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This study establishes a reaction–diffusion system to capture the dynamics of rumor propagation, considering two possibilities of contact transmission. The sufficient and necessary conditions for a positive equilibrium point are provided, and the Turing instability conditions for this equilibrium point are derived. Furthermore, utilizing variational inequalities, a first-order necessary condition for parameter identification based on optimal control is established. During the numerical simulation process, the correctness of the Turing instability conditions is verified, and optimal control-based parameter identification is applied to the target pattern. Additionally, statistical methods are employed for pattern parameter identification. The identification results demonstrate that optimal control-based parameter identification exhibits higher efficiency and accuracy. Finally, both theories' parameter identification principles are extended to a small-world network, yielding consistent conclusions with continuous space. [ABSTRACT FROM AUTHOR]

This paper takes a reaction-diffusion predator-prey system with ratio-dependent Holling III functional response function and Leslie-Gower term into consideration. First of all, the system model is proposed on the basis of basic biological assumptions and previous work, and the existence conditions of the equilibrium point of the system are discussed. Secondly, under the assumption of the existence of equilibrium point, the Turing instability necessary conditions induced by diffusion are investigated. Thirdly, optimal control theory is derived, and the adjoint system and the first-order optimization condition are established. Fourthly, parameter identification based on optimal control is studied, and the technique is extended to network structure. Finally, extensive numerical simulations, including Turing pattern, Normalized Population High Distribution Area (NPHDA) diagram and parameter identification, are carried out to illustrate and validate the analytical results. For the system dynamics phenomena, the results from different perspectives effectively demonstrate that the theoretical findings, numerical simulations and natural reality are identical. In terms of the parameter identification of continuous model and network model, the efficiency and accuracy of various algorithms are fully tested. • This paper proposes and investigates index normalized population height distribution area. • The optimal control problem of reaction-diffusion predator-prey system is formally derived. • The parameter identification research is extended to population network structure space. [ABSTRACT FROM AUTHOR]

The harm caused by rumors is immeasurable. Studying the dynamic characteristics of rumors can help control their spread. In this paper, we propose a nonsmooth rumor model with a nonlinear propagation rate. First, we utilize the positive invariant regions to prove the boundedness of solutions. Second, we analyze the conditions for the existence of equilibrium points in both the left and right systems. Additionally, we confirm the occurrence of saddle-node bifurcation in the left system. Next, by considering the influence of spatial diffusion, we establish the conditions for Turing instability. Then we discuss the conditions for spatial homogeneous and inhomogeneous Hopf bifurcations in the left and right systems, respectively. We differentiate between supercritical and subcritical bifurcations using the Lyapunov coefficient. Furthermore, we examine the conditions for the existence of discontinuous Hopf bifurcation at the demarcation point. Finally, in the numerical simulation section, we validate our theorems on Turing patterns. We also investigate the impact of parameter changes on rumor propagation and conclude that an increase in the psychological inhibitory factor significantly reduces the rate of rumor propagation, providing an effective strategy for curbing rumors. To that end, we fit actual data to our system and the results are excellent, confirming the validity of the system. [ABSTRACT FROM AUTHOR]

In this work, the global dynamic behavior of a new reaction–diffusion multi-group SVEIR (Susceptible-Vaccinated-Exposed-Infectious-Recovered) rumor propagation model is studied. Compared with the traditional SVIR and SEIR models, the new model takes into account the latent of rumors, that is, believing rumors does not mean spreading rumors, and the impact of official rumor refutation on rumor propagation, which will lead to positive changes in the results of rumor propagation. The expression of basic reproduction number R 0 obtained through the next generation matrix method. By constructing Lyapunov function and applying graph-theoretic approach, we have obtained that the rumor eliminating equilibrium point E 0 is globally asymptotically stable when R 0 ≤ 1 , and the rumor spreading equilibrium point E ∗ is globally asymptotically stable when R 0 > 1 . The conclusion is verified by numerical simulation. In addition, the effects of time delay and the total number of plates are also given by numerical simulation. [ABSTRACT FROM AUTHOR]

