The impact of Lévy noise on a stochastic and fractal-fractional Atangana–Baleanu order hepatitis B model under real statistical data

The main focus of this paper revolves around the analysis of Lévy noise-driven Hepatitis B virus (HBV) infectious disease by considering the vaccination effect on the dynamical behaviour of epidemic. For accomplishing this, the existing and uniqueness techniques have been chosen for the feasible solution. In the nexus, a theoretical analysis of the stochastic model is led by the suitable Lyapunov function that broadly includes the existence and unique-ness of the positive solution, the dynamic properties around the disease-free equilibrium and the endemic equilibrium. To exterminate the diseases, a stochastic basic reproduction number “R₀” for the extinction is construed with the condition, if “R₀<1”, the disease could be extinct. Consequently, the fractional-order system is obtained by the model conversion process; the converted model lies under the Atangana–Baleanu derivative in the sense of Caputo with a fractal dimension of time and non-integer order. Moreover, the qualitative analysis is made by further probing the fractal fractional version of the proposed model. For further in-depth analysis and validation, the numerical simulations for both problems have been offered, in conjunction with comparing the stochastic and fractal-fractional approaches with the deterministic system. We believe that this study would provide a strong theoretical basis for understanding the spread of an epidemic, the adaptation of control strategies, and real-world problems in several academic fields.