Analysis of Rumor Propagation Model Based on Coupling Interaction Between Official Government and Media Websites

Abstract

The COVID-19 pandemic has not only brought a virus to the public, but also spawned a large number of rumors. The Internet has made it very convenient for media websites to record and spread rumors, while the official government, as the subject of rumor control, can release rumor-refutation information to reduce the harm of rumors. Therefore, this study took into account information-carrying variables, such as media websites and official governments, and expanded the classic ISR rumor propagation model into a five-dimensional, two-level rumor propagation model that interacts between the main body layer and the information layer. Based on the constructed model, the mean field equation was obtained. Through mathematical analysis, the equilibrium point and the basic reproduction number of rumors were calculated. At the same time, stability analysis was conducted using the Routh Hurwitz stability criterion. Finally, a numerical simulation verified that when the basic regeneration number was less than 1, rumors disappeared in the system; when the basic regeneration number was greater than 1, rumors continued to exist in the system and rumors erupted. The executive power of the official government to dispel rumors, that is, the effectiveness of the government, played a decisive role in suppressing the spread of rumors.

1. Introduction

In the era of rapid technological advancement, the Internet has become an indispensable part of modern life. Based on social platforms, such as Twitter, WeChat, and Facebook, which enable people to interact instantly with others, social networks have become a window for people to communicate with the world. However, malicious rumors and erroneous information spread rapidly through social networks from time to time. The proliferation of such content not only disrupts daily life but also incites widespread anxiety, diminishing the overall quality of life and even posing threats to social and political stability. In times of crisis, the impact of these rumors can be particularly devastating, affecting the world on a profound level [1,2,3].

The study of rumor propagation has been going on for more than half a century, and many scholars made a series of attempts in this field [4,5]. So far, a large number of research results have been produced, gradually developing into a relatively complete disciplinary system. Drawing parallels with the transmission of diseases [6,7,8], the study of rumor spread has been a significant area of focus since the 1960s, yielding a wealth of insights. Daley and Kendal [9] constructed a classic rumor propagation model using mathematical methods, known as the model. Assuming that individuals are uniformly mixed and that the probability of contact between individuals is constant, they divided the population into three subcategories: people who have never heard of rumors (ignorant), people who know and spread rumors (spreaders), and people who know but do not spread rumors (stiflers). Subsequently, Maki and Thomson [10] refined their model with the stipulation that upon contact between two spreaders, it is the initial spreader who ceases to disseminate and transitions into a stifler. In recent years, with the enrichment of research achievements in complex network theory, scholars have established a large number of rumor propagation models based on network topology, and combined small world networks [11,12] and scale-free networks [13] to conduct more in-depth research on the combination of rumor propagation and complex network theory. Nekovee [14] calculated the rumor propagation threshold through the mean field theory and introduced the threshold theory into the rumor propagation model. Beyond the traditional models of rumor propagation, a multitude of scholars around the world introduced numerous innovations within rumor models that categorize individuals and integrate complex network theory, thereby significantly enriching the theoretical framework of rumor dissemination. They introduced individual activities [15,16], incubation periods [17,18], popular science education [19], counterattack mechanisms [20,21,22], hesitation mechanisms [23], forgetting and memory mechanisms [24], social reinforcement mechanisms [25], trust mechanisms [26], and rumor refutation mechanisms [27,28,29] into rumor propagation research.

In recent years, scholars significantly broadened the scope of population dynamics models to encompass more complex scenarios. Beyond the traditional in-depth exploration of dynamic modeling and propagation mechanisms of intertwined phenomena, such as epidemics, knowledge dissemination, and information spread [30,31,32], the application of propagation models has indeed expanded into areas such as network information security, urban traffic congestion, and Internet of Things (IoT) security. This demonstrates the versatility and relevance of these models across various domains.

For instance, in the field of cybersecurity, rumor propagation models are employed to study the spread of malware and viruses within networks, as well as how to prevent the proliferation of these threats through technical means [33,34]. Specifically, a study by Zhong et al., published in Science China: Information Science, introduced a hybrid stochastic control strategy using a two-layer network to alleviate urban traffic congestion [35]. This innovative approach showcases how propagation models can be adapted to address real-world challenges in traffic management and urban planning. Furthermore, a study highlighted in the IEEE Internet of Things Journal underscores the application of propagation models in IoT security. The research discusses the fractional optimal control for malware propagation in the Internet of Underwater Things [36]. This study emphasizes the importance of propagation models in safeguarding the security and integrity of IoT networks, particularly in critical infrastructures, such as underwater communication systems. This model aids in understanding how misinformation can spread rapidly among populations, underscoring the need for effective control strategies to mitigate its impact. However, there were still relatively few dynamic model studies that involved information variables, such as government and media. When an emergency occurs, it will inevitably lead to a flood of rumors, which will bring great harm. On the one hand, the government can release some official information to dispel the rumors, thereby guiding the masses to recognize the facts [37,38,39]. On the other hand, media outlets can enrich rumor propagation channels by collecting and documenting public reports of rumors on their platforms. The antagonistic role of this pair of opposite information subjects in the spread of rumors has become a topic worthy of in-depth discussion.

