1. Introduction
A rumor about iodized salt that can protect against radiation has spread rapidly throughout China since the Fukushima nuclear power plant exploded after Japan’s earthquake occurred on 11 March 2011. The rumor triggered a panic purchase behavior of iodized salt. Large crowds doubted the rumor refutation information and persisted in buying salt, leading to social and economic chaos. Due to the suddenness, urgency, high uncertainty, and destructiveness of an emergent event (such as an earthquake), if the comprehensive and accurate information cannot be timely spread to people, the formation and propagation of rumors can be caused. In emergency management, the process of rumor propagation is also a process of negative emotion diffusion. Rumors can mislead people to make irrational behaviors, hinder the smooth progress of emergency management, and expand the losses triggered by the emergent events. As a consequence, rumor spreading control is a crucial issue in emergency management.
An army of scholars investigated the rules of rumor propagation by constructing propagation models from different dimensions, among which the Maki and Thomson (DK) [
1] model and the Maki and Thomson (MT) [
2] model are two classical rumor propagation models. In these classical models, people are subdivided into three groups: ignorants, spreaders, and stiflers. The ignorants refer to people who are not aware of rumor information, the spreaders refer to people who are spreading rumor information, and the stiflers refer to people who have heard the rumor but do not propagate it anymore. The transformation mechanism between different groups is as follows: the spreader–ignorant contact will convert the ignorant into the spreader, spreader–spreader contact will convert both spreaders into stiflers since both spreaders lose interest in spreading rumor information, and spreader–stifler contact will stifle the spreader. The MT model differs from the DK model in the termination mechanism of rumor spreading, namely, when a spreader and another spreader contact in the MT model, only the initial spreader becomes the stifler.
A series of modified rumor propagation models have been developed based on the classical models mentioned above. On the one hand, some researchers focused on the fundamental processes involved in rumor propagation. They introduced the social reinforcement mechanism [
3], hesitation mechanism [
4], etc. It’s worth noting that time delay [
5,
6,
7] may also influence rumor propagation. Huo et al. [
8] believed that when the spreader contacts the ignorant, the ignorant individual first experiences an incubation period before turning into a spreader. Time delay not only affects the crowds, but also expert intervention and government policy [
9], the effect of delay made an important contribution to the stability of the system. Moreover, some researchers redefined and redivided the groups in the rumor propagation model. Wang et al. [
10] argued that there are rationals who can verify the rumor in the initial state of the rumor spreading system. With the exception of rationals, the incubators and wise individuals were introduced by Huo et al. [
3,
8]; the counterattack group was introduced by Zan et al. [
11]; the exposed individuals by Xia et al. [
12]. In addition, considering the rumor propagation system has close contact with the outside world, some scholars take the population migration into account [
6,
7,
13]. They first assumed that the rumor propagation system has constant immigration and emigration, then studied the impact of other influencing factors, such as delayed on the rumor spreading and aimed to figure out the stability of the equilibrium point and the final condition of rumor spreading.
On the other hand, many scholars explored the rumor propagation model from the perspective of individuals and the rumor itself. Huo et al. [
14] reckoned that the transmission rate of ignorant to the spreader is affected by psychological effects in the context of emergencies. In addition, the forgetting mechanism and the remembering mechanism of crowds were considered [
4,
15,
16,
17]. Wang et al. [
18] claimed that whether the ignorants trust the spreaders or not determines the conversion rate of ignorants into spreaders after the contact of spreaders and ignorants. Moreover, the transmission rate is higher when the spreaders are credible. Similarly, Qiu et al. [
19] found that a spreader with a high reputation can promote the spread of rumors. Furthermore, the education rate is a significant factor that may affect rumor propagation. Afassinou [
20] built the Spreader, Educated Ignorant, Non-educated Ignorant, Stiflers (SEIR) rumor spreading model, and explored the impact of the education rate on rumor propagation. In this rumor propagation model, ignorants were distinguished by educated ignorants and non-educated ignorants. The educated ignorants have a lower probability of becoming spreaders and a higher probability of becoming stiflers after contacting the spreaders. Nevertheless, the uneducated ignorants have a higher probability of turning into spreaders and a lower probability of turning into stiflers after contacting spreaders. Additionally, in a like manner, Hui et al. [
21] concluded that the higher-educators class is more likely to be immune to rumors with a higher probability and the lower-educators class is more inclined to accept and spread the rumors. With the development of information science, rumors have altered their spread mode to Online Social Networks (OSNs); hence, Hosni et al. [
22] investigated the effect of OSN addiction on rumor propagation. They found that addicted individuals may accelerate the dissemination under specific conditions. As for the rumor itself, Chen et al. [
23] suggested that rumor credibility and the correlation between rumor and people’s lives will influence the dissemination of rumors. The higher the credibility of a rumor is, the larger the final scale of rumor spreading is, and the rumor propagation speed and spreading scale are proportional to the relevance between a rumor and people’s lives. Xia et al. [
12] maintained that when rumors had high attractiveness, the exposed individuals would decide to spread rumors and become the spreader; when the rumor was fuzzy, spreaders would cease spreading rumors and transform into exposed individuals; when the fuzziness of rumors was low, the spreaders would lose interest in spreading rumors and transform into stiflers. Moreover, when it comes to eradicating rumors, rumor intensity will contribute to the identification of the stop condition for anti-rumor propagation and the initial spreaders of anti-rumors [
24].
Typically, numerous scholars investigated the relationship between rumor spreading and emergency management as well as developed the corresponding interplay model. Zhang et al. [
25] found that rumor spreading has the effect of restricting the rapid expansion of the impact of emergencies. Huo et al. [
26] stated that in emergencies, the official responses to emergencies produce positive social utility, while the negative social utility brought by the spread of rumors weaken the positive social utility. The authorities can take actions to control the spread of rumors to downplay negative social utility. However, if the emergency response is not enough to achieve the expected effect of the public, the negative social utility will be maximized. Zhao et al. [
27] stated that the authorities can control the situation by using trusted media outlets with high audience ratings to disseminate information.
Existing research mainly focused on the definition, causes, influencing factors, and the final spreading state, but neglected to give a reasonable description of the development process of rumors. In addition, though some works of literature considered the educational level of individuals [
3,
21,
22], an individual’s ability to resist rumor spreading is not fully considered. People’s critical ability to refute rumors does not just pertain to theoretical knowledge acquired through education, but also many other factors, such as conscious behavior [
3]. Individuals with critical ability can obtain the evidence of false rumor information through effective ways (i.e., rumor refutation information) and can further suppress the rumor propagation through effective feedback and rumor refutation information. Furthermore, in the era of the internet, people are connected under one virtual society, people on the network migrate frequently, namely, people can easily participate in a discussion of a topic, as conduct the spreading of rumors. Hence, the impact of population migration on rumor propagation should be comprehensively analyzed. However, in the very few studies that incorporate population migration, in most cases, population migration is served as a prerequisite to study the influence of other influencing factors on the spread of rumors. However, the influence of population migration on rumor spreading has not been studied in depth. Based on the above analysis, this paper establishes a new compartmental model of rumor propagation, and the main contributions of this paper can be summarized as follows:
- (1)
The rumor refutation mechanism and information feedback mechanism are considered. The groups in the rumor propagation system are redivided and redefined, a new kind of people—the skeptic is introduced into the model, and the stiflers are divided into stiflers who believe the rumor and stiflers who do not believe the rumor.
- (2)
The rumor propagation model is analyzed in both the closed system and open system, and the development process of rumor spreading is comprehensively described, and the general rules of rumor propagation under the influence of population migration are studied.
- (3)
Multiple influencing factors besides effective feedback mechanisms are comprehensively considered in this paper.
The rest of the article is structured as follows: In
Section 2, the rumor propagation model in a closed system and an open system are established, the group division in the rumor propagation model and the behavior law of each group are analyzed, the mean-field equations are established. In
Section 3, the general rules of rumor propagation and the rumor propagation laws under different influencing factors of a closed system are analyzed through the data obtained numerically. In
Section 4, the general rules of rumor propagation and the rumor propagation laws under different influencing factors of an open system were analyzed through the data obtained numerically. Finally,
Section 5 summarizes the paper.