Rumors may cause serious harm to social productivity and life. Studying the dynamics of rumor propagation can help provide corresponding countermeasures to control communication effectively. Based on the classic S I infectious disease model, this paper studies a rumor spreading model with a time delay caused by communication delay between rumor communicators and nonsmooth threshold propagation function in reality. In this model, we solve non-negative equilibrium points firstly. By analyzing the characteristic equations of smooth and nonsmooth rumor systems, we establish local stability at non-negative equilibrium points and find the existence of Hopf bifurcation from two aspects of diffusion coefficient and delay, respectively. In addition, considering the discontinuity of rumor system, we separate it into two systems and get the condition of Hopf bifurcation when the system is discontinuous. We further verify the feasibility of the theoretical results by simulation results. [ABSTRACT FROM AUTHOR]

With the development of the times, rumors spread rapidly on the Internet. Firstly, this paper establishes a reaction-diffusion system with Allee effect to describe the rumor spreading process and derives the necessary conditions for the emergence of Turing bifurcation. Next, a parameter identification approach utilizing optimal control theory is shown. Ultimately, the impact of the magnitude of the certain parameters in the objective function on parameter identification is examined through numerous parameter identifications in continuous space and various complex networks. Additionally, the convergence rates and error magnitudes of different algorithms for parameter identification are studied across different spatial structures. • The necessary conditions for the emergence of Turing bifurcation are investigated. • This paper applies optimal control theory to achieve parameter identification. • The effects of the three algorithms and topologies on parameter identification are studied. [ABSTRACT FROM AUTHOR]

In this paper, we establish a predator–prey model with Beddington–Deangelis functional response. For this model, firstly, the existence of the positive equilibrium point and the conditions for the Turing instability are studied. Then the amplitude equation is derived through weakly nonlinear analysis, and the relationship between the selection of pattern and the coefficients of the amplitude equation is obtained. At the same time, through a large number of numerical simulations, we verify the accuracy of the theoretical analysis. Actually, we mainly choose to change the values of parameters r and d 2 to study the sensitivity of the pattern to them. When the pattern tends to be stable, there could be the pattern of spots, coexistence of spots and stripes or stripes. Even if they are spot pattern, the spot density will also be different due to the selection of parameters. Finally, we simulate and compare that the network structure(mainly BA and WS) has a certain influence on the time required for pattern stabilization and the distribution of node density. The final results show that the growth rate of prey, diffusion coefficients and network structure all play an important role in the formation of Turing pattern. [ABSTRACT FROM AUTHOR]

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 virtue of identifying the influence of nodes, the spatial distance of rumor propagation is defined with the partition and clustering in the network. Considering the temporal and spatial propagation characteristics of rumors in online social networks, we establish a delayed rumor propagation model based on the graph theory and partial functional differential equations. Firstly, the unique existence and uniform boundedness of the nonnegative solution are explored. Secondly, we discuss the existence of positive equilibrium points sufficiently. Thirdly, stabilities of the rumor-free and rumor-spreading equilibrium points are investigated according to the linearization approach and Lyapunov function. Finally, we perform several numerical simulations to validate theoretical results and show the influence of time delay on rumor propagation. Experimental results further illustrate that taking forceful actions such as increasing the time delay in the rumor-spreading process can control rumor propagation due to the timely effectiveness of the information. [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]

Mathematical modeling is an important approach to research rumor propagation in online social networks. Most of prior work about rumor propagation either carried out empirical studies or focus on ordinary differential equation models with only consideration of temporal dimension; little attempt has been given on understanding rumor propagation over both temporal and spatial dimensions. This paper primarily addresses an issue related to how to define a spatial distance in online social networks by clustering and then proposes a partial differential equation model with a time delay to describing rumor propagation over both temporal and spatial dimensions. Theoretical analysis reveals the existence of equilibrium points, a priori bound of the solution, the local stability and the global stability of equilibrium points of our rumor propagation model. Finally, numerical simulations have analyzed the possible influence factors on rumor propagation and proved the validity of the theoretical analysis. [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]

The study of rumor propagation dynamics is of great significance to reduce false news and ensure the authenticity of news information. In this paper, a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied. First, through stability analysis, we obtain the conditions for the existence and local stability of the positive equilibrium point. By selecting suitable variable as the control parameter, the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained. Then, using the above theorem and multi-scale standard analysis, the expression of amplitude equation around Turing bifurcation point is obtained. By analyzing the amplitude equation, different types of Turing pattern are divided such as uniform steady-state mode, hexagonal mode, stripe mode and mixed structure mode. Further, in the numerical simulation part, by observing different patterns corresponding to different values of control variable, the correctness of the theory is verified. Finally, the effects of different network structures on patterns are investigated. The results show that there are significant differences in the distribution of users on different network structures. [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]