The rest of this paper is arranged as follows: In Section 2, we combine complex network theory and system dynamics theory, introduce the guidance and intervention role of the government and the participation role of media websites into the classic ISR rumor propagation model, and establish a multi-agent participation rumor propagation model. In Section 3, the threshold value of rumor propagation is calculated, and the threshold condition for continuous rumor diffusion in the system is obtained through mathematical analysis. In Section 4, the results of simulation experiments are presented to explore the impacts of various parameter variables on rumor propagation and diffusion to verify the above analysis. Finally, the Section 5 summarizes the conclusions and puts forward some management suggestions.

2. Model Construction

Taking into account the intervention role of the government and the participation role of media websites, this section constructs a multi-dimensional $ISR-GM$ network rumor propagation control model and bifurcates the subjects of propagation and information into two distinct layers. The upper layer of the model is the propagation subject layer, and the lower layer is the information participation layer. In the model, the groups of rumor propagation subjects and information participation subjects are regarded as nodes, and the interactive connections between different levels and individuals are regarded as edges. The undirected social network $G=\left(V,E\right)$ is obtained, where $V$ represents the combination of nodes and $\mathrm{E}$ represents the combination of edges. The main layer of propagation subjects is divided into the ignorant ($I$), the spreaders ($S$), and the stiflers ($R$). The groups involved in the control layer of information include the official government ($G$) and the news media ($M$), where $G$ represents the number of anti-rumor messages published by the official websites during the rumor propagation process. Conversely, when the public disseminates rumors, the media records this information in real time. The variable $M$ is used to denote the number of rumors documented by the media. In general, after contact with a spreader, an ignorant subject transforms into a spreader with the probability $\mathsf{\alpha }$. When an ignorant subject receives media information on a news media website, they will be influenced by the media, and then become a spreader with a probability of $\beta$. When a spreader searches or posts rumor information on a website, the media website will store the rumor information with probability ${\lambda }_1$. At the same time, we assumed that the spreader forgets the rumor with probability $\delta$. When the ignorant and spreaders learn the official rumor refutation information, they will turn into stiflers with probability $\eta$.

Individuals exhibit varying degrees of openness to external information. Some are resistant to change, while others are more receptive and adjust their beliefs accordingly. We refer to this attribute of adhering to one’s own beliefs as individual confidence, expressed by $\phi \left(i\right)$, where $\phi \left(i\right)=e^{-{\displaystyle \frac{150}{\rho \left(i\right)}}}$. Here, 150 is the distribution of the number of Twitter users’ “fans” and “followers” counted by scholars, such as Kwak H [40]. Research shows that the number of “followers” of most Weibo users hovers around 150. $\rho \left(i\right)$ indicates the number of “fans” of individual $\mathrm{i}$ on Sina Weibo.

In the process of the propagation of ideas, whether the topic of an official government information release can be adopted by the masses or not is also affected by its own reliability. Simultaneously, the influence of the effectiveness of official government refutations, specifically their intensity, merits consideration and should not be overlooked. Therefore, the government interference rate $\eta$ is considered as a comprehensive impact of confidence and validity, expressed by the function $\eta =\kappa e^{-{\displaystyle \frac{\chi }{\omega }}}$. Here, $\chi$ is a statistical constant that represents the statistical mean value of “followers” and “fans” in Microblog websites by different official governments. $\omega$ indicates the confidence level of the government. $\kappa$ indicates the validity of the official information; as the government increases its efforts to refute rumors, the higher the validity, where its value range is assumed to be [0, 1]. When the value of information validity $\kappa =0$, it means that the government has not published the refuting information, and the government has no influence on the spread of groups. When the validity $\kappa =1$, it indicates that the official government has used the maximum intensity to refute rumors.

In view of the impact of the density of rumor spreaders on the release of official rumor-refuting information, this paper posits that the influence rate of spreaders on the issuance of such information is denoted by ${\lambda }_2$. The value of the parameter ${\lambda }_2$ characterizes its impact intensity, and its magnitude is adjusted dynamically with the density of rumor spreaders in the environment between [0, 1]. As the Internet becomes ubiquitous, the proliferation of rumor information on online platforms poses a significant and undeniable threat to our social fabric. To suppress the spread of rumors, authorities possess the authority to mandate the retraction of media-related misinformation. It is presumed that the penalty rate for compelling the removal of such rumor-laden content is denoted by $d$. In the propagation group layer, the ignorant enter the system with a certain probability ${\Lambda }_1$ (similar to the natural birth rate), and individuals participating in the propagation spontaneously move out of the system with a probability ${\mu }_1$ (similar to the natural mortality). In the information layer, the official government releases rumor refutation information with the probability ${\Lambda }_2$, the media websites information and official information are submerged in the network with a certain probability ${\mu }_2$, and $d$ is the natural cancellation rate of media information.

The rumor propagation process of the improved $ISR-GM$ model is shown in Figure 1.

In order to improve the readability of this article, the definitions of each parameter are shown in Table 1.