2. Rumor Propagation Models Considering Rumor Refutation and Information Feedback
From the perspective of managing the public sentiment crisis in emergency management, the population in the rumor propagation system was subdivided into five groups in accordance with the rumor propagation process: ignorants (X), spreaders (Y), skeptics (W), stiflers who believed the rumor (Z
1), and stiflers who did not believe the rumor (Z
2). In emergency management, public panic is easily caused by rumor spreading. The negative impact scale of rumors can be described as the proportion of stiflers who believe the rumor at the equilibrium state of rumor spreading. Then, ignorants were divided into two groups on the ground of whether they possess critical ability: ignorants with critical ability and ignorants without critical ability. Additionally, the proportion between the two groups was constant in the initial state of a specific rumor spreading model. Different ignorants had diverse behavioral responses to rumors. Ignorants with critical ability could not easily accept rumors and would become skeptics after contacting the spreaders. However, ignorants without critical ability accepted rumors readily and became spreaders after contacting the spreaders. Skeptics did not spread rumor information and actively obtained the refutation information. If refutation information was not available, skeptics became stiflers who believed the rumor; if refutation information was available, skeptics became stiflers who did not believe the rumor. When spreaders contacted other spreaders or stiflers who believed the rumor, they found that the rumor information was obsolete and stopped spreading the rumor information, and became stiflers who believed the rumor. While spreaders contacted stiflers who did not believe the rumor, the latter would persuade the spreaders to become stiflers who did not believe the rumor under the effective rumor refutation information feedback. Lastly, in terms of the method of Maki [
2], it was thought that a rumor propagated when a spreader directly contacted others [
28]. Additionally, each spreader completed a random contact process within a time step, and the skeptic took at least a time step to obtain the refutation information.
2.1. Rumour Propagation Model XYWZ1Z2-C in a Closed System
In this part, a closed rumor propagation system which has no population immigration and emigration was considered. Additionally, the rumor propagation model XYWZ1Z2-C in a closed system was established, as shown in
Figure 1.
According to the mean-field theory, the mean-field equations of the rumor propagation model XYWZ1Z2-C in a closed system were derived as follows:
The parameters in the mean-field equations of rumor propagation were defined as follows: N refers to the total population in the rumor propagation system; X, Y, W, Z1, and Z2, respectively, represent the population of ignorants, spreaders, skeptics, stiflers who believe the rumor, and stiflers who do not believe the rumor; φ denotes the proportion of ignorants without critical ability in all ignorants; (1 − φ) represents the proportion of ignorants with critical ability in all ignorants; α refers to the conversion rate of the ignorants into the spreaders or the skeptics when the ignorants contact the spreaders; δ is the conversion rate of the first spreader into the stifler who believes the rumor when the spreader contacts other spreaders or stiflers who believe the rumor; η is the rate of the skeptics change into stiflers who believe the rumor or stiflers who do not believe the rumor, which is related to the importance of rumor information identification; θ represents the ratio of the conversion rate of the skeptics into stiflers who believe the rumor to the total conversion rate of the skeptics; (1 − θ) represents the ratio of the conversion rate of the skeptics into stiflers who do not believe the rumor to the total conversion rate of the skeptics, which is related to rumor discrimination capability; λ is the conversion rate of the spreaders into stiflers who do not believe the rumor when contacting the stiflers who do not believe the rumor, which indicates the intensity of the rumor refutation information feedback mechanism.
Let
x =
X/
N,
y =
Y/
N, w =
W/
N,
z1 =
Z1/
N,
z2 =
Z2/
N, and
x,
y,
w,
z1,
z2, respectively, represent the population density of ignorants, spreaders, skeptics, stiflers who believe the rumor, and stiflers who do not believe the rumor in the rumor dissemination system. Then, Equation (1) can be expressed as:
Equation (2) should satisfy the constraint condition:
x + y + w + z1 + z2 = 1. To obtain the equilibrium point of Equation (2), the right side of Equation (2) should be equal to zero according to the definition of the equilibrium point of the differential equation, that is:
By solving Equation (3), we can obtain:
where
x*,
z1*,
z2* are positive values satisfying the constraint condition
x* +
z1* +
z2* = 1. According to Equation (2), the dissemination and developing process of the rumor spreading system has no concern with the independent variable, i.e., time parameter. Thus, the rumor spreading system is autonomous, and the rumor spreading process is deterministic. When a spreader appears in the system, the rumor spreading process starts and eventually reaches the equilibrium state. At the equilibrium point,
y = 0,
w = 0, namely, there are no spreaders and skeptics in the system. Therefore, the system finally reaches the rumor-free equilibrium state, corresponding to the rumor-free equilibrium point
E*= (
x*, 0, 0,
z1*,
z2*).
