A Rumor Propagation Model Considering Media Effect and Suspicion Mechanism under Public Emergencies

Abstract

In this paper, we collect the basic information data of online rumors and highly topical public opinions. In the research of the propagation model of online public opinion rumors, we use the improved SCIR model to analyze the characteristics of online rumor propagation under the suspicion mechanism at different propagation stages, based on considering the flow of rumor propagation. We analyze the stability of the evolution of rumor propagation by using the time-delay differential equation under the punishment mechanism. In this paper, the evolution of heterogeneous views with different acceptance and exchange thresholds is studied, using the standard Deffuant model and the improved model under the influence of the media, to analyze the evolution process and characteristics of rumor opinions. Based on the above results, it is found that improving the recovery rate is better than reducing the deception rate, and increasing the eviction rate is better than improving the detection rate. When the time lag τ < 110, it indicates that the spread of rumors tends to be asymptotic and stable, and the punishment mechanism can reduce the propagation time and the maximum proportion of deceived people. The proportion of deceived people increases with the decrease in the exchange threshold, and the range of opinion clusters increases with the decline in acceptance.

1. Introduction

In the era of online information, rumors are no longer limited to specific groups, spaces, and communication scopes [1]. Rumors have diversified, changing from face-to-face communication to spreading with the help of social networks [2]. The intensification of the speed and scope of transmission can easily lead to social instability. Rumors on social media represent risks to information transmission and the crisis management of public emergencies in modern society [3]. Considering that the propagation process of public opinion rumors in social networks has similar characteristics and rules to those of the propagation process of transmissible viruses, some researchers have begun to use the warehouse modeling method to study the regulations of rumor propagation [4,5,6,7]. The spread of rumors has a great social risk, so it is the key focus of public opinion security research.

Nowadays, the security analysis of rumor propagation modeling has become one of the most popular interdisciplinary research fields. The research mainly focuses on three aspects: the first is to study the rumor propagation of network public opinion in complex networks [8,9,10,11], the second is to improve the classical rumor propagation model according to specific factors [12,13,14,15], and the third is to analyze the rumor propagation in online social platforms by integrating other disciplinary theories [16,17,18,19]. The study of rumor propagation began with the classical theory established in the 60s of the 19th century. In 1965, Daley and Kendall proposed the Daley–Kendall theory [20], pointing out the differences between infectious disease transmission and rumor propagation models. They used mathematical modeling methods to realize the combination of the mathematical theory and the rumor propagation model for the first time. Since then, Zanette [21] has applied complex network theory to the study of rumor propagation for the first time and found that the process of rumor propagation is also affected by the topology of social networks. Ye et al. [22] established a fractional response–diffusion rumor propagation model in a multilingual environment. They discussed the global asymptotic stability of the equilibrium point by using the Lyapunov function method. Govindankutty et al. [23] introduced psychological bias into social networks and analyzed its effects. Hong Wei et al. [24] added individual states to the traditional rumor propagation model and established a SIRT model. Zhang Yaming et al. [25] constructed a model of rumor propagation dynamics based on the dual social reinforcement mechanism and simulated the propagation of rumors on this basis. Kumar et al. [26] established a node model of the simultaneous occurrence of rumor propagation and control. They applied it to the modeling of rumor propagation and the control of rumors. Teng Jie et al. [27] established a rumor diffusion model based on heterogeneous cellular automata. They proposed that the information interaction strategy, removal rate, and initial distribution of rumor-dispelling users determine the information diffusion effect. Based on the theory of gravity, Tan Zhenhua et al. [28] analyzed the participants and rumor information of public opinion quantitatively and established a new rumor propagation model.

At present, viewpoint evolution models are mainly divided into discrete viewpoint model and continuous viewpoint models. The earliest continuous viewpoint evolution model can be traced back to the Degroot model in 1974 [29]. The model was initially applied to the problem of the consistency of expert opinions in group decision-making and later was widely applied to the research of opinion dynamics. The bounded trust model in the continuous viewpoint propagation model focuses on the influence of the viewpoint difference between nodes on the propagation process. The most classic of these models is the HK model proposed by Hegselmann and Krause [30] and the Deffuant model proposed by Deffuant et al. [31]. Among the discrete view models, one of the most representative models is the Voter model [32]. The agent has only two discrete point of view states to choose from. The agent exchanges views with randomly selected neighbors and updates its point of view states according to the views of neighbors. This model is suitable for studying the change in discrete point of view states among agents. At present, the Deffuant model can simulate the information transmission and attitude change of people in social networks well, and the model can intuitively and concisely simulate the exchange process of any two individuals in a viewpoint group. Experts in this field have widely used the bounded trust model of continuous views to study the evolution of public opinion.

In summary, the main purpose of this paper is to study the rumor propagation model, considering the media effect and the mechanism of suspicion in public emergencies. The main work and contributions are reflected in the following:

(1)

Propose an improved rumor propagation model, considering the suspicion mechanism. Analyze the characteristics of propagation and the evolution process of propagation at different stages to provide some theoretical support for the mathematical model for rumor governance.

(2)

Use the quadratic matrix method to obtain the basic reproduction number. Based on the Jacobian matrix, Lyapunov’s stability theory, and Lassalle’s invariance principle, discuss the local and global stability of the no-rumor equilibrium point.

(3)

Based on the above model, we establish the time-delay differential equation to study the delay in transforming the susceptible into the infectious, caused by punishment and anti-fraud awareness.

(4)

To study the Deffuant opinion exchange model under the influence of the media, and to simulate the evolution process of rumor propagation under the same and different views.

