Global dynamics and Impact of Gaussian noise intensity on the stochastic epidemic model with local fractional derivative

Society must understand, model, and forecast infectious disease transmission patterns in order to prevent pandemics. Mathematical models and computer technology may help us better understand the pandemic and create more systematic and effective infection management strategies. This study offers a novel perspective through a compartmental model that incorporates fractional calculus. The first scenario is based on proportional fractional definitions, considering compartmental individuals of susceptible, moving susceptible, exposed, infected, hospitalized, and recovered. Through an extension of this derivative, they decimated the model to integer order. We extended the deterministic model to a stochastic extension to capture the uncertainty or variance in disease transmission. It can develop an appropriate Lyapunov function to detect the presence and uniqueness of positive global solutions. Next, we discuss how the epidemic model might have become extinct. In our theoretical study, we demonstrated that a sufficiently outrageous amount of noise can cause a disease to become extinct. A modest level of noise, on the other hand, promotes the persistence of diseases and their stationary distribution. The Khasminskii method was used to determine the stationary distribution and ergodicity of the model.