This paper is devoted to investigating the impact of the recurrence of rumors and individual behaviors and control strategies related to rumor spreading in online social networks. To do this, a novel susceptible-hesitating-infected-latent-recovered (SHILR) rumor propagation model in heterogeneous networks is presented. Firstly, based on the relevant mean-field equations of the model, the threshold value is examined to demonstrate the existence and stability of rumor-free/spreading equilibrium with the help of the algebraic method. Secondly, the global stabilities of the equilibria are analyzed through applying Lyapunov stability theory and LaSalle’s invariance principle. Next, the optimal control is proposed by taking advantage of Pontryagin’s maximum principle for reducing the number of infected individuals with minimum cost. Moreover, some numerical examples are carried out to test the theoretical results. Finally, combined with practice, a model application is presented.

The ubiquitous Wsp (wrinkly spreader phenotype) chemosensory system and DSF (diffusible signal factor) quorum sensing are two important chemically associated signaling systems that mediate bacterial communications between the host and environment. Although these two systems individually control biofilm formation in pathogenic bacteria via the ubiquitous second messenger c-di-GMP, their crosstalk mechanisms remain elusive. Here we present a scenario from the plant-beneficial and antifungal bacterium Lysobacter enzymogenes OH11, where biofilm formation favors the colonization of this bacterium in fungal hyphae. We found that the Wsp system regulated biofilm formation via WspR-mediated c-di-GMP signaling, whereas DSF system did not depend on the enzymatic activity of RpfG to regulate biofilm formation. We further found that WspR, a diguanylate cyclase (DGC) responsible for c-di-GMP synthesis, could directly bind to one of the DSF signaling components, RpfG, an active phosphodiesterase (PDE) responsible for c-di-GMP degradation. Thus, the WspR-RpfG complex represents a previously undiscovered molecular linker connecting the Wsp and DSF systems. Mechanistically, RpfG could function as an adaptor protein to bind and inhibit the DGC activity of unphosphorylated WspR independent of its PDE activity. Phosphorylation of WspR impaired its binding affinity to RpfG and also blocked the ability of RpfG to act as an adaptor protein, which enabled the Wsp system to regulate biofilm formation in a c-di-GMP-dependent manner by dynamically integrating the DSF system. Our findings demonstrated a previously uncharacterized mechanism of crosstalk between Wsp and DSF systems in plant-beneficial and antifungal bacteria.

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Considering the media coverage and age‐dependent education, a deterministic and a stochastic class‐age‐structured rumor propagation models are studied, respectively. First, the deterministic rumor propagation model is characterized by a coupled system of ordinary and partial differential equations. The positivity and boundness of solutions are proved, and the basic reproduction number is derived. Second, the stochastic rumor propagation model is formulated by stochastic differential equations. The existence of global positive solutions in model is discussed with the Itô's formula and stochastic Lyapunov function. Additionally, by utilizing comparison principle of stochastic differential equations and the strong law of large numbers, several sufficient conditions for extinction and persistence of the rumor are derived. Finally, numerical simulations are carried out for illustrating the results.