2. Materials and Methods

2.1. SCIR Model under Media Influence

The traditional SIR model [33] divides the population into susceptible S(t), infectious I(t), and compromised R(t), where S(t) represents the number of people who have not been deceived by rumors at time t in the propagation process but are likely to be deceived by such rumors, I(t) represents the number of people who have been deceived by rumors at time t in the propagation process and have the ability to deceive, and R(t) indicates the number of people who have been diverted from the infectious scam at the t-moment of the propagation process. However, the SIR model does not have the characteristics of population mobility, emotional neutrality, and rumors. In addition, compared with the original SI model, the main propagation channel of rumors is considered in the media, and the two states of recovered and compromised are added correspondingly, which is detailed further in the state classification. The SCIR model considers the mutual influence and transition rules between individuals in different states, and uses parameters for immigration rate, recovery rate, rumor transmission rate, rumor propagation refresh rate, rumor propagation incidence rate, and rumor propagation mortality rate. The dynamics of the model are more complex and diverse. We divide a group composed of N individuals into four types: S (susceptible), I (infectious), R (recovered), C (compromised), a new SCIR rumor propagation model is established, and the rumor propagation flow chart is shown in Figure 1. Considering the influence of the media on the rumor propagation process, the model is close to the actual situation and accurately reveals the state transition process between the susceptible, infectious, compromised and recovered. Compared with SNA technology, it pays more attention to node centrality and the key node aspects of social network topology. It helps to make a quantitative analysis and prediction through more specific numerical results, which has the advantages of a simple and easy-to-operate algorithm. Through the improvement of communication rules, the SCIR model can be more flexibly applied to modern scenarios, which can be used to evaluate the impact of different intervention strategies on rumor propagation. It also provides more targeted intervention suggestions and provides a more comprehensive perspective and analytical framework for the study of rumor propagation.

ξ₁ is the incidence rate, ξ₂ is the mortality rate, β is the transmission rate, Λ is the immigration rate, μ is the emigration rate, θ is the recovered rate, γ is the refresh rate, all of the above parameters are positive. The description is shown in

Table 1. Paraments of SCIR model.

Parameter	Description	Abbreviation
S(t)	The number of Susceptible at t moment	Susceptible
C(t)	The number of Compromised at t moment	Compromised
I(t)	The number of Infectious at t moment	Infectious
R(t)	The number of Recovered at t moment	Recovered
ξ1	The probability that a Compromised believe and become a Infectious after being exposed to rumor or influenced by the media	Incidence rate
ξ2	The probability that Infectious becomes Compromised after contact with other Infectious or being influenced by the media	Mortality rate
β	The probability that a Susceptible becomes Infectious after being exposed to rumors	Transmission rate
Λ	The probability of people migrating into the system after exposure to rumors	Immigration rate
μ	The probability of moving out of the system after the population moves	Emigration rate
θ	The probability that the Compromised will see through the rumor and become Recovered after being affected	Recovered rate
γ	The probability that the Infectious becomes a Recovered after being indifferent to the rumor or after being aware of the rumor	Refresh rate

.

The rules of rumor propagation in this paper are as follows:

(1)

When a susceptible person hears the rumor, he becomes infectious with probability β or compromised with 1 − β.

(2)

Infectious, due to their own reasons or the influence of the media, forgets or senses the rumor with probability γ and becomes recovered.

(3)

Infectious diseases become compromised with a probability of ξ₂ after contact with other infectious diseases or exposure to the media.

(4)

The compromised recover through their own experience or the influence of the media to detect the rumor with a probability of θ, or to believe that the rumor becomes infectious with a probability of ξ₁.

The initial propagation conditions are as follows:

(1)

Crowd attributes: the types of crowds are evenly mixed, and the impact of regional differences on crowd density is not considered. Considering the mobility of the population, the corresponding parameters are set to reflect the incoming and outgoing of the population.

(2)

Mode of communication: The population is generally susceptible, and the mode of communication considers the mode of human-to-human transmission, the reasons of the people themselves, and the influence of the media.

(3)

Subject characteristics: the suspects in the crowd are transformed from the easily deceived and the deceived, and the deceived are contagious. In addition, once a person is confirmed to have been recovered, they are deemed to be incapable of spreading rumors.

According to the rumor propagation rule, the mean-field equation of the SCIR rumor propagation model can be obtained, as shown in Equation (1).

$$\{\begin{array}{l}\frac{dS(t)}{dt}=\Lambda -I(t)\frac{S(t)}{N(t)}-\mu S(t)\\ \frac{dI(t)}{dt}=\beta I(t)\frac{S(t)}{N(t)}-(\mu +\gamma +{\xi }_2)I(t)+{\xi }_{1}C(t)\\ \frac{dC(t)}{dt}=(1-\beta )I(t)\frac{S(t)}{N(t)}-(\mu +{\xi }_1+\theta )C(t)+{\xi }_{2}I(t)\\ \frac{dR(t)}{dt}=\gamma I(t)+\theta C(t)-\mu R(t)\end{array}$$

(1)

The positive invariant set of the system is determined $\Omega =\{(S,C,I,R)\in R_{+}^4\};S+C+I+R\le \frac{\Lambda }{\mu }$ The following differential equation can be obtained by adding the propagation mean field Equation (2)

$$\frac{dN(t)}{dt}=\Lambda -\mu N(t)$$

(2)

This equation is a first-order linear differential equation, and the total number of people in the whole propagation can be obtained by solving the equation as shown in Equation (3)

$$N(t)=\frac{\Lambda }{\mu }-\frac{e^{-\mu t}}{\mu }$$

(3)

The formula shows that, with the increased rumor propagation time, the total number of people in the system tends to be a fixed value. To study the stability of the basic regeneration number and the equilibrium point [34], the parameters in the equation are simplified by i = I/N, s = S/N, c = C/N, and r = R/N, in this case, i + s + c + r = 1. The simplified formula is shown in Equation (4)

$$\{\begin{array}{l}\frac{ds}{dt}=\mu -is-\mu s\\ \frac{di}{dt}=\beta is-(\mu +\gamma +{\xi }_2)i+{\xi }_{1}c\\ \frac{dc}{dt}=(1-\beta )is-(\mu +{\xi }_1+\theta )c+{\xi }_{2}i\\ \frac{dr}{dt}=\gamma i+\theta c-\mu r\end{array}$$

(4)

Using this equation and the next-generation matrix method [35,36,37,38] to calculate the basic regeneration number, it can be seen from $E_0=(1,0,0,0)$ that there is no rumor-free equilibrium point. If an equilibrium point is asymptotically stable, then all solutions that start near the equilibrium point will not only remain near that point but will tend to be infinite over time, as shown below.

$$X^{\prime }=(I,C,S,R)=F(x)-V(x)$$

(5)

$$F(x)=\left(\begin{array}{c}\beta is\\ (1-\beta )is\\ 0\\ 0\end{array}\right)\hspace{1em}V(x)=\left(\begin{array}{c}(\mu +\gamma +{\xi }_2)i-{\xi }_{1}c\\ (\mu +{\xi }_1+\theta )c-{\xi }_{2}i\\ is+\mu s-\mu \\ -\gamma i-\theta c+\mu r\end{array}\right)$$

(6)

Compute the Jacobian matrix of F(x) at E₀, denoted as F₀:

$$F_0=\left(\begin{array}{cccc}\beta & 0& 0& 0\\ 1-\beta & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$

(7)

Compute the Jacobian matrix of V(x) at E₀, denoted as V₀:

$$V_0=\left(\begin{array}{cccc}\mu +\gamma +{\xi }_2& -{\xi }_1& 0& 0\\ -{\xi }_2& \mu +{\xi }_1+\theta & 0& 0\\ 1& 0& \mu & 0\\ -\gamma & -\theta & 0& \mu \end{array}\right)$$

(8)

$$\left|V_0\right|={\mu }^2(\mu +\gamma +{\xi }_2)(\mu +{\xi }_1+\theta )-{\xi }_1{\xi }_2$$

(9)

Calculate the inverse matrix of V₀, denoted as $V_0^{-1}$:

$$V_0^{-1}=\frac{1}{\left|V_0\right|}\left(\begin{array}{cccc}{\mu }^2(\mu +{\xi }_1+\theta )& {\mu }^2{\xi }_1& 0& 0\\ {\mu }^2{\xi }_2& {\mu }^2(\mu +{\xi }_2+\gamma )& 0& 0\\ -\mu (\mu +{\xi }_1+\theta )& -\mu {\xi }_1& y_1& 0\\ \mu \gamma (\mu +{\xi }_1+\theta )+\mu {\xi }_2\theta )& \mu {\xi }_1\gamma & 0& y_1\end{array}\right)$$

(10)

$$y_1=\mu (\mu +{\xi }_2+\gamma )(\mu +{\xi }_1+\theta )-\mu {\xi }_1{\xi }_2$$

(11)

Next-generation matrices:

$$F_0{V}_0^{-1}=\frac{1}{\left|V_0\right|}\left(\begin{array}{cccc}\beta {\mu }^2(\mu +{\xi }_1+\theta )& \beta {\mu }^2{\xi }_1& 0& 0\\ \left(1-\beta \right){\mu }^2(\mu +{\xi }_1+\theta )& \left(1-\beta \right){\mu }^2{\xi }_1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$

(12)

The characteristic polynomial is as follows:

$$\left|F_0{V}_0^{-1}-\lambda E\right|={\lambda }^3\left(\lambda -\frac{\beta (\mu +\theta )+{\xi }_1}{(\mu +{\xi }_2+\gamma )(\mu +{\xi }_1+\theta )-{\xi }_1{\xi }_2}\right)=0$$

(13)

The maximum eigenvalue of the polynomial is the basic reproduction R₀:

$$R_0=\frac{\beta (\mu +\theta )+{\xi }_1}{(\mu +{\xi }_2+\gamma )(\mu +{\xi }_1+\theta )-{\xi }_1{\xi }_2}$$

(14)

To analyze the global stability of the rumor-free equilibrium point E₀ when R₀ < 1, we can define a Lyapunov function [39] according to the rumor propagation system of SCIR as follows:

$$V(t)=\left(\beta +\frac{(1-\beta ){\xi }_1}{\mu +{\xi }_1+\theta }\right)\left(s-s^0-s^0\ln \frac{s}{s^0}\right)+i+\frac{{\xi }_1}{\mu +{\xi }_1+\theta }c$$

(15)

For the analysis of the above constructor $f(x)=x-1-\ln x$, it is clear that $f(1)=0$ when $x=1$, and when $x\ne 1$, $f(x)>0$ set up. Therefore, only at the no-rumor equilibrium point E₀ there is $V(t)=0$, outside of which there is $V(t)>0$, and the derivative of V(t) along the system trajectory is shown below.

$$\frac{dV(t)}{dt}=\left(\beta +\frac{(1-\beta ){\xi }_1}{\mu +{\xi }_1+\theta }\right)\left(s^{\prime }-\frac{s^{\prime }}{s}\right)+i^{\prime }+\frac{{\xi }_1}{\mu +{\xi }_1+\theta }{c}^{\prime }$$

(16)

Finish into Equation (17):

$$\frac{dV(t)}{dt}=\mu \left(\beta +\frac{(1-\beta ){\xi }_1}{\mu +{\xi }_1+\theta }\right)\left(2-s-\frac{1}{s}\right)+\left(\beta +\frac{(1-\beta ){\xi }_1+{\xi }_1{\xi }_2}{\mu +{\xi }_1+\theta }-\mu -\gamma -{\xi }_2\right)i$$

(17)

When $R_0<1$, $\frac{dV(t)}{dt}\le 0$ if and only if there is no rumor equilibrium point E₀,$\frac{dV(t)}{dt}=0$. According to the Lasalle invariant set principle, the rumor-free equilibrium point E₀ is asymptotically stable.

Next, analyze the local stability of the positive equilibrium point of the rumor $E_1(s*,i*,c*)$ when $R_0>1$, and firstly, if E₁ is the steady state of the system, then E₁ should satisfy the following equation, Equation (18):

$$\{\begin{array}{l}\mu -i*s*-\mu s*=0\\ \beta i*s*-(\mu +\gamma +{\xi }_2)i*+{\xi }_{1}c*=0\\ (1-\beta )i*s*-(\mu +{\xi }_1+\theta )c*+{\xi }_{2}i*=0\end{array}$$

(18)

It is solved from the above equation

$$E_1\left(\frac{\mu }{\mu +i*},\mu (R_0-1),\frac{i*}{(\mu +{\xi }_1+\theta )}\left({\xi }_2+\frac{\mu (1-\beta )}{\mu +i*}\right)\right)$$

(19)

The Jacobian matrix at the positive equilibrium point E₁ of the rumor is:

$$J(E_1)=\left(\begin{array}{ccc}-i*-\mu & -s*& 0\\ \beta i*& \beta s*-(\mu +{\xi }_2+\gamma )& {\xi }_1\\ (1-\beta )i*& (1-\beta )s*+{\xi }_2& -(\mu +{\xi }_1+\theta )\end{array}\right)$$

(20)

Its corresponding feature polynomial is:

$${\lambda }^3+a{\lambda }^2+b\lambda +c=0$$

(21)

$$\{\begin{array}{l}a=\mu (2+R_0)+{\xi }_2+{\xi }_1+\theta +\gamma -\frac{\beta }{R_0}\\ b=\mu R_0(2\mu +{\xi }_2+{\xi }_1+\theta +\gamma )+(\mu +\gamma +{\xi }_2)(\mu +{\xi }_1+\theta )-\frac{\beta }{R_0}(2\mu +\theta )-{\xi }_1(\frac{1}{R_0}+{\xi }_2)\\ c=\mu R_0(\mu +{\xi }_1+\theta )\left(\mu +{\xi }_2+\gamma -\frac{\beta }{R_0}\right)+\mu \left(1-\frac{1}{R_0}\right)\left(\beta \mu +\beta \theta +{\xi }_1\right)-\mu {\xi }_1{R}_0\left(\frac{1-\beta }{R_0}+{\xi }_2\right)\end{array}$$

(22)

Analytically, we know that $a>0,b>0,c>0$, $ab>c$. According to the Hurwitz criterion, the positive equilibrium point of the rumor is locally asymptotically stable.

2.2. Time-Delayed Rumor Propagation Model under Penalty Mechanism

With the deepening of fraud propaganda and the increase in punishment, there will be a certain delay in converting rumors from susceptible to infectious, called time lag [40,41] Therefore, the delay differential equation is established to study the delay effect of this penalty. The rumor propagation process of this analysis is the same as that of the SCIR model, and the established mean field equation of rumor propagation is shown below.

$$\{\begin{array}{l}\frac{dS(t)}{dt}=\Lambda -\epsilon I(t-\tau )\frac{S(t)}{N(t)}-\mu S(t)\\ \frac{dI(t)}{dt}=\beta \epsilon I(t-\tau )\frac{S(t)}{N(t)}-(\mu +\gamma +{\xi }_2)I(t)+{\xi }_{1}C(t)\\ \frac{dC(t)}{dt}=(1-\beta )\epsilon I(t-\tau )\frac{S(t)}{N(t)}-(\mu +{\xi }_1+\theta )C(t)+{\xi }_{2}I(t)\\ \frac{dR(t)}{dt}=\gamma I(t)+\theta C(t)-\mu R(t)\end{array}$$

(23)

where ε represents the punishment intensity of the rumor, expressed by the proportion of infectious in the system; as shown in Equation (24), when ε approaches 0, the punishment intensity of the rumor is the greatest, and when ε approaches 1, the punishment intensity of the rumor is the least.

$$\epsilon =I(t)/N(t)$$

(24)

By simplifying the above equation, the parameters in the formula are simplified by i = I/N, s = S/N, c = C/N, r = R/N, and there is i + s + c + r = 1. The simplified formula is shown in Equation (25).

$$\{\begin{array}{l}\frac{ds}{dt}=\mu -i(t-\tau )i(t)s(t)-\mu s(t)\\ \frac{di}{dt}=\beta i(t-\tau )i(t)s(t)-(\mu +\gamma +{\xi }_2)i+{\xi }_{1}c\\ \frac{dc}{dt}=(1-\beta )i(t-\tau )i(t)s(t)-(\mu +{\xi }_1+\theta )c+{\xi }_{2}i\\ \frac{dr}{dt}=\gamma i+\theta c-\mu r\end{array}$$

(25)

For the local stability of the rumor positive equilibrium point $E_1(s*,i*,c*)$, E₁ is assumed to be the stable state of the system, and then E₁ should satisfy the following equation.

$$\{\begin{array}{l}\mu -i{*}^{2}s*-\mu s*=0\\ \beta i{*}^{2}s*-(\mu +\gamma +{\xi }_2)i*+{\xi }_{1}c*=0\\ (1-\beta )i{*}^{2}s*-(\mu +{\xi }_1+\theta )c*+{\xi }_{2}i*=0\end{array}$$

(26)

Solve the system and substitute i* for the other two terms, as follows:

$$\{\begin{array}{l}s*=\frac{\mu }{\mu +i{*}^2}\\ c*=\frac{1}{{\xi }_1}(\mu +{\xi }_2+\gamma )i*-\frac{\beta \mu i{*}^2}{{\xi }_1(\mu +i{*}^2)}\end{array}$$

(27)

Substituting Equation (27) into Equation (26) to solve the rumor positive equilibrium point $E_1(s*,i*,c*)$, and the Jacobian matrix at the rumor positive equilibrium point E₁ likes Equation (28)

$$J(E_1)=\left(\begin{array}{ccc}-i{*}^2-\mu & -s*i*(1+e^{-\lambda \tau })& 0\\ \beta i{*}^2& \beta s*i*(1+e^{-\lambda \tau })-(\mu +{\xi }_2+\gamma )& {\xi }_1\\ (1-\beta )i{*}^2& (1-\beta )s*i*(1+e^{-\lambda \tau })+{\xi }_2& -(\mu +{\xi }_1+\theta )\end{array}\right)$$

(28)

Its corresponding feature polynomial is ${\lambda }^3+A_2{\lambda }^2+A_1\lambda +A_0+(B_2{\lambda }^2+B_1\lambda +B_0)e^{-\lambda \tau }=0$; this paper uses the Hurwitz criterion to analyze the local stability of the positive equilibrium point of rumors.

