Abstract
In this work, we develop and analyze a fractional-order stochastic model for COVID-19 transmission, incorporating the effects of vaccination. The model is formulated using the Atangana-Baleanu fractional derivative in the Caputo sense, which captures memory and hereditary properties of disease transmission more accurately than classical derivatives. We first examine the positivity and boundedness of the deterministic fractional model and determine its equilibrium points. The model is then extended to a fractional stochastic differential equation (FSDE) to account for random fluctuations and uncertainties in disease dynamics. We establish the existence and uniqueness of solutions for the FSDE model using stochastic analysis techniques. To numerically solve the FSDE, we develop a novel numerical scheme that accommodates the non-local nature of the Atangana-Baleanu derivative. The real data of COVID-19 in Pakistan have been used to estimate the model parameters. Numerical simulations are presented for both the deterministic fractional model and its stochastic counterpart. These simulations illustrate the impact of the fractional-order parameter on disease dynamics, showing how different orders influence the rate of infection and convergence to equilibrium. Additionally, we analyze scenarios under varying parameter values to explore conditions for disease elimination, highlighting the role of vaccination and stochasticity. Our findings demonstrate that fractional stochastic modeling provides deeper understanding of COVID-19 transmission dynamics and can be a valuable tool in assessing control strategies under uncertainty. The originality of this work stems from the integration of Atangana-Baleanu fractional derivatives with stochastic modeling, supported by a new numerical solution method providing a more realistic and flexible approach to modeling COVID-19 transmission under uncertainty.