Stability and Bifurcation Analysis of Differential Equations and its Applications
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Dynamical Behaviors of Rumor Spreading Model with Control Measures
Abstract
Rumor has no basis in fact and flies around. And in general, it is propagated for a certain motivation, either for business, economy, or pleasure. It is found that the web does expose us to more rumor and increase the speed of the rumors spread. Corresponding to these new ways of spreading, the government should carry out some measures, such as issuing message by media, punishing the principal spreader, and enhancing management of the internet. In order to assess these measures, dynamical models without and with control measures are established. Firstly, for two models, equilibria and the basic reproduction number of models are discussed. More importantly, numerical simulation is implemented to assess control measures of rumor spread between individualstoindividuals and mediumtoindividuals. Finally, it is found that the amount of message released by government has the greatest influence on the rumor spread. The reliability of government and the cognizance ability of the public are more important. Besides that, monitoring the internet to prevent the spread of rumor is more important than deleting messages in media which already existed. Moreover, when the minority of people are punished, the control effect is obvious.
1. Introduction
As a typical social phenomenon, rumor has no basis in fact and flies around, especially when major public events happen and people do not have exact information and knowledge about the events; the rumor is dispersed by some people for achieving the specific purpose. It has been described in detail by some pieces of literature [1–6]. In the modern society, the rumor not only has not disappeared but also, with the development of the communication transmission modes, such as internet, telephone, and advanced information technology, spreads more quickly and the scope involved is much broader. Thus, the internet rumors become an important factor that influences the current social harmony and stability in emergencies and all kinds of crisis, and it is becoming the focus of the netizens and governments at all levels.
The classical models to study the spread of rumor were given by Daley and Kendall and Maki and Thomson [7, 8]. Since the dissemination process of rumor is similar to the spreading of infectious disease, epidemic models have usually been applied to investigate the spread of rumors [9–13]. The ignorant, the spreader, and the stifler are equivalent to the susceptible, the infected, and the recovered. Some models are established based on network [14–19]. Some are built on the basis of the random theory [20–23].
With rapid development of today's society, besides propagation by word of mouth, rumors also can be spread by public homepage, SMS, emails, or blogging that provide faster velocity of transmission [9, 24, 25]. The new type of transmission mode has been studied dynamically by [26]. It established an dynamical system including spreading between individuals and mediumtoindividuals to describe the actual pattern of transmission. With regard to the internet rumor, the government should share real information in a timely manner with the public to avoid the public hazard [27]. In 1953, the formula that describes the generation of rumor was proposed by Cross. , where is the importance of events, is ambiguity of events, and is the critical ability of the public. There are some models to assess the control measures [28, 29]. In order to control the rumor spreading, we can focus on the credibility of the authorities' media [30–32] and increasing the cognizance ability of the public. Besides, the government should give a certain punishment for the spreader. Therefore, this paper mainly assesses the effect of these measures.
2. A Dynamical System for Rumor Spreading
Without consideration of government measures, the dynamical system we establish will include the following four classes: the susceptible individual (), the spreader (), the stifler (), and the message in media (). Here, the bilinear incidence rate is considered. The interpretation of parameters can be seen in Table 1. The model we employ is as follows:

3. Dynamical Behaviors of System (1)
It is easy to know that . So, the positive invariant set is , , , , , , , , , . The diseasefree equilibrium is , where .
The basic reproduction number, that is, the expected number of secondary spreaders produced by a spreader in a completely ignorant population [33–35], can be calculated as follows:
The detailed calculation method can be seen in [35] and the behavior of (1) is discussed in Theorem 1.
Theorem 1. (a) When , the diseasefree equilibrium is globally asymptotically stable.
(b) When , the diseasefree equilibrium is unstable.
Proof. (a) Define a Lyapunov function
When , the Lyapunov function satisfies
It is easy to know that only hold when . As a result, the diseasefree equilibrium point is the only fixed point of the system. By applying the LyapunovLaSalle asymptotic stability theorem [36, 37], the diseasefree equilibrium point is globally asymptotically stable.
(b) The Jacobian matrix at the is the Jacobian
With regard to this matrix, the eigenvalues are the roots of the polynomial equation
It is easy to know that and are two of the eigenvalues. When ,; that is, there must exist a positive root. That means that is unstable.
With regard to the positive equilibrium , it should satisfy
By calculating the equations, we have where
The analysis about , and is more complex and we list the result in Table 2.

4. A Dynamical System for Rumor Spreading with Government Measures
Now, we add the measures of government to the system, especially issuing the actual message through the medium and punishment for the spreaders, which are reflected in and . Moreover, the ability of cognizance of the public is reflected in . The higher the cognizance ability the smaller the . These interpretations can be seen in Table 2. can be adopted as different term according to different situation. The system has the following form:
5. Dynamical Behavior of System (10)
What this paper mainly discusses is the effect of measures carried out by authority. At first, we can assume that the authority will release quantitative trustworthy message per time. So, the behavior of authority is independent of the rumor spreading; that is, . The parameters of system (10) are in Table 3.

Case 1 (). In this case, the diseasefree equilibrium , where ,,. Let us look at the basic reproduction number of the spreading of rumor. For the rumor spread, one has
So, we derive
Then, the basic reproduction number is = +. From expression of , we can see that the measures of government reduce the basic reproduction number.
Under some situations, once the rumor emerges, the government will issue the news to clarify the rumor and the message released by the authority. The more the spreaders there are the more the message the authority should issue. Thus, we can let , .
Case 2 (). In Case 2, the diseasefree equilibrium . Similarly, the basic reproduction number is , which does not relate to the parameters . From the expression of , we can know that is the same as . Thus, if the government adopts measure after the appearance of rumor during the early stage of the rumor spread, the measure cannot change the value of the basic reproduction number.
Case 3 (). In this case, the diseasefree equilibrium , where ,,. And the basic reproduction number is , which is similar to Case 1.
6. Sensitivity Analysis
This paper mainly discusses the effect of measures adopted by government. On the one hand, in the early stage, the sensitivity of the basic reproduction