Based on the above assumptions, the model differential equations could be derived to obtain the following mean field equations:

(1)$\left\{\begin{array}{l}{\displaystyle \frac{dI}{dt}}={\Lambda }_1-\alpha IS-\beta IM-\eta IG-{\mu }_{1}I\\ {\displaystyle \frac{dS}{dt}}=\alpha IS+\beta IM-\eta SG-\delta S-{\mu }_{1}S\\ {\displaystyle \frac{dR}{dt}}=\eta IG+\eta SG+\delta S-{\mu }_{1}R\\ {\displaystyle \frac{dG}{dt}}={\Lambda }_2+{\lambda }_{2}S-{\mu }_{2}G\\ {\displaystyle \frac{dM}{dt}}={\lambda }_{1}S-dM-{\mu }_{2}M\end{array}\right.$

3. Model Analysis

3.1. Rumor-Free Equilibrium Point and Basic Regeneration Number

From Equation (1), there is a unique rumor-free equilibrium point $E_0$ for the dynamic system, as shown below:

$\{I_0\to {\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}},S_{\mathbf{0}}\to 0,R_{\mathbf{0}}\to {\displaystyle \frac{{\Lambda }_1{\Lambda }_2\eta }{{\mu }_1\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)}},G_{\mathbf{0}}\to {\displaystyle \frac{{\Lambda }_2}{{\mu }_2}},M_{\mathbf{0}}\to 0\}$

The expression of the propagation threshold can be accurately calculated based on the condition that the rumor-free equilibrium point of the system is locally asymptotically stable. Therefore, for general complex network propagation dynamics models, the Jacobian matrix at the disease-free equilibrium point can be calculated first, and then the propagation threshold can be obtained by deducing the condition where all eigenvalues of the Jacobian matrix are less than zero. However, for relatively complex differential equation systems, such as Equation (1), solving all eigenvalues of the Jacobian matrix at the rumor-free equilibrium point is computationally intensive. Therefore, the next-generation matrix method can be attempted to calculate the rumor propagation threshold [41,42]. As an extremely important concept, the basic reproduction number refers to the number of ignorant individuals that the disseminator can infect per unit of time in the early stages of rumor dissemination when each individual is ignorant. Usually, ${\Re }_0=1$ can be used as a threshold to determine whether a rumor has disappeared. To calculate ${\Re }_0$, it is important to distinguish new spreaders from all other changes in the population. For the next-generation matrix methods, we mainly focused on new infection flows and transfer flows between infected individuals.

First, this section uses a fixed-weight model to calculate the basic reproduction number. System (1) can be rewritten as follows:

(2)$dx/dt=f\left(x\right)=\mathcal{F}\left(x\right)-𝓋\left(x\right)$

where $x={\left(x_1,x_2,x_3,x_4\right)}^T={\left(S,R,G,M\right)}^T$. According to the method of the next generation matrix [37,38], the system (1) has a rumor-free equilibrium point $E_0=(0,{\displaystyle \frac{{\Lambda }_1{\Lambda }_2\eta }{{\mu }_1\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)}},{\displaystyle \frac{{\Lambda }_2}{{\mu }_2}},0)$.

Let $\mathcal{F}\left(x\right)$ be the probability of new entrants into the spreader community per unit time:

$\mathcal{F}\left(x\right)=\left[\begin{array}{c}\begin{array}{c}\alpha IS+\beta IM\\ \eta IG+\eta SG\\ 0\\ 0\end{array}\end{array}\right]$

And $𝓋\left(\mathrm{x}\right)$ represents the probability of an individual being removed from the group of the spreader subclass per unit time:

$\mathbf{𝓋}\left(x\right)=\left[\begin{array}{c}\begin{array}{c}\eta SG+\delta S+{\mu }_{1}S\\ -\delta S+{\mu }_{1}R\\ -{\Lambda }_2-{\lambda }_{2}S+{\mu }_{2}G\\ -{\lambda }_{1}S+dM+{\mu }_{2}M\end{array}\end{array}\right]$

According to the concept of a next-generation matrix and basic regeneration number given in Refs. [41,42], the propagation threshold can be expressed as ${\Re }_0=\rho \left(F{V}^{-1}\right)$, where $\rho \left(F{V}^{-1}\right)$ denotes the spectral radius of matrix $F{V}^{-1}$, and matrix $F{V}^{-1}$ is called the next-generation matrix. And the Jacobian matrices of $\mathcal{F}\left(\mathrm{x}\right)$ and $𝓋\left(\mathrm{x}\right)$ at the rumor-free equilibrium $E_0$ are $F=D\mathcal{F}\left(E_0\right)$ and $V=D\mathbf{𝓋}\left(E_0\right)$. The respective derivatives at the equilibrium point ${\mathrm{E}}_0$ are calculated to give

$F={\displaystyle \frac{\partial \mathcal{F}\left(x\right)}{\partial E_{\mathbf{0}}}}=\left[\begin{array}{cccc}\alpha I_0& 0& 0& \beta I_0\\ \eta G_0& 0& \eta I_0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$

$V={\displaystyle \frac{\partial \mathbf{𝓋}\left(x\right)}{\partial E_{\mathbf{0}}}}=\left[\begin{array}{cccc}\eta G_0+\delta +{\mu }_1& 0& 0& 0\\ -\delta & {\mu }_1& 0& 0\\ -{\lambda }_2& 0& {\mu }_2& 0\\ -{\lambda }_1& 0& 0& d+{\mu }_2\end{array}\right]$

