Modeling the dynamics of information propagation in the temporal and spatial environment

Abstract

In this paper, we try to establish a non-smooth susceptible–infected–recovered (SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R 0 to study the relationship between R 0 and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I c , we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem.