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In recent years, rumor propagation in social networks attracts more researchers' attention. In this paper, we have established I2S2R rumor spreading models in both homogeneous networks and heterogeneous networks considering the effect of time delay. In the homogeneous network model, we obtain the basic reproduction number by means of the next-generation matrix. Besides, the local stability and the global stability of the equilibrium points are discussed by linearization approach of nonlinear systems and Lyapunov function. In the heterogeneous network model, we calculate the basic reproduction number through algebraic method. In addition, Lyapunov functional method and Lasalle invariance principle are applied to study the stability of equilibrium points in the complex network model. Further, we put forward some useful strategies to control the spreading of rumor based on the complex network theory. Finally, we take advantage of numerical simulations to verify the theory above and come up with necessary conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we improve an Ignorant-Lurker-Spreader-Removal (ILSR) rumor propagation model as in [Yang et al., 2019] in social networks with consideration to Logistic growth and two discrete delays. First, we prove the existence of equilibrium points by calculating the basic reproduction number according to the next generation matrix. Regarding the two discrete delays as bifurcating parameters, the local asymptotical stability and Hopf bifurcation of the positive equilibrium point are discussed for six different scenarios by analyzing the characteristic equations of linearized systems. Applying the normal form theory and the center manifold theorem, some important conclusions about the stability and direction of bifurcating periodic solution are given when the two time delays are equal. Subsequently we study the global stability of the equilibrium points by constructing Lyapunov functions when the two delays disappear. Finally, we verify the conclusions through numerical simulations and perform sensitivity analysis on the basic reproduction numbers. [ABSTRACT FROM AUTHOR]

• A network epidemic model with nonlinear incidence rate and time delay is regarded. • Stability is proved in a strict mathematical way. • Four different control strategies are investigated and compared in this paper. • Numerical simulations provide a new insight into epidemic diffusion on complex networks. In this paper, we establish a susceptible-infected-susceptible (SIS) epidemic model with nonlinear incidence rate and time delay on complex networks. Firstly, according to the existence of a positive equilibrium point, we work out the threshold values R 0 and λ c of disease propagation. Secondly, we demonstrate the stabilities of the disease-free equilibrium point and the disease-spreading equilibrium point by constructing Lyapunov function and applying delay differential equations theorem. Thirdly, four different control strategies are investigated and compared, including uniform immunization control, acquaintance immunization control, active immunization control and optimal control. Finally, we perform representative numerical simulations to illustrate the theoretical results and further discover that the nonlinear incidence rate can more accurately reflect individual psychological activities when a certain disease outbreaks at a high level. [ABSTRACT FROM AUTHOR]

Abstract With rapid emergence of online social networks, the rumor transition has a growing significant impact on social stability and human lives. This paper investigates the local stability and optimal control of a rumor propagation model in online social network with changing number of total users, which is proposed in Ref. Dong et al. (2018). In addition, we set up a model with a delay based on the reality, and analyze the sufficient conditions for Hopf bifurcation. Finally, the rumor spreading model is simulated numerically. The results of simulation are shown the process of rumor propagation and prove the correctness of the theoretical analysis. Highlights • This paper presents an important improvement in theory and numerical analysis for Ref. Dong et al. (2018). • Bifurcation and stability switches are proved in a strict mathematical way. • Optimal control strategy of rumor propagation is investigated and compared in this paper. [ABSTRACT FROM AUTHOR]

A series of online rumours have seriously influenced the normal production and living of people. This paper aims to study the combined impact of psychological factor, propagation delay, network topology and control strategy on rumour diffusion over the online social networks. Based on an online social network, which is seen as a scale-free network, we model the spread of rumours by using a delayedSIS(Susceptible and Infected) epidemic-like model with consideration of psychological factor and network topology. First, through theoretical analysis, we illustrate the boundedness of the density of rumour-susceptible individuals and rumour-infected individuals. Second, we obtain the basic reproduction numberR0and prove the stability of the non-rumour equilibrium point and the rumour-spreading equilibrium point. Third, control strategies, such as uniform immunisation control, proportional immunisation control, targeted immunisation control and optimum control, are put forward to restrain rumour diffusion. Meanwhile, we have compared the differences of these control strategies. Finally, some representative numerical simulations are performed to verify the theoretical analysis results. [ABSTRACT FROM AUTHOR]

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder's fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Mobile wireless sensor networks (MWSNs) have become an area of intense research activity due to technical advances in sensors, wireless communications and networking, and signal processing. Many applications, including environment monitoring, battlefield surveillance, and urban search and rescue especially in hazardous situations, are envisaged. However, MWSNs may be vulnerable to malicious interference because of the large-scale characteristics. When a contaminated node communications with its neighbors, multiple copies of the malware are transmitted to its neighbors, which may destroy, block regular communications, or even damage the integrity of regular data packets. Modeling spatial distribution of malware propagation over time is the first step to predict the trend of malware propagation in MWSNs. We propose a novel wireless malware propagation model with the discrete time delay based on reaction–diffusion equations in mobile wireless sensor networks, and study its dynamic behaviors. By analyzing the stability and Hopf bifurcation of the equilibrium of our model, we search for the sufficient conditions, which leads to the malware propagation disappears or continues. Furthermore, we demonstrate that oscillations in this model occur through the destabilization of the stationary solution at a Hopf bifurcation point. And formulas for determining the stability of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidence shows that the spatial–temporal dynamic characteristics of malware propagation in MWSNs are closely related to the packet transmission rate, the communication rang and the mobile behavior of nodes. [ABSTRACT FROM AUTHOR]

With the rapid development of network information technology, information networks security has become a very critical issue in our work and daily life. This paper investigates a nonlinear malware propagation model in wireless sensor networks (WSNs) based on SIR epidemic model. Sufficient conditions for the local stability of the positive equilibrium point and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation. Moreover, formulas for determining the properties of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Furthermore, with the help of the Maximum Principle of Pontryagin, we design an optimal control strategy for the previous model to extend the region of stability and reduce the density of infected nodes in WSNs. Finally, we conduct extensive simulations to evaluate the proposed model. Numerical evidence shows that the dynamic characteristics of malware propagation in WSNs are closely related to the immune period of a recovered node and the rate constant for nodes becoming susceptible again after recovered. Besides, we obtain that the optimal control strategy effectively improves the performance of the networks. [ABSTRACT FROM AUTHOR]

In the new social media era, it is becoming increasingly important to explore the propagation rules for rumors in social networks. This article is concerned with investigating a diffusive susceptible-infected rumor propagation model with a nonlinear propagation function in a spatially heterogeneous environment. We establish the uniform persistence and analyze the asymptotic behavior of the rumor-spreading steady state for the spatially heterogeneous model when one of the diffusion coefficients tends to zero. Moreover, to better reflect the effect of a time delay on the process of rumor propagation, we establish a spatially homogeneous model with a time delay and prove the existence and local stability of the corresponding equilibrium point. Furthermore, the optimal control in the spatially homogeneous environment case is derived. Finally, several numerical simulations are performed to verify the theoretical results in both spatially heterogeneous and spatially homogeneous systems. [ABSTRACT FROM AUTHOR]

The emergence of rumors is subjective and aggressive which will bring disturbance and misleading to people's lives. This paper is dedicated to comprehensively investigating the influence of forced silence function, time delay and network topology on rumor propagation in online social networks. Considering the network supervision such as forced silence and network topology, a novel delayed SIR(susceptible-propagating-recovery) epidemic-like rumor propagation model in both homogeneous and heterogeneous networks is proposed. By using the mean-field theory, we analyze the propagation dynamics of the model in detail. Firstly, we demonstrate the boundedness of the solutions, then we obtain the basic reproduction number and the existence of the rumor-prevailing equilibrium points. Secondly, the forward and backward bifurcation as well as the local and global stabilities of the equilibrium points in both homogeneous and heterogeneous networks are theoretically analyzed and revealed. Finally, the possible impact is analyzed through numerical simulations, and the factors affecting rumor propagation are theoretically analyzed, which prove the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Considering the ability of network users to distinguish and refute the rumor, we establish a Susceptible–Believed–Denied (SBD) rumor spreading model with nonlinear incidence rate and time delay in complex networks. Specifically, two dynamical systems of rumor propagation are given based on the mean field theory in homogeneous and heterogeneous networks respectively. Then, we obtain the equilibrium points of the systems and calculate the basic reproduction number R 0 according to the next generation matrix in homogeneous networks and the existence of a positive equilibrium point in heterogeneous networks. The local and global asymptotic stabilities of the rumor-free equilibrium points are demonstrated by constructing Lyapunov functions under the given condition R 1 ≤ 1. Further, we perform representative numerical simulations to illustrate the theoretical results. Through simulation experiments, the dynamics around the rumor-spreading equilibrium points when R 0 > 1 are also discussed to complement some dynamical properties of our models. • This paper presents a novel SBD rumor spreading model combined with classifying thought. • Comparative analysis of homogeneous and heterogeneous networks is numerically given. • Stability is proved in a strict mathematical way. [ABSTRACT FROM AUTHOR]

With the development of electronic technology and the diversification of network platforms, the in-depth study of rumor spreading is becoming more and more popular. In this paper, we find the basic reproduction number in a different way with Dong et al. (2017). Meanwhile, we analyze the local stability, the global stability and the optimal control of the SEIR rumor spreading model based on Dong et al. (2017). Besides, we should not neglect that time delay in the process of rumor spreading. Therefore, we propose a modified rumor spreading model and discuss its local and global stability. Finally, we perform the numerical simulations to prove the exactness of theoretical analysis and redress the simulation in Dong et al. (2017). • This paper presents an important improvement in theory and numerical analysis for Dong et al. (2017). • Global stability is proved in a strict mathematical way. • Optimal control strategy of rumor propagation is investigated and compared in this paper. [ABSTRACT FROM AUTHOR]