For the problem that qualitative analysis methods are mostly adopted for the study of spam information propagation and it is difficult to reveal intrinsic propagation rules of spam information propagation, an ILDR (Ignorant-Lurker-Disseminator-Removed) model of spam information was proposed based on the idea of virus propagation modeling by considering the actual factors such as different input rates and removal rates. Firstly, the equilibrium point and the propagation threshold were calculated, and the stability conditions of the equilibrium point were given. Secondly, the local stability of non-spam information and spam information was proved by the Routh-Hurwitz criterion, then the global stability of non-spam information was certified via the invariance principle of LaSlle, which was proved based on the Bendixson criterion. Theoretical research shows that the non-spam information equilibrium is global asymptotically stable when propagation threshold is less than 1; the spam information equilibrium is global asymptotically stable when propagation threshold is greater than 1. The numerical simulation validates that the value of the propagation threshold can be decreased when decreasing the transfer rate from the lurker to the disseminator, increasing the transfer rate from the ignorant to the remover and the transfer rate from the lurker to the remover; the value of the disseminator can be decreased via increasing the proportionality coefficient from the ignorant to the lurker, or increasing the transfer rate of the disseminator to the remover and the system removal rate.

The impact of individual interest and refutation mechanism on rumor spread was investigated, and a new IWSR (Ignorant-Weak spreader-Strong spreader-Removal) rumor spread model was proposed. The basic reproduction number and equilibrium points of the model were calculated. Using Lyapunov stability theorem, Hurwitz criterion and LaSalle invariance principle, the local stability and global stability of some equilibrium points were proved. Through numerical simulations, it is concluded that increasing the effectivity of the governments' refutation actions or improving people' ability of rumor judgement can effectively suppress rumor spread. Finally, numerical simulations were conducted in WS (Watts-Strogatz) small-world network and BA (Barabási-Albert) scale-free network, showing that the network topology exerts significant influence on rumor spread.