Under what conditions can fake news be eradicated using corrective information? What mechanisms allow for the coexistence of conflicting rumors? Addressing these questions is crucial. The spread of rumors has been conventionally investigated using epidemic models, whereas changes in opinions have been analyzed using opinion models. However, almost no frameworks integrate both.

To establish such a framework, the standard susceptible–infected–susceptible (SIS) model is first considered. In this baseline model, individuals are either “susceptible” (S, not spreading a rumor) or “infected” (I, actively spreading a rumor). A rumor—serving as an infectious disease—infects an S individual and causes a transition to I with a certain probability per unit time per I individual in a population. Meanwhile, an I individual loses interest and reverts to S with a certain probability per unit time. The fate of the outbreak is dictated by the basic reproduction number—the expected number of secondary infections generated by a single I in a completely susceptible population, which is determined solely by transition probabilities. A reproduction number < 1 causes the infection to shrink and die out, whereas a reproduction number > 1 causes exponential growth, thus resulting in a pandemic.

In this study, an SIS competition model is proposed to capture the dynamic interactions between rumor spread and opinion changes. The model introduces two conflicting rumors, A and B. Accordingly, individuals are classified into one of four states: S_A, I_A, S_B, or I_B. An S_A individual supports rumor A but is not spreading it, whereas an I_A individual supports rumor A and is actively spreading it. If an S_A individual is infected by I_B, they then change their opinion from A to B and become I_B. Importantly, the infection probability of a rumor differs depending on the rumor currently supported by the individual; for example, the transition probability for S_A → I_A is different from that for S_B → I_A. Through such mechanisms, individuals change their opinions based on the rumors they hear; conversely, a rumor’s transmissibility changes depending on their opinions.

Linear stability analysis was performed to determine all the steady states and their local stabilities. Crucially, in the proposed model, rumor A can survive even if its reproduction number for S_A → I_A is less than 1. This is because new spreaders can emerge through S_B → I_A transitions. If the reproduction number for S_A → I_A is less than 1 but still relatively high, then rumor A depletes I_B individuals. Because rumor A relies on rumor B to sustain its spread, eradicating I_B causes I_A individuals to become extinct. Therefore, rumor A can paradoxically survive and coexist because of its low transmissibility.

(Written by Yu Takiguchi on behalf of all authors.)

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