When $\tau =0$, the characteristic equation is as follows: ${\lambda }^3+A_2{\lambda }^2+A_1\lambda +A_0+(B_2{\lambda }^2+B_1\lambda +B_0)=0$; according to the Hurwitz criterion, only hypothesis 1 needs to be satisfied to determine the local stability.

$$H1:\{\begin{array}{l}{A}_2+B_2>0\\ (A_2+B_2)(A_1+B_1)>A_0+B_0>0\end{array}$$

(29)

When $\tau >0$, the characteristic equation is as follows: ${\lambda }^3+A_2{\lambda }^2+A_1\lambda +A_0+(B_2{\lambda }^2+B_1\lambda +B_0)e^{-\lambda \tau }=0$, since the root and imaginary parts of the equation are both 0, set $\lambda =\omega i$ and simplify the equation as follows:

$$\{\begin{array}{l}(B_2{\omega }^2-B_0)\cos \omega \tau -B_1\omega \sin \omega \tau =A_2{\omega }^2-A_0\\ (B_2{\omega }^2-B_0)\sin \omega \tau +B_1\omega \cos \omega \tau =A_1\omega -w^3\end{array}$$

(30)

Square the above equation according to the trigonometric relationship to reduce the equation to the following equation:

$${\omega }^6+\left(A_2-2A_1-B_2^2\right){\omega }^4+(A_1^2-2A_0{A}_2-B_1^2+2B_0{B}_2){\omega }^2+A_0^2-B_0^2=0$$

(31)

In this case, hypothesis 2 can be determined according to the zero-point existence theorem, and ω has a finite number of positive real solutions.

$$H2:A_0^2-B_0^2<0$$

(32)

By solving the above equation, the positive real number solution ω_i can be obtained, and the corresponding τ_i real number solution can be obtained by substituting the solution into the original equation.

$${\tau }_i=\frac{1}{{\omega }_i}\arccos \frac{\left(B_1-A_2{B}_2\right){\omega }_i^4+(A_2{B}_0+A_0{B}_2-A_1{B}_2){\omega }_i^2-A_0{B}_0}{B_1^2{\omega }_i^2+{(B_0-B_2{\omega }_i^2)}^2}$$

(33)

${\tau }_0=\min \{{\tau }_i|i=0,1,2,\dots ,k\}$, the following equation can be obtained by using λ to derive τ in the eigenequation

$${\left(\frac{d\lambda }{d\tau }\right)}^{-1}=-\frac{3{\lambda }^2+2A_2\lambda +A_1}{\lambda ({\lambda }^3+A_2{\lambda }^2+A_1\lambda +A_0)}+\frac{2B_2\lambda +B_1}{\lambda (B_2{\lambda }^2+B_1\lambda +B_0)}-\frac{\tau }{\lambda }$$

(34)

$$H3={\left[\frac{dRe(\lambda )}{d\tau }\right]}_{\lambda ={\omega }_{0i}}^{-1}\ne 0$$

(35)

Based on the conditions for establishing the above assumptions, we draw the following conclusions:

(1)

When H1 holds, i.e., τ = 0, the model is locally stable.

(2)

When H2 and H3 are true, if 0 < τ < τ₀, the propagation model will tend to be asymptotically stable. If τ = τ₀, the propagation model starts to have a Hopf bifurcation; if τ > τ₀, the propagation model will oscillate, and the model will no longer be stable.

2.3. Deffuant Model

The Deffuant model [42,43] is a mathematical model of the sustainable exchange of opinions. It holds that when two individuals in the whole opinion group exchange opinions, the difference between the evolutionary viewpoint value in the opinion group can only be exchanged between any two individuals within a fixed interval. The maximum difference of the evolutionary viewpoint value exchange is called the exchange threshold, expressed as δ.

The opinion group in the mathematical model contains N different individuals, and each individual in the opinion group will produce its own evolutionary viewpoint value $O_i(i=1,2,\dots ,N),O_i\left[0,1\right]$. For the evolution of opinions, there are only two experiences in the unit time of the evolution of opinions, and there is an exchange of opinions.

This study argues that the spread of rumors among individuals is mainly affected by the degree of trust, and the degree of trust of the heterogeneous evolutionary viewpoint value is generally dominated by poor information, that is, by the media’s guidance of public opinion. The more people rumors deceive, the less influence the media will have on the spread of public opinion, which is manifested in the fact that the views of the deceived people are not easy to change. Individuals with negative views in groups, such as the susceptible ones, are more likely to accept the views of the deceived people. Assuming that individual i and individual j exchange opinions at time t, when the evolutionary viewpoint value of the two is less than or equal to the exchange threshold $\left|O_i-O_j\right|\le \delta$, the law of opinion exchange between individual i and j is shown in Equation (36).

$$\{\begin{array}{l}{O}_i(t+1)=O_i(t)-K\mu (O_i-O_j)\\ O_i(t+1)=O_i(t)+K\mu (O_i-O_j)\end{array}$$

(36)

$$K=e^{-0.001I(t)}$$

(37)

μ is the acceptance of individual opinions, $\mu \in \left[0,0.5\right]$. K denotes the effect on the media’s degree of trust in opinions. When heterogeneous participants in the system begin to exchange opinions, as shown in Equation (37), I(t) is the number of negative emotions.

In the Deffuant model, the meaning of individual opinion acceptance μ is the stubbornness of the attitude of the participants involved in the exchange of views in the process of spreading rumors on the online social platform; when μ = 0, the evolutionary viewpoint value of the two participants participating in the exchange of opinions does not change after the interaction, when μ = 0.5, the evolutionary viewpoint value of the two participants in the exchange of views will be taken as the average of the views of the exchangers, and when μ > 0.5, the two participants in the exchange of views support each other’s views in terms of opinion tendency after the interaction, that is, the opinion tendency of the two opinion participants has changed.

3. Results

3.1. The Division of Rumor Life Cycle

Referring to Steven Fink’s [44] and Li Zhihong et al.’s [45] research on the life cycle of public opinion, this paper divides the life cycle of rumors into five stages: initiation, outbreak, fluctuation, climax, and declining, and analyzes the characteristics of each period.

By collecting the basic public opinion discussion data of the ‘Taocheng Hengshui Middle School Incident’ for 15 consecutive days, taking the Weibo Topic readership number, the number of discussions and the number of original people as the life cycle division which is shown in Figure 2, the emotional words were extracted through the emotion dictionary [46], and some of the emotion words, under the artificial annotation of specific comments, were taken as the emotion base words [47]. Then, calculate the semantic similarity [48]. Identify the emotion polarity [49] of the emotion target words according to the semantic similarity between the selected emotion base words and the target words.