The system is in the initial equilibrium state when there is no spreader in the system, where
x* = 1,
z1* = 0,
z2* = 0. Let the equilibrium point corresponding to this equilibrium state be the initial equilibrium point E
0= (1,0,0,0,0). Subsequently, the Jacobian matrix of Equation (2) was established to analyze the stability of the initial equilibrium point
where
.
The Jacobian matrix of the system at the initial equilibrium point E
0 = (1,0,0,0,0) is:
The characteristic equation is:
In the above characteristic equation, there is a characteristic root (φ*α), which is a positive real part. Consequently, the system is unstable at the initial equilibrium point E0. When being disturbed at the initial equilibrium point, the system is far away from the equilibrium state until there is no spreader or skeptic in the system. In other words, the rumor spreading ends, and the system reaches a new equilibrium state. We can imply that the closed rumor propagation model XYWZ1Z2-C does not have critical point leading to rumor propagation. Provided that a spreader appears in the system, it will inevitably lead to the spread of rumors in a certain range; thus, yielding some negative effects.
2.2. Rumor Propagation Model XYWZ1Z2-O in an Open System
With the advent of OSNs, people are connected under one virtual society [
24]. On a practical level, the rumor propagation system has close contact with the outside world. Hence, the population in each group could have changed. Therefore, there was a stable immigration rate and emigration rate in the rumor propagation system [
6]. Then, an open system rumor spreading model XYWZ1Z2-O was established, as shown in
Figure 2.
The rumor spreading law of XYWZ1Z2-O can be expressed by mean-field equations as follows:
Equation (5) follow the parameter definition in Equation (2), and the new parameters are defined as follows:
Nt refers to the total population of the system at the t time step;
κ represents the immigration rate of the rumor propagation system, and
μ represents the emigration rate of the rumor propagation system. To simplify the rumor spreading model, we assumed that the immigration rate was equal to the emigration rate [
6,
7,
8,
13], namely,
κ =
μ. From Equation (5), the total population of the rumor propagation system did not change over time, that is
Nt =
N. Let
x = X/N,
y = Y/N,
w = W/N,
z1 = Z1/N, z2 = Z2/N, and
x, y, w, z1, z2, respectively, represent the population density of ignorants, spreaders, skeptics, stiflers who believe the rumor, and stiflers who do not believe the rumor in the rumor spreading system. Given that τ =
μt,
α0 =
α/
μ,
δ0 =
δ/
μ,
λ0 =
λ/
μ,
η0 =
η/
μ, then Equation (5) can be transformed into:
To attain the equilibrium point of the system, let the right side of Equation (6) be equal to zero, and Equation (7) is obtained:
From the first equation of Equation (7), we can obtain:
Inserting Equation (8) into the third equation of Equation (7), we can obtain:
Inserting Equations (8) and (9) into the fourth equation of Equation (7), we can obtain:
Inserting Equations (8)–(10) into the second equation of Equation (7), we can obtain:
Equation (11) has a constant zero solution, corresponding to the rumor-free propagation equilibrium point EF= (1, 0, 0, 0, 0). According to Equation (12), it was found that a > 0; when φ > μ/α, c < 0, a positive solution was added for Equation (11), corresponding to the rumor propagation equilibrium point EE= (xE, yE, wE, z1E, z2E). Given that R0= (φ*α)/μ, if R0 > 1, the system had a rumor-free equilibrium point and a rumor-endemic equilibrium point; if R0 ≤ 1, the system only had a rumor-free equilibrium point and did not have a rumor-endemic equilibrium point.
The rumor-free equilibrium point E
F took the same value as the initial equilibrium point E
0= (1,0,0,0,0). The stability of the system at the initial equilibrium point was analyzed below, and at the outset, the Jacobian matrix of Equation (6) was obtained:
where
.
After solving the Jacobian matrix of Equation (6) at the initial equilibrium point E
0,
J0 was obtained as follows:
Then, the characteristic equation was obtained:
.
It was simple to figure out that the characteristic roots of the characteristic equation were −1, −1, −1, (−1 − η0), (−1 + φ*α0), i.e., −1, −1, −1, (−1 − η/μ), (−1 + φ*α/μ). If φ > μ/α, i.e., R0 > 1, all characteristic roots had strictly negative real parts, and the system was asymptotically stable at the initial equilibrium point E0. Since a spreader appeared in the system, the system deviated from the equilibrium state and, subsequently, returned to the initial equilibrium state. If φ ≤ μ/α, i.e., R0 ≤ 1, the characteristic root had a positive real part; accordingly, the system was unstable at the initial equilibrium point E0. The system was far from the equilibrium state after the spreader appeared in the system. Therefore, if the proportion of ignorants without critical ability in the rumor spreading system was less than a certain value, the system could not reach the threshold point R0 of rumor spreading, and rumors could not continue to spread and yield continuous negative influence.
5. Discussion and Conclusions
5.1. Discussion about the Application
From the above analysis, we could find that the development process of rumor dis-semination could be divided into four stages. It would make a significant contribution to reducing the negative effect of rumor propagation. Accordingly, one strategy may be proposed to reduce the losses brought by rumor propagation combined with artificial intelligence technology. When an emergency event happens, rumors also appear. The authority should determine the values of the parameters in the rumor propagation model. Subsequently, the authority can obtain the estimated population of ignorants, spreaders, skeptics, stiflers who believe the rumor, and stiflers who do not believe the rumor in different stages. Additionally, the authority can also obtain the possible development of a rumor. Then, according to the number of views, comments, and comment content of rumor information on social media, such as Twitter and Weibo, the authority can also obtain the real population in five groups. Thus, the authority may evaluate the actual situations and take appropriate measures accordingly. The discussion above is only a general framework. There are still many details to be refined and much work to be performed in future.
5.2. Conclusions
In this paper, we established a rumor propagation model based on the rumor refutation mechanism. In this model, the roles of individuals in controlling rumor spreading and the individuals’ behavioral regularities of rumor spreading were fully considered in both the closed system and the open system. The equilibrium points in both the closed system and open system were discussed. The rumor spreading in the closed system was autonomous, and the spreading process was deterministic. While in the open system, the critical threshold existed in the open rumor propagation system. In addition to the theoretical analysis, several numerical simulations were performed in both the closed system and open system. We selected appropriate parameters for Equations (1) and (5) to conduct a simulation to verify the correctness of the theories analysis. Additionally, it was found that the development process of rumor dissemination could be divided into four stages: latent period, progressive period, intense period, and recession period. It was helpful for the authorities to take measures at the appropriate time. We also analyzed the rumor propagation law under the different influencing factors. Some measures, such as providing open and efficient information queries and exchange platforms by the authorities, could be taken to stop the propagation of rumors. Under the impact of population migration, rumor spreading could exceed the threshold, and the rumor spreading process in an open system presented a fluctuating development. The rumor did not disappear in this autonomous system. Moreover, the negative impact of rumor propagation could be minimized by population immigration and emigration. Furthermore, spreaders were required for the dynamic equilibrium state of rumor propagation and we should notice that the number of rumor spreaders increased with the increase in the population migration rate.
In this article, the law of rumor propagation based on the mean-field equation was primarily studied. However, the impact of the actual network structure on rumor dissemination was not considered. Moreover, the rumor spreading process was influenced by many factors inside and outside the system with a complicated influencing mechanism. When formulating the control measures of rumor propagation in emergent events, it is necessary to combine the characteristics of the actual rumor propagation system and take comprehensive optimization measures to effectively reduce the negative impact of rumor propagation and control rumor propagation.