By calculations, it can be concluded that

(3)${\Re }_0=\rho \left(F{V}^{-1}\right)={\displaystyle \frac{\left(\beta {\lambda }_1+\alpha \left(d+{\mu }_2\right)\right)I_0}{\left(\delta +\eta G_0+{\mu }_1\right)\left(d+{\mu }_2\right)}}={\displaystyle \frac{{\Lambda }_1{{\mu }_2}^2\left(\beta {\lambda }_1+\alpha \left(d+{\mu }_2\right)\right)}{\left(d+{\mu }_2\right)\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)\left(\eta {\Lambda }_2+\left(\delta +{\mu }_1\right){\mu }_2\right)}}$

The propagation threshold ${\Re }_0$ represents the average time it takes for a rumor to be disseminated from one individual to another within a coupled network relative to the time it takes for individuals to forget the rumor. The propagation threshold ${\Re }_0$ is an important parameter that portrays the expectation of rumor propagation, which indicates the number of second-generation cases that can be spread without external intervention when a spreader enters the population of the ignorant. A pivotal threshold is ${\Re }_0=1$, which serves as a critical demarcation. The higher the value of ${\Re }_0$, the more challenging it becomes to manage the spread of rumors, as it delineates the conditions under which rumors persist in a two-tiered coupled network, as referenced in [41,43]. And if ${\Re }_0<1$, the number of rumor propagators will gradually decrease to 0, i.e., the rumor will gradually disappear in the coupling network. If ${\Re }_0>1$, the rumor will spread in the coupled network.

3.2. Stability Analysis of Rumor-Free Equilibrium Point

Theorem 1.

For system (1), if ${\Re }_0<1$, the rumor-free equilibrium point is locally asymptotically stable. If ${\Re }_0>1$, the rumor-free equilibrium point is unstable.

Proof of Theorem 1.

Considering the five-dimensional system (1), the Jacobian matrix at the rumor-free equilibrium point $E_0$ is

$J\left(E_0\right)=\left[\begin{array}{ccccc}-\eta G_0-{\mu }_1& -\alpha I_0& 0& -\eta I_0& -\beta I_0\\ 0& \alpha I_0-\eta G_0-\delta -{\mu }_1& 0& 0& \beta I_0\\ \eta G_0& \eta G_0+\delta & -{\mu }_1& \eta I_0& 0\\ 0& -{\lambda }_2& 0& -{\mu }_2& 0\\ 0& -{\lambda }_1& 0& 0& -{\mu }_2\end{array}\right]$

The characteristic polynomial is as follows:

$\left|J\left(E_0\right)-\gamma E\right|=\left[\begin{array}{ccccc}-\eta G_0-{\mu }_1-\gamma & -\alpha I_0& 0& -\eta I_0& -\beta I_0\\ 0& \alpha I_0-\eta G_0-\delta -{\mu }_1-\gamma & 0& 0& \beta I_0\\ \eta G_0& \eta G_0+\delta & -{\mu }_1-\gamma & \eta I_0& 0\\ 0& -{\lambda }_2& 0& -{\mu }_2-\gamma & 0\\ 0& -{\lambda }_1& 0& 0& -{\mu }_2-\gamma \end{array}\right]$

where $E$ is the unit matrix. By setting the characteristic polynomial to 0, the eigenvalue $\gamma$ is calculated as follows:

${\gamma }_1=-{\mu }_1,{\gamma }_2=-\eta G_0-{\mu }_1,{\gamma }_3=-{\mu }_2,{\gamma }_{4,5}={\displaystyle \frac{-B\pm \sqrt{B^2-4AC}}{2A}}$

where $A=1$, $B=-I_0\alpha +\delta +\eta G_0+{\mu }_1+{\mu }_2$, and $C=\beta {\lambda }_1{I}_0+\left(-\alpha I_0+\delta +\eta G_0+{\mu }_1\right){\mu }_2$. And in these expressions, $I_0$ and $G_0$ are ${\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}$ and ${\displaystyle \frac{{\Lambda }_2}{{\mu }_2}}$, respectively.

The first three eigenvalues (${\gamma }_1,{\gamma }_2,\mathrm{and}{\gamma }_3$) are obviously less than 0. Substituting the free equilibrium point $E_0$ and simplifying obtains the last two eigenvalues (${\gamma }_4\mathrm{and}{\gamma }_5$), which can be regarded as the solution of a system of quadratic equations with one variable, and their signs can be determined according to the Vieta theorem.

When ${\Re }_0<1$, then ${\Re }_0-1={\displaystyle \frac{\left(\beta {\lambda }_1+\alpha \left(d+{\mu }_2\right)\right)I_0}{\left(\delta +\eta G_0+{\mu }_1\right)\left(d+{\mu }_2\right)}}-1={\displaystyle \frac{\beta {\lambda }_1{I}_0+\left(\alpha I_0-\delta -g\eta -{\mu }_1\right)\left(d+{\mu }_2\right)}{\left(\delta +\eta G_0+{\mu }_1\right)\left(d+{\mu }_2\right)}}<0$ holds constant. From this, it can be concluded that $\alpha I_0-\delta -\eta G_0-{\mu }_1<0$ is constant. In contrast, $-I_0\alpha +\delta +\eta G_0+{\mu }_1>0$. Therefore, $B=-I_0\alpha +\delta +\eta G_0+{\mu }_1+{\mu }_2>0$ and $C=\beta {\lambda }_1{I}_0+\left(-\alpha I_0+\delta +\eta G_0+{\mu }_1\right){\mu }_2>0$ are constant. Finally, all eigenvalues are less than zero, which holds constant. □

Theorem 2.

When ${\Re }_0<1$, the rumor-free equilibrium point $E_0\left(I_0,S_0,R_0,G_0,M_0\right)$ in the region D is globally asymptotically stable.

Proof of Theorem 2.

Define the following Lyapunov function:

$V\left(I,S,R,G,M\right)=a_1{I}^2+a_2{S}^2+a_3{R}^2+a_4{G}^2+a_5{M}^2$

where $a_1,a_2,a_3,a_4,a_5>0$. By calculating the time derivative of $V\left(I,S,R,G,M\right)$ along the trajectory of system (1), we can obtain

$V^{\prime }\left(t\right)={\displaystyle \frac{\partial V}{\partial I}}{\displaystyle \frac{dI}{dt}}+{\displaystyle \frac{\partial V}{\partial S}}{\displaystyle \frac{dS}{dt}}+{\displaystyle \frac{\partial V}{\partial R}}{\displaystyle \frac{dR}{dt}}+{\displaystyle \frac{\partial V}{\partial G}}{\displaystyle \frac{dG}{dt}}+{\displaystyle \frac{\partial V}{\partial M}}{\displaystyle \frac{dM}{dt}}$

$$\begin{gathered} V^{\prime }\left(t\right)=2a_{1}I\left({\Lambda }_1-\alpha IS-\beta IM-\eta IG-{\mu }_{1}I\right)+2a_{2}S\left(\alpha IS+\beta IM-\eta SG-\delta S-{\mu }_{1}S\right)+2a_{3}R\left(\eta IG+\eta SG+ \\ \delta S-{\mu }_{1}R\right)+2a_{4}G\left({\Lambda }_2+{\lambda }_{2}S-{\mu }_{2}G\right)+2a_{5}M\left({\lambda }_{1}S-dM-{\mu }_{2}M\right) \end{gathered}$$

At the equilibrium point $E_0$, we have $I_0={\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}, S_0=0,R_0={\displaystyle \frac{{\Lambda }_1{\Lambda }_2\eta }{{\mu }_1\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)}},G_0={\displaystyle \frac{{\Lambda }_2}{{\mu }_2}},$ and $M_0=0$.

By substituting these values into $V^{\prime }\left(t\right)$ and simplifying them, we can obtain

$V^{\prime }\left(E_0\right)=2a_1{I}_0\left({\Lambda }_1-\eta I_0{G}_0-{\mu }_1{I}_0\right)+2a_3{R}_0\left(\eta I_0{G}_0-{\mu }_1{R}_0\right)+2a_4{G}_0\left({\Lambda }_2-{\mu }_2{G}_0\right)$

We need to prove that the above expression is negative for all $\left(I,S,R,G,M\right)\ne \left(I_0,S_0,R_0,G_0,M_0\right)$. Analyzing the term of $V^{\prime }\left(E_0\right)$, for the first term $2a_1{I}_0\left({\Lambda }_1-\eta I_0{G}_0-{\mu }_1{I}_0\right)$, since $I_0={\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}$, it can be simplified to obtain ${\Lambda }_1-\eta I_0{G}_0-{\mu }_1{I}_0={\Lambda }_1-\eta {\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}{\displaystyle \frac{{\Lambda }_2}{{\mu }_2}}-{\mu }_1{\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}=0$.

For the second term $2a_3{R}_0\left(\eta I_0{G}_0-{\mu }_1{R}_0\right)$, since $R_0={\displaystyle \frac{{\Lambda }_1{\Lambda }_2\eta }{{\mu }_1\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)}}$, it can be simplified to obtain $\eta I_0{G}_0-{\mu }_1{R}_0=\eta {\displaystyle \frac{{\Lambda }_1{\mu }_2}{{\Lambda }_2\eta +{\mu }_1{\mu }_2}}{\displaystyle \frac{{\Lambda }_2}{{\mu }_2}}-{\mu }_1{\displaystyle \frac{{\Lambda }_1{\Lambda }_2\eta }{{\mu }_1\left({\Lambda }_2\eta +{\mu }_1{\mu }_2\right)}}=0$.

For the third term $2a_4{G}_0\left({\Lambda }_2-{\mu }_2{G}_0\right)$, since $G_0={\displaystyle \frac{{\Lambda }_2}{{\mu }_2}}$, it can be simplified to obtain ${\Lambda }_2-{\mu }_2{G}_0={\Lambda }_2-{\mu }_2{\displaystyle \frac{{\Lambda }_2}{{\mu }_2}}=0$.

Since all terms are zero at the equilibrium point $E_0$, we need to consider small perturbations of $I,R$, and $G$. For any $I\ne I_0$, the term $2a_1{I}_0\left({\Lambda }_1-\eta I_0{G}_0-{\mu }_1{I}_0\right)$ will be non-zero, and since $\eta$ and ${\mu }_1$ are positive, the term $2a_1{I}_0\left({\Lambda }_1-\eta I_0{G}_0-{\mu }_1{I}_0\right)$ will be negative. Similarly, for $R\ne R_0$ and $G\ne G_0$, the terms $2a_3{R}_0\left(\eta I_0{G}_0-{\mu }_1{R}_0\right)$ and $2a_4{G}_0\left({\Lambda }_2-{\mu }_2{G}_0\right)$ are negative.

