Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives, when emergencies occur, rumors will spread widely on the Internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens' behavior and attitude, which make the transmission rates of these information among social network groups being not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation that (1)increasing noise intensity can effectively reduce rumor propagation when $\widehat{\mathcal{R}}_{0}>1$. That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens' opinions, thus reducing the degree of rumor propagation; (2) Speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen(i.e. increase the value of $\beta$ and decrease the value of $\alpha$ and $\gamma$ ) can effectively curb rumor propagation.