Rumors are abnormal children of public opinion, they usually arise with environmental crisis and have serious negative functions. Hence, how to control rumor propagation has become an urgent topic for the government. Since the communication between people in the real world is inevitably interfered by random factors, and time delay exists in the process of rumor propagation not only influence rumor-forwarding individuals by rumor-hesitant individuals but also in government decision-making. Based on these premises, we propose a stochastic delayed rumor propagation model with generalized incidence function. By setting the suitable Lyapunov functions, we show that the asymptotic behavior of stochastic rumor spreading model is constrained by some conditions. Finally, our analytical results are supported by numerical simulations. The results show that (1) the increase of noise intensity in social environment will reduce rumor propagation; (2) time delay has an impact on rumor propagation to a certain extent. Extending the thinking time may reduce the number of rumor-forwarding individuals and even lead to the disappearance of rumors; (3) improve scientific literacy and accelerate rumor refutation (i.e. reduce the value of β), strengthen rumor suppression (i.e. increase the value of γ) can effectively control rumor transmission.

The rapid development of information society highlights the important role of rumors in social communication, and their propagation has a significant impact on human production and life. The investigation of the influence of uncertainty on rumor propagation is an important issue in the current communication study. Due to incomprehension about others and the stochastic properties of the users' behavior, the transmission rate between individuals on social network platforms is usually not a constant value. In this paper, we propose a new rumor propagation model on homogeneous social networks from the deterministic structure to the stochastic structure. First, a unique global positive solution of the rumor propagation model is obtained. Then, we verify that the extinction and persistence of the stochastic rumor propagation model are restricted by some conditions. If $R̂0*<1$ and the noise intensity $σi(i=1,2,3)$ satisfies some certain conditions, rumors will extinct with a probability one. If $R0*>1$⁠, rumor-spreading individuals will persist in the system, which means the rumor will prevail for a long time. Finally, through some numerical simulations, the validity and rationality of the theoretical analysis are effectively verified. The numerical results show that (1) on the premise that other parameters are determined, the increase of noise intensity can effectively control the spread of rumors; (2) cut off the way of spreading rumors and reduce the contact between ignorant and rumor-spreading individuals (i.e., reduce the value of α); popularize scientific knowledge, reducing the attraction of rumors (i.e., increase the value of β) or replacing rumors of emergencies with other hot topics (i.e., increase the value of η) can effectively curb rumor propagation.

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number ${\mathcal{R}}_0$ and the basic immunity reproduction number ${\mathcal{R}}_1$. The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters.

In daily lives, when emergencies occur, rumors will spread widely on the internet. However, it is quite difficult for the netizens to distinguish the truth of the information. The main reasons are the uncertainty of netizens’ behavior and attitude, which make the transmission rates of these information among social network groups be not fixed. In this paper, we propose a stochastic rumor propagation model with general incidence function. The model can be described by a stochastic differential equation. Applying the Khasminskii method via a suitable construction of Lyapunov function, we first prove the existence of a unique solution for the stochastic model with probability one. Then we show the existence of a unique ergodic stationary distribution of the rumor model, which exhibits the ergodicity. We also provide some numerical simulations to support our theoretical results. The numerical results give us some possible methods to control rumor propagation. Firstly, increasing noise intensity can effectively reduce rumor propagation when ℛ ^ 0 > 1 . That is, after rumors spread widely on social network platforms, government intervention and authoritative media coverage will interfere with netizens’ opinions, thus reducing the degree of rumor propagation. Secondly, speed up the rumor refutation, intensify efforts to refute rumors, and improve the scientific quality of netizen (i.e., increase the value of β and decrease the value of α and γ), which can effectively curb the rumor propagation.