As can be seen from the Figure 2, the number of people related to the topic is only 40 on the first and second days. Public opinion has not yet broken out at this stage, and the topic discussed has nothing to do with ‘Taocheng Hengshui Middle School’, so the first and second days are regarded as the beginning of the rumor life cycle. The number of people related to the topic grew rapidly and peaked on the 2nd to 3rd day. The number of topic associated people began to decline quickly on the 3rd to 4th day, so the 2nd to 4th day, when public opinion broke out rapidly and fell back, was regarded as the outbreak period of the rumor life cycle. The number of people related to the topic began to grow again on the 4th to 7th day, and the number of readers of the topic reached 300 million, the number of topic discussions reached 210,000, and the original number of the topic reached 20,000. The topic readership number peaks on the 7th–9th day, when the number of readers reaches 340 million, and the public opinion at this stage is mixed with rumors. It causes a lot of negative remarks, so the 7th–9th day is regarded as the climax of the rumor life cycle. The number of people related to the topic gradually decreased to 150 from the 9th to 15th day, and both public opinion and rumors at this stage were greatly reduced due to the clarification of the public security department, so the 9th to 15th day was regarded as the dissipation period of the rumor’s life cycle.

In summary, days 1–2 are the beginning of the rumor life cycle, days 2–4 are the outbreak period of the rumor life cycle, days 4–7 are the fluctuation period of the rumor life cycle, days 7–9 are the climax of the rumor life cycle, and days 9–15 are the dissipation period of the rumor life cycle.

3.2. SCIR Propagation Model Simulation

3.2.1. Simulation of SCIR Model Propagation under Media Influence

After the above stability analysis, we used more than 20,000 comments collected on the Weibo platform for 15 consecutive days, and finally retained 8440 data texts through screening and integration for follow-up analysis. The collected public opinion sentiment data, combined with the rumor propagation cycle, were used to simulate the rumor propagation. The transmission rate in the system was changed to β′ to consider the influence of the media effect on rumor propagation, as shown in Equation (38), where m represents the media impact factor.

$${\beta }^{\prime }=e^{-mI\left(t\right)}\beta$$

(38)

In addition, we use the numerical calculation software Matlab R2018b (MathWorks, Natick, MA, USA) to simulate the numerical solution of differential equations, as well as using the Euler method and the code to implement iterative loops, calculate the state of the next moment according to the state of the current moment, update the value of each variable, and finally draw the corresponding graph. The value of the propagation parameters in the system is as follows: β transmission rate is the proportion of negative emotions among all affective polarities, γ refresh rate is the proportion of positive affective among all affective polarities, ζ₁ incidence rate is the proportion of negative bonus points among all neutral affective polarities, ζ₂ mortality rate is the proportion of positive bonus points among all negative affective polarities, and θ recovered rate is the proportion of positive bonus points among all neutral affective polarities.

The propagation parameters of the initial period of the rumor were calculated as follows: β = 0.14, γ = 0.03, Λ = 6, μ = 0.01, ξ₁ = 0, ξ₂ = 0, θ = 0, m = 0.001. Due to the low number of overall discussions at the beginning of the period, the total population of this stage was set at N = 1000, and the simulation is shown in Figure 3.

This phase of full transmission takes 500 days for the entire transmission process, with the maximum number of infectious reaching 50 on day 60 and the maximum number of recovered reaching 360 on day 90. At the beginning of the rumor, due to the consideration of neutral emotional polarity, the deception rate is greatly reduced, so the scale of infectious is very small. Still, those who recovered from the rumor represent the largest in the whole propagation process. From the perspective of the impact of public opinion dissemination, the initial propagation period will produce more rumors, increasing the risk of public opinion in the later stage of rumor propagation.

The propagation parameters of the rumor outbreak period were calculated as follows: β = 0.16, γ = 0.05, Λ = 10, μ = 0.01, ξ₁ = 0.007, ξ₂ = 0.002, θ = 0.007, m = 0.001. As the overall number of people discussing during the outbreak began to increase significantly, the total number of people in this phase was set at N = 1,000,000 and simulated accordingly. The results of the simulation are shown in Figure 4.

For the entire transmission process, the full transmission time for this phase takes 500 days, with the number of insensitive reaching a maximum of 700 on day 140 and the maximum number of recovered reaching 3100 on day 150. During the outbreak of rumors, the number of victims increased due to the increase in the rate of deception and incidence rate due to the spread of the topic, so the number of victims also increased compared to the previous stage. From the perspective of the influence of public opinion dissemination, spreading rumors in the outbreak period provides infectious and recovered for further spreading rumors. Still, its propagation time is longer than the initial period, and it is more likely to be blocked in the propagation process.

The propagation parameters of rumors during the fluctuation period are calculated as follows: β = 0.18, γ = 0.05, Λ = 10, μ = 0.01, ξ₁ = 0.025, ξ₂ = 0.028, θ = 0.025, m = 0.001. Due to the large number of people discussing the topic overall during the fluctuation period, the total population was set at N = 1,000,000 and simulated on this basis. The results of the simulation are shown in Figure 5.

For the entire transmission process, this phase of full transmission takes 500 days, with the number of infectious reaching a maximum of 1000 on day 100 and the recovered reaching a maximum of 2500 on day 98. During the fluctuation period of rumors, the number of victims increased compared to the previous stage due to the discussion caused by the further spread of the topic, resulting in an increase in the number of victims and a decrease in the number of recovered people compared to the previous stage. At the same time, the number of compromised people increased significantly. From the perspective of the impact of public opinion dissemination, the spread of fluctuating periods increased the number of recovered for the spread of rumors and increased the population flow between recovered, compromised, and infectious, so that the number of recovered began to decrease.

The propagation parameters of the high tide of rumors were calculated as follows: β = 0.8, γ = 0.03, Λ = 10, μ = 0.01, ξ₁ = 0.005, ξ₂ = 0.003, θ = 0.005, m = 0.001. Due to the large number of people discussing the topic overall during the climactic period, the total population of this period was set at N = 1,000,000 and simulated accordingly. The results of the simulation are shown in Figure 6.

For the entire transmission process, this phase of full transmission takes 250 days, with a maximum of 500,000 infectious on day 34 and 140,000 recovered on day 30. In the climax period of rumors, due to the influence of the existing infectious and recovered in the process of rumor propagation and the secondary spread of rumors, the deception rate increases significantly. In contrast, the recovery rate, incidence rate, mortality rate, and infection rate decrease, so the number of infectious people increases greatly and exceeds the number of recovered. In this stage, measures aimed at the mortality rate can reduce the size of some of the most prominent victims, but the impact on the victims in the whole process of rumor propagation is small, and the impact on the risk of propagation is small. From the perspective of the impact of public opinion dissemination, the climactic period is the most harmful stage in the whole communication process, which involves a large number of people, and a large scale of infectious and recovered, so it is easy to cause various injuries and the loss of credibility, and the time spent on the communication process at this stage is only half of that of other communication processes, which means that rumors can spread quickly and cause harm at this stage.

