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We propose a multi-stage structured rumor spreading model that consists of ignorant, new spreader, old spreader, and stifler. We derive a mean field equation to obtain the multi-stage structured model on homogeneous networks. Since rumors spread from a few people, we consider a large population by setting the number of initial spread to one in total population n and limiting n to ∞. We investigate a threshold phenomenon of rumor outbreak in the sense of the large population limit by studying the driven multi-stage structured model. The main conclusion of this paper is that the proposed model has a threshold phenomenon in terms of a basic reproduction number which is similar to the SIR epidemic model. We present numerical simulations to show the developed theory numerically.

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The spread of rumors is a phenomenon that has heavily impacted society for a long time. Recently, there has been a huge change in rumor dynamics, through the advent of the Internet. Today, online communication has become as common as using a phone. At present, getting information from the Internet does not require much effort or time. In this paper, the impact of the Internet on rumor spreading will be considered through a simple SIR type ordinary differential equation. Rumors spreading through the Internet are similar to the spread of infectious diseases through water and air. From these observations, we study a model with the additional principle that spreaders lose interest and stop spreading, based on the SIWR model. We derive the basic reproduction number for this model and demonstrate the existence and global stability of rumor-free and endemic equilibriums.

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