4. Numerical Simulation

In this section, we provide some simulation examples to validate the accuracy of our previous analysis. In the numerical simulation of rumor propagation, this study employed a scale-free network topology for the simulation, as it effectively captures the connection patterns found in actual social networks. The abundance of hub nodes in these networks, characterized by an exceptionally high degree of connectivity, enhances the rapid spread of information. Keeping in line with the proposed model, the simulations assumed that the two-layer network composed of the propagation group layer and the information participation layer was a uniformly mixed network whose degree distribution could be expressed as $p\left(k\right)={\displaystyle \frac{e^{-⟨k⟩}⟨k⟩}{k!}}$, where $⟨k⟩$ is the average degree of the network. For the initial setup, it was assumed that there was a single rumor spreader within a network where the propagation group layer consisted of $\mathrm{N}=1000$ nodes, and all others were ignorant. In the information participation layer, there was some related rumor information on both the official government and media websites, and their initial probability densities were assumed to be 0.1. That is, $I\left(0\right)=0.98$, $S\left(0\right)=0$, $R\left(0\right)=0$, $G\left(0\right)=0.1$, $M\left(0\right)=0.1$. The propagation coefficient, forgetting rate coefficient, entry and exit rate coefficients, and coupling coefficient between the different layers in the network took random values between (0, 1). According to the threshold condition ${\Re }_0=1$, we assumed two scenarios with values as shown in Table 2, and Scenario 1 corresponded to the condition ${\Re }_0>1$, while Scenario 2 corresponded to the condition ${\Re }_0<1$.

Given the primary innovation of our constructed model lies in the introduction of control and intervention effects by the government and media, we focused our study on the impact of the scale of spreaders in the propagation layer on the two types of outside entities, government and media (parameters ${\lambda }_1$ and ${\lambda }_2$); the impact of the government’s refutation intensity (parameter $\kappa$ and $\omega$) on the propagation layer groups; and the mediating effect of the media (parameter $\beta$) on the propagation layer. We then validated the effectiveness of the established model in two parts. First, we examined the population’s evolutionary trajectory across various scenarios. Subsequently, we performed a sensitivity analysis on the parameters identified in the initial part.

4.1. Evolutionary Trajectory

This section analyzes the evolution trajectories of different populations under different threshold conditions ${\Re }_0>1$ (Figure 2a) and ${\Re }_0<1$ (Figure 2b).

Figure 2a provides a detailed overview of the temporal dynamics within the network system when the basic reproduction number ${\Re }_0$ exceeded 1, indicating a favorable condition for rumor propagation.

Initially, as time progressed, rumors infiltrated the network, which caused a steady decline in the population of individuals classified as ignorant. This decline was attributed to two primary transformations: a portion of the ignorant became spreaders, where they actively participated in the dissemination of the rumor, while another portion adopted the role of rumor stiflers, likely due to their exposure to counter-information from the government or personal skepticism. The evolution of the rumor spreaders was characterized by an initial surge in the number of spreaders, which reached a peak as the rumor’s influence expanded. However, this growth was checked by the strategic release of official emergency rumor refutation information. This intervention was pivotal, as it resulted in a substantial rise in the number of internet users who gained knowledge of the facts and the truth underlying the rumor. Consequently, the population of spreaders began to dwindle, as individuals were swayed by the official refutation and shifted to become stiflers. This transition was marked by a gradual decline in the number of spreaders, which eventually settled at a stable, constant value. Simultaneously, the stiflers’ population experienced a steady increase, which reflected the success of the refutation campaign in curbing the spread of the rumor. This increase eventually plateaued, indicating a new equilibrium within the network.

To mitigate the adverse effects of the rumor, the official government adjusted both the volume and substance of the information it disseminated. In the early stages of the rumor propagation, there was a rapid escalation in the government’s information release, which mirrored the urgency of the situation. However, as time elapsed, the volume of rumor-refuting information issued by the government reached a saturation point and gradually stabilized, suggesting a containment of the situation. Media websites, which served as recorders and disseminators of the rumor information, exhibited a synchronized fluctuation with the changing dynamics of the spreaders. As the spreaders’ population swelled and subsequently receded, the media’s coverage and the amount of rumor information it carried also followed a similar pattern, which underscored the media’s dual role as a conduit and a barometer of public sentiment and dialogue.

Figure 2b offers an in-depth look at the trajectory of different population groups within the network system under the condition where the basic reproduction number ${\Re }_0$ was less than 1, signifying that the rumor propagation was not sustainable in the long term.

Upon further inspection of the figure, it is evident that at the outset, the rumors spread quickly through the network. This rapid spread corresponded with a sharp decline in the number of individuals classified as ignorant. The reduction in the number of ignorant individuals happened when they came into contact with spreaders, either through personal interactions or indirectly by being exposed to rumor-related content published on media platforms. This exposure prompted them to adopt the role of spreaders, which contributed to the dissemination of the rumor. Conversely, a portion of the ignorant population was immediately transformed into stiflers. This transformation was the result of their exposure to official channels that disseminated rebuttal messages. The timely release of such information was instrumental in preventing the ignorant from becoming spreaders and instead aligned them with the stiflers who worked to suppress the rumor propagation.