The propagation parameters of the rumor dissipation period are calculated as follows: β = 0.17, γ = 0.04, Λ = 10, μ = 0.01, ξ₁ = 0.004, ξ₂ = 0.004, θ = 0.004, m = 0.001. As the overall number of people discussing the topic during the dissipation period began to decrease, the total population of this phase was set at N = 1,000,000 and simulated accordingly. The results of the simulation are shown in Figure 7.

In terms of the overall transmission process, this phase of full transmission takes 500 days, with the number of infectious reaching a maximum of 500 on day 130 and the maximum number of recovered reaching 3700 on day 140. The number of victims has also decreased in the dissipation period of the rumor. The number of recovered people has gradually increased due to the spread of public opinion on the topic due to the influence of the rumors announced by the relevant departments, resulting in a decrease in the deception rate and an increase in the mortality rate. From the perspective of the influence of public opinion dissemination, the dissipation period no longer has a greater impact on public opinion and social harm, and the influence of the entire rumor spread gradually dissipates.

3.2.2. Analysis of Propagation Parameters of SCIR Model

Basic reproduction is a key parameter used to analyze the severity of rumor propagation, so it is also necessary to explore the impact of parameter changes on basic reproduction. This paper selects incidence, mortality, infection, and recovery rates as variables to study the trend of basic reproduction. The value range is set to [0, 1] when studying the effects of incidence rate and mortality rate, infection rate, and recovery rate. The rest of the propagation parameters are selected as the propagation parameters of the climax period of rumors. The data are as follows: β = 0.8, γ = 0.03, μ = 0.01, ξ₁ = 0.005, ξ₂ = 0.003, θ = 0.005. The results of the impact of the incidence rate and mortality rate are shown in Figure 8.

The basic reproduction of the whole rumor propagation system decreases with the increase in incidence rate ξ₁ and the increase in mortality rate ξ₂, which is basic, as most of the regions of reproduction are in the high risk state of R₀ > 1 rumor propagation, which is very likely to cause large-scale rumor damage. The incidence rate and mortality rate are very low in this state. Only when the rate approaches 0 and the mortality rate approaches 1 can the basic reproduction be controlled below 3. Therefore, the decrease in incidence rate and increase in mortality rate can only exert a small influence on the suppression of rumors during the climax period of rumor propagation. Figure 9 shows the influence results of the infection rate and the recovery rate.

The whole rumor propagation system’s basic reproduction increases with the recovery rate μ and decreases with the rise in infection rate θ. Some areas of the basic reproduction are in the low-risk state of rumor propagation at R₀ < 1, and most are at R₀ > 1 only when the recovery rate tends to 0. The high-risk rumor spreading state is likely to cause large-scale rumor damage. This state’s recovery rate is extremely high, and can keep most of the basic reproduction area below 3. In contrast, the lower impact of the infection rate is not noticeable compared to the recovery rate. Therefore, the increase in recovery rate has a major impact on the suppression of rumors during the climax period of rumor propagation.

To sum up, there are the following measures to control rumors: In the stage of susceptible transforming into infectious, the main reasons are the lack of correct rumor cognition, such as the lack of scientific common sense, blind conformity, and luck mentality. Expanding the scope of popular science can affect the cognition of rumors. In the stage of infectious transforming into recovered, the main reason is the implementation of rumor supervision, such as the closure of rumor propagation accounts, official statements to clarify facts, etc. To affect the supervision of rumors, we should increase the intensity of control, speed up the publication of facts, and expand the scope of supervision. In the stage of compromised transforming into infectious, the main reason is the exaggeration of rumor propaganda. In the stage of infectious transforming into compromised, the main reasons for the transformation are the improvement of personal vigilance and suspicion awareness, or a more dialectical and rational view of rumor information after experiencing rumor deception. At the same time, they tend to be more scientific and reliable channels in terms of information acquisition, and official or authoritative institutions provide more authoritative information and explanations to reduce the spread and influence of rumors. In the stage of compromised transforming into recovered, the main reason is that the rumor is found to be contrary to common sense, the rumor has become a typical case, and the impact of the rumor can be systematically analyzed by the recovered.

3.3. Time-Lag Propagation Model Simulation

To study the propagation of rumors in a time lag, the propagation parameters of rumors in the climax period were selected for propagation simulation, and the propagation parameters were as follows: β = 0.8, γ = 0.03, λ = 10, μ = 0.01, ξ₁ = 0.005, ξ₂ = 0.003, θ = 0.005. From this, it can be calculated that τ₀ is about 110, τ = 100 is used to simulate the spread of rumors, and the results of the simulation are shown in Figure 10.

The graph and stability analysis shows that when τ = 100 satisfies the condition of τ < τ₀, the propagation model tends to be asymptotically stable. The complete propagation time at this stage is 140 days, the minimum proportion of susceptible is only 18% on day 21, and the final proportion is 40% after the system stabilizes. Infectious accounted for the largest, at 23% on day 11 and 10% after the system stabilized, and recovered accounted for the highest, at 12.4% on day 26 and 7.4% after the system stabilized. Compared with the SCIR propagation model without penalty time lag, the punishment mechanism can shorten the rumor propagation time by nearly 50% and reduce the maximum proportion of infectious by 27%, which has an important impact on inhibiting the harm of rumor propagation. τ = 150 is used to simulate the spread of rumors, and the simulation results are shown in Figure 11.

The diagram and the above stability analysis show that when τ = 150 satisfies the condition of τ > τ₀, the propagation model will produce oscillations and is no longer stable. At this stage, the propagation begins to fluctuate periodically. The system is no longer stable, with the proportion of susceptible oscillating between 22% and 60% from day 200, the proportion of infectious from day 300 onwards between 2.5% and 25%, and the proportion of recovered from day 280 onwards from 3.5% to 11.5%. Compared with the SCIR propagation model without penalty time lag, the long penalty lag will lead to the periodic oscillation of rumor propagation, and effective measures cannot be taken to curb the harm of rumor propagation, so it is necessary to avoid the long penalty lag caused by too severe punishment.

