}\mathcal{C}$-operator produces better dynamics of the proposed equation than the Caputo-Fabrizio one.