Unlike the scenario where ${\Re }_0<1$, the number of spreaders, after it reached a peak, was ultimately controlled and declined to zero in this scenario. This indicates that the circumstances were not conducive to the rumor’s sustained propagation, which led to its eventual dissipation, as the spreaders were either swayed by the debunking information or their interest waned with time. The evolution of the stiflers and the information variables $G$ (government-released refutation information) and $M$ (media-circulated rumor information) followed a similar pattern that was observed when ${\Re }_0<1$. The stiflers’ population increased as more individuals were swayed by the refutation information, which contributed to the decline in the spreaders. The variables $G$ and $M$ fluctuated in tandem, which reflected the interplay between the government’s efforts to quell the rumor and the media’s role in either spreading or suppressing the rumor, which depended on the content they chose to circulate.

In summary, Figure 2b illustrates the critical role of the refutation information in curbing the rumors when the basic reproduction number was less than 1, highlighting the interdependence between the government’s response and the media’s influence on the network system’s population dynamics.

4.2. Sensitivity Analysis

4.2.1. The Influence of Communicators on the Government and Media

This section explores the sensitivity analysis of the variables $G$ and $M$ with respect to the parameters ${\lambda }_1$ and ${\lambda }_2$ The simulation results are shown in Figure 3.

Upon close examination of Figure 3a, a direct positive correlation emerged between the volume of the government-released rumor refutation information ($G$) and the parameter ${\lambda }_2$. This correlation underscored that an intensified impact of rumor spreaders on the government correlated with an increased issuance of refutation information. It suggests that as the influence of spreaders on the government escalated, so did the government’s effort to counteract the rumors through official channels. Additionally, an upward trend was observable in $G$ with an increase in the parameter ${\lambda }_1$. This trend may be attributed to the amplifying effect that the spreaders had on media activity, which, in turn, prompted the government to step up its refutation efforts. A higher ${\lambda }_1$ implies a more significant influence of spreaders on the media landscape, which appeared to bolster the government’s response in terms of the quantity of refutation information released.

Figure 3b delves into the nuances of how the number of rumors ($M$) present on media websites was influenced by the parameters ${\lambda }_1$ and ${\lambda }_2$. The figure reveals that as ${\lambda }_1$ increased, the prevalence of rumors on the media websites also rose. This implies that an increased presence of spreaders in the community prompted media outlets to cover more stories related to the rumor, likely because of the perceived newsworthiness or the public’s engagement with these subjects. Conversely, an increase in ${\lambda }_2$ was associated with a decrease in the number of rumors ($M$) on the media websites. This inverse relationship indicates that when the government was significantly affected by the activities of rumor spreaders, it responded by releasing information that could indirectly pressure media websites to reduce the circulation of rumor information. The coupling effect within the system, where government actions influenced the media behavior, was evident in the sharp decline in rumor information as ${\lambda }_2$ increased.

In essence, the system’s coupling effect was pivotal: when the spreaders significantly influenced the government, the government’s response could lead to a substantial reduction in the amount of rumor information on the media websites. This highlights the media’s responsiveness to the government’s refutation efforts and the importance of the government’s role in shaping the media narrative during rumor propagation events.

4.2.2. The Influence of the Government’s Refutation Intensity on the Propagation Layer

This section discusses the influence of the government refutation intensity on the propagation layer, and the simulation results are shown in Figure 4.

Figure 4 shows the analysis of the effects of the validity $\kappa$ and reliability $\mathsf{\omega }$ of the rumor refutation information on the spreaders $S$ and stiflers $R$. As can be seen from Figure 4a, when the system tended to stabilize, the spreaders decreased as the validity $\kappa$ of the rumor refutation information increased. In other words, the more vigorously the refutation information was publicized, the lower the steady-state density of spreaders within the system became. When the rumor refutation information is fully implemented, the spreaders in the stable system will no longer exist, and the rumor will disappear. The weaker the validity $\mathsf{\kappa }$ of refutation information, the greater the final density of spreaders. Similarly, the reliability $\omega$ of the official rumor refutation information showed a positive correlation with the density of spreaders, indicating that the higher the reliability $\omega$ of the official government, the easier it was for the spreaders to collapse in the system. In other words, the credibility of the refutation information was crucial in dissuading individuals from spreading rumors.

Figure 4b shows that when the system tended to stabilize, the stiflers $\mathrm{R}$ increased as the validity $\kappa$ of the rumor refutation information increased, that is, the higher the promotion degree of rumor refutation information, the greater the execution force, the more the public could grasp the truth, and then turn into stiflers. Moreover, Figure 4b demonstrates a positive correlation between the reliability ($\omega$) of the official rumor refutation information and the final size of the stiflers. This correlation indicates that the trustworthiness of the information played a pivotal role in converting spreaders into stiflers. When the official information was trusted, it had a greater chance of convincing people to stop spreading rumors and join the ranks of the stiflers, thereby hastening the rumor’s decline.

4.2.3. The Influence of Media on the Propagation Layer

To explore the impact of information variables, such as media websites, on the spread of rumors, this section mainly explores the impact of the parameter $\beta$ on the evolutionary trends of rumor-spreading groups.

Figure 5 describes the impact of the amount of rumor information on the media websites on the ignorant, spreaders, and stiflers through the parameter $\beta$, where the parameter $\beta$ depicts the impact intensity of rumor information in media websites on the ignorant. From Figure 5, it can be seen that there was a threshold value ${\beta }_c$ (approximately 0.08). This threshold was instrumental in delineating the propagation dynamics of the rumors within the system.