3.4. The Evolution of Views Simulation

To analyze the evolution of rumor propagation in heterogeneous views, the acceptance μ were set as 0.5, 0.3, and 0.2, the exchange thresholds were 0.8, 0.5, 0.3, and 0.2, the total number of participants was N = 100, the degree of trust followed $K=e^{-0.001I(t)}$, and the evolution step was 4000 steps. The individual opinion values were evenly distributed in the interval of [0, 1] and simulated. The results of the simulation at acceptance μ = 0.5 are shown in Figure 12.

As seen from (a–d), when the acceptance μ = 0.5 and the exchange threshold δ = 0.8, the number of evolutions is 1600 times, and the final evolutionary view is 0.52. When μ = 0.5 and δ = 0.5, the number of evolutions is 1600 times, and the final evolutionary view is 0.52. When μ = 0.5 and δ = 0.3, the number of evolutions is 3400, and the final evolutionary views are 0.71 and 0.26. When μ = 0.5 and δ = 0.2, the number of evolutions is 3500, and the final evolutionary views are 0.80, 0.49, and 0.15, respectively.

The results of the simulation at acceptance μ = 0.3 are shown in Figure 13.

As seen from (a–d), when the acceptance μ = 0.3 and the exchange threshold δ = 0.8, the number of evolutions is 1900 times, and the final evolutionary view is 0.51. When μ = 0.3 and δ = 0.5, the number of evolutions is 1700 times, and the final evolutionary view is 0.55. When μ = 0.3 and δ = 0.3, the number of evolutions is 2300, and the final evolutionary views are 0.71 and 0.20. When μ = 0.3 and δ = 0.2, the number of evolutions is 4000, and the final evolutionary views are 0.80, 0.48, and 0.19, respectively.

The results of the simulation at acceptance μ = 0.2 are shown in Figure 14.

As seen from (a–d), when the acceptance μ = 0.2 and the exchange threshold δ = 0.8, the number of evolutions is 1900 times, and the final evolutionary view is 0.52. When μ = 0.2 and δ = 0.5, the number of evolutions is 2300 times, and the final evolutionary view is 0.52. When μ = 0.2 and δ = 0.3, the number of evolutions is 3200, and the final evolutionary views are 0.72 and 0.21. When μ = 0.2 and δ = 0.2, the number of evolutions is 4000, and the final evolutionary views are 0.89, 0.50, and 0.13, respectively.

Regarding the influence of the rumor threshold, when the public’s views on rumors are at a high threshold, it is not easy to form negative views. Still, it promotes the neutralization of views in communication, which increases the proportion of compromised and reduces the proportion of infectious for the spread of rumors among the susceptible. When the exchange threshold is reduced, it is difficult for the masses with strong positive views to exchange views with the masses with strong negative views, which leads to the overall evolutionary viewpoint value gradually becoming neutral and negative. When the exchange threshold is further reduced, the range of exchangeable views is further reduced, resulting in an increase in the number of evolutions and the polarization of opinions, resulting in a higher positive evolutionary viewpoint value and a lower negative evolutionary viewpoint value, which increases the proportion of infectious for the spread of rumors among the susceptible. When the exchange threshold is extremely low, it will further increase the number of evolutions and a pluralistic differentiation of views, resulting in a very high positive, a neutral, and a very low negative evolutionary viewpoint value. Such a result will create irreconcilable contradictions in the same view, which will increase the rate of infectious and reduce the recovery rate. In terms of the influence of rumor acceptance, the decrease will lead to a step-size reduction in the rumor opinion change, which will increase the number of opinion evolutions. Furthermore, it enlarges the range of viewpoints under the same evolutionary step, which is not conducive to forming the final viewpoint.

4. Conclusions

This paper collects the basic information data of online rumors and public opinion with highly topical public opinions. We analyzed the spread of online public opinion rumors under the suspicion mechanism by considering the flow of rumor propagation. The number of compromised decreases, and recovered increases; the secondary spread of rumors in the climax period leads to a large increase in the number of infectious and exceeds the number of compromised. The number of infectious decreases while susceptible increases due to the influence of rumor refutation during the rumor dissipation period. The analysis parameters show that reducing the incidence rate and increasing the mortality rate can reduce the size of the largest number of infectious, and increasing the emigration rate is better than improving recovered rate. Using the time-delay differential equation under the punishment mechanism to analyze the stability of the evolution of rumor propagation, it can be found that when τ < τ₀ tends to be asymptotically stable, the punishment mechanism can shorten the propagation time and reduce the maximum proportion of deceived people, and when τ > τ₀ there is a periodic oscillation and the system is no longer stable, and no effective measures can be taken to suppress rumors. Finally, the improved Deffuant model, under the influence of the media, will be used to analyze the evolution of heterogeneous views with different acceptance degrees and exchange thresholds. It can be found that the high threshold reduces the proportion of infectious, the deception rate increases when the exchange threshold is extremely low, and the unity of views decreases when the acceptance is reduced. The rumor propagation model can be applied to the public opinion monitoring and management platform of social media platforms in public security and network security to block or guide the communication of public opinion. To restrain the spread of rumors at the practical level, efforts should be made toward public opinion guidance in education and social environment governance. On the one hand, education should strengthen the public’s ability to identify information, help the public reinforce their ability to distinguish truth from falsehood and increase their vigilance against rumors. On the other hand, scientific knowledge and common sense should be promoted to reduce the spread of rumors by improving public cognition. Public opinion can be guided by establishing a rumor detection and management mechanism to promptly identify and deal with rumor information to reduce the public’s impact. In addition, official organizations and influential media also need to strengthen their credibility and release more authoritative information to provide more reliable sources to suppress the effect of rumors. The direction of social environmental governance needs to improve the improvement of laws and regulations, clear punishment measures, and increase the cost of illegal. Establish a sound rumor supervision system to reduce the impact of rumors. Regularly crack down on bad media to improve the authenticity and objectivity of news reports. Strengthen the network supervision, standardize the network speech, purify the network environment and work together to deal with the negative impact of rumors.