Abstract
In this research, we investigate the dynamical behaviour of a HPAI epidemic system featuring a half-saturated transmission rate and significant evidence of crossover behaviours. Although simulations have proposed numerous mathematical frameworks to portray these behaviours, it is evident that their mathematical representations cannot adequately describe the crossover behaviours, particularly the change from deterministic reboots to stochastics. Furthermore, we show that the stochastic process has a threshold number R-0(S) that can predict pathogen extermination and mean persistence. Furthermore, we show that if R-0(S) > 1, an ergodic stationary distribution corresponds to the stochastic version of the aforementioned system by constructing a sequence of appropriate Lyapunov candidates. The fractional framework is expanded to the piecewise approach, and a simulation tool for interactive representation is provided. We present several illustrated findings for the system that demonstrate the utility of the piecewise estimation technique. The acquired findings offer no uncertainty that this notion is a revolutionary viewpoint that will assist mankind in identifying nature.