When the actual value of $\beta$ fell short of this critical threshold (${\beta }_c>\beta$), the system exhibited a stagnation in rumor dissemination. In these instances, the media’s influence on rumor information was inadequate to catalyze a substantial transformation of the ignorant populace into active spreaders, which inhibited the rumor propagation.

In contradistinction, when $\beta$ surpassed the threshold (${\beta }_c<\beta$), a pronounced escalation in rumor propagation ensued. Herein, the magnitude of rumor information on the media platforms was sufficiently potent to induce a significant segment of the ignorant to turn into spreaders, and thus, trigger the outbreak of rumors.

The existence of the threshold ${\beta }_c$ accentuated the nonlinear characteristics of rumor propagation. It underscored the sensitivity of the system to minor fluctuations in online rumor information, which could precipitate either the containment or the rampant dissemination of rumors.

5. Conclusions

This paper established a rumor propagation model that accounts for the interaction between online media participation and official government intervention. Then, the interference effects of online media recording of rumor information and the reliability $\omega$ and validity $\kappa$ of government release of rumor refutation information on the process of rumor dissemination are discussed. By employing the next-generation matrix method, we determined the propagation threshold value (i.e., the basic reproduction number ${\Re }_0$), which measures the potential spread of rumors within the $ISR-GM$ model. Subsequently, we calculated and analyzed the existence and stability of equilibrium points using the stability theory of dynamical systems. Ultimately, simulation experiments were conducted to investigate the effects of different parameter variables on the rumor propagation.

Mathematical analysis and numerical simulation showed that there was a threshold value for rumor propagation in the $ISR-GM$ system, as represented by the basic reproduction number ${\Re }_0$. When ${\Re }_0$ exceeded 1, rumors were likely to break out in the system. When ${\Re }_0$ was less than 1, rumors were likely to die out.

The extent to which the information layer affected the communication layer was significantly determined by the prevalence of the spreaders within the system. Our findings indicate that when these spreaders exerted a substantial influence on the official government, the system’s inherent coupling mechanisms came into play. As a result, the information disseminated by the government could have an indirect yet profound impact on the media websites. This dynamic could lead to a substantial decline in the prevalence of rumor information on these platforms.

The effectiveness of the official government’s efforts to debunk rumors is pivotal in curbing the proliferation of misinformation. In other words, government departments and other entities should enhance public opinion supervision to mitigate the societal harm caused by rumors and misinformation.

The decisive role of government in combating rumors cannot be overstated. The efficacy of government initiatives is crucial in curbing the proliferation of misinformation. To this end, it is imperative for official government bodies to enhance their oversight of public opinion. By doing so, they can effectively mitigate the societal damage caused by the spread of rumors and false information.

Author Contributions

Y.C.: conceptualization, methodology, data curation, validation, and writing—original draft. T.Y.: supervision, writing—review and editing, overall revision of this article, and validation. B.X.: mathematical analysis. Q.Y.: software, visualization, and investigation. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data sharing is not applicable to this article, as no data sets were generated or analyzed during the current study.

Conflicts of Interest

No conflicts of interest exist in the submission of this manuscript.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Schematic diagram of rumor propagation with the participation of multiple subjects.

Figure 2. (a) Time evolution of various groups when [Forumla omitted. See PDF.]. (b) Time evolution of various groups when [Forumla omitted. See PDF.].

Figure 3. (a) Variation in the information variable [Forumla omitted. See PDF.] with respect to the parameters [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] (b) Variation in the information variable [Forumla omitted. See PDF.] with respect to the parameters [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 4. (a) Variation in the spreaders [Forumla omitted. See PDF.] with respect to the parameters [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] (b) Variation in the stiflers [Forumla omitted. See PDF.] with respect to the parameters [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 5. Effect of parameter [Forumla omitted. See PDF.] change on different groups.

Table 1

Meaning of different parameters.

Parameter	Definitions
${\Lambda }_1$	Natural birth rate of increase in the ignorant
${\Lambda }_2$	The refutation rate of government
${\mu }_1$	The natural mortality of propagation layer
${\mu }_2$	The natural mortality of information layer
$\alpha$	The basic spreading rate
$\beta$	The impact rate of media on the ignorant
$\kappa$	The validity of official information
$\chi$	The statistical mean value of “followers” and “fans”
$\omega$	The confidence level of the government
$\eta$	The rate of government interference
${\lambda }_1$	The impact rate of spreaders on the media
${\lambda }_2$	The impact rate of spreaders on the government
$\delta$	The natural forgetting rate of spreaders
$d$	The natural cancellation rate of media information

Table 2

Parameter values of numerical simulations.

Parameter	Scheme 1	Scheme 2
${\Lambda }_1$	0.1	0.1
${\Lambda }_2$	0.1	0.1
${\mu }_1$	0.1	0.2
${\mu }_2$	0.1	0.2
$\alpha$	0.8	0.5
$\beta$	0.2	0.2
$\kappa$	0.2	0.5
${\displaystyle \frac{\chi }{\omega }}$	5	0.8
${\lambda }_1$	0.5	0.5
${\lambda }_2$	0.5	0.5
$\delta$	0.1	0.2
$d$	0.1	0.05
${\Re }_0$	${\Re }_0>1$	${\Re }_0<1$

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