• This paper presents a novel non-smooth rumor spreading model. • Complex dynamics are studied by using the theory of partial differential equations. • Nonlinear control function is adopted in modeling and we have proved its effectiveness. To reflect the government and media refutation to rumor propagation, we have established a reaction-diffusion rumor propagation model by considering a non-smooth control function. Firstly, we obtain the rumor propagation threshold according to the next generation matrix theory, prove the existence and uniqueness of solution and discuss the existence of the equilibrium points. Secondly, we discuss the stability of the equilibrium points with the impact of the spatial diffusion. Thirdly, the conditions for spatially homogeneous and discontinuous Hopf bifurcation are presented. Finally, several numerical simulations are given to show the possible impact, and the factors affecting rumor propagation are theoretically analyzed, which proves the validity of the theoretical analysis. [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]

In this paper, a reaction-diffusion model is established to study the dynamic behavior of rumor propagation. Firstly, we consider the existence of the positive equilibrium points. Then, we perform a stability analysis to study the conditions for the occurrence of Turing instability. Secondly, we use multiscale analysis to derive the expression of the amplitude equation. In the process of numerical simulation, the reality is considered. It shows that controlling the spread rate of rumor and the number of new Internet users have a great effect on curbing the spread of online rumor. Furthermore, it is proved that the analysis of amplitude equation plays a decisive role in the formation of Turing patterns. We also discuss the phenomenon of Turing patterns when the network structure changes and verify the rationality of the model by Monte Carlo method. Finally, we consider two methods based on statistical principle and convolutional neural network severally to identify the parameters of the reaction-diffusion system with Turing instability by using stable patterns. The statistical principle-based method offers superior accuracy, whereas the convolutional neural network-based approach significantly reduces recognition time and cuts down time costs. • This paper investigates a rumor propagation system with reaction-diffusion behavior. • The properties of Turing pattern with or without network background are studied. • We compare different methods to select the optimal parameter identification for system. [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]

In this paper, a homogeneous-mixing population fractional model for human immunodeficiency virus (HIV) transmission, which incorporates anti-HIV preventive vaccines, is proposed. The dynamics of the model indicate that the basic reproduction number being the unity is a strict threshold for disease eradication when there is no vaccine. However, it has been shown that when the efficacy or dosage of vaccines is low, the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium point (DFE) coexists with a stable endemic equilibrium point (EE) when the associated reproduction number is less than unity. Therefore, driving the basic reproduction number below the unity is not enough to eradicate the disease. A new critical value at the turning point should be deduced as a new threshold of disease eradication. We have generalized the integer LaSalle invariant set theorem into fractional system and given some sufficient conditions for the disease-free equilibrium point being globally asymptotical stability. Mathematical results in this paper suggest that improving the efficiency and dosage of vaccines are all valid methods for the control of disease. [ABSTRACT FROM AUTHOR]

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence. [ABSTRACT FROM AUTHOR]

The myocyte enhancer factor-2 (MEF2) is a member of the MADS-box family. It controls the expression of genes that are critical for biological processes such as proliferation, cell death, and differentiation. Some studies have shown that MEF2 expression is enhanced in osteogenic progenitor cells established from bone marrow stromal cells with other types of mesenchymal progenitor cells. However, the effect of MEF2 on dental pulp stem cells (DPSCs) is unclear. In this study, we investigate the effect of MEF2 on regulating osteogenic differentiation and proliferation of DPSCs. We find that MEF2 is stably expressed in DPSCs, and the expression is increased time-dependently along with cell osteogenic differentiation. MEF2 expression also increases the alkaline phosphatase (ALP), runt-related transcription factor 2 (Runx2) activity, and enhances mineralization in DPSCs. SB202190, inhibitor of p38, blocks the p38/MEF2 pathway and osteogenic differentiation. In addition, MEF2 overexpression inhibits DPSC proliferation. In summary, our data indicate that MEF2 not only regulates DPSCs as an inhibitor of cell proliferation but is also a promoter of osteogenic differentiation through the p38/MEF2 signaling pathway. [ABSTRACT FROM AUTHOR]