Dynamics of a multi-platform reaction–diffusion IESR rumor propagation model with general incidence and time-delay

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The rampant rumors on the Internet pose a serious challenge to the social order. This paper delves into the propagation mechanism of rumors on multiple social platforms, creatively incorporating the generalized correlation function and time-delay into the reaction–diffusion IESR rumor propagation model, which provides a new perspective for the study of rumor propagation. First, the positivity of the solution is proved by using the upper and lower solution theory of the proposed mixed monotonic system. The next generation matrix method is used to calculate the basic reproduction number (BRN) [Formula: see text]. Furthermore, by constructing a Lyapunov function, we prove that the rumor free equilibrium point [Formula: see text] is globally asymptotically stable (GA-stable) when [Formula: see text], and the rumor equilibrium point is globally asymptotically stable (GA-stable) when [Formula: see text]. In addition, the sensitivity analysis is carried out on parameters associated with [Formula: see text], which can enable us to gauge the influence of diverse factors on rumor dissemination and gain deeper insights into the dynamic nature of rumors. Finally, we employ numerical simulations to validate the accuracy of our conclusions and illustrate the model’s applicability through real-world cases.