New insights of a stochastic nonlinear differential equations model with ergodic stationary distribution via fractal-fractional derivatives involving random perturbations

Abstract

Rumor circulation has been a serious challenge to preserving the established societal structure. Recently, significant attempts have been made to explore the patterns of rumors using epidemic models, assuming that only rumor streams for a particular incident occur in the network. In this research, the rumor dynamics during catastrophes are discussed using a unique rumor spreading mechanism via the novel fractal-fractional derivative operators (Caputo and Caputo-Fabrizio) sense by proposing white noise in the system. By strengthening a comprehensive stochastic Lyapunov function, which gives us a reasonable representation of permanence. Furthermore, we established the existence-uniqueness of the non-negative solution of the model. Besides that, we are capable of demonstrating the necessary requirements for the persistence and exclusivity of an ergodic stationary distribution for the aforesaid model. Numerical configurations were investigated to illustrate the effect of model parameters, power law and exponential decay kernel with fractional-order and fractal-dimension on the rumor metastatic phenomenon. Finally, the corresponding management solutions with the fractal-fractional operators for preventing the proliferation of rumors during catastrophes are also covered in this research. AMS Subject Classification: 26A51; 26A33; 26D07; 26D10; 26D15.