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Turing pattern for explaining the spatial distribution in nature mostly focused on continuous media and existing networks but there are few attempts at studying them on the systems with high-order interactions. Considering that high-order interactions have a particularly significant impact on rumor propagation, this article establishes a generalized reaction-diffusion rumor propagation model based on a multiplex network, where simplicial complexes are employed to describe the high-order structures. It aims to provide the spatial distribution patterns of the population participating in rumor propagation and identify structural factors that affect such patterns. We theoretically give the necessary conditions for Turing instability in the single-layer and multiplex networks with consideration of high-order interactions. In the numerical simulation, we demonstrate that Turing pattern is controlled by adjusting the diffusion coefficient, high-order structure intensity, and average degree of the network. The results conclude that: (i) in a single-layer network, Turing pattern only exists when high-order interactions appear, and the difference in diffusion rate plays a decisive role, (ii) in a multiplex network, Turing pattern can still be observed under the same diffusion rates, which are affected by the difference of higher-order intensity between two layers, and (iii) in the existing networks, the average degree of the network takes an important impact on Turing pattern. All these findings contribute to comprehending the impact of network structure on pattern formation, especially the high-order interactions on Turing pattern.

In this paper, a fractional-order susceptible-infected-recovered-susceptible (SIRS) model is studied, focusing on delay effects and high-order interactions. Two types of time delays are considered to describe latent period and healing cycle, respectively. From the ecological point of view, we found that the increasing delays caused by either the latent period or the healing cycle lead to the periodic outbreak of disease. The finding provided us with an important implication to preventing periodic outbreaks of disease by reducing the time delay, like accelerating the healing process with effective medication and medical intervention. Specifically, taking the time delays as bifurcation parameters, the stability of endemic equilibria and the existence of Hopf bifurcation are studied by analyzing the characteristic equation of the SIRS model. From a general point of view, based on the establishment of a fractional-order SIRS model, we found that the order of the fractional order is critical for describing the dynamic behavior of the model. Typically, the decrease of the order appears to bring about the disappearance of the periodic phenomenon (i.e. the periodic oscillation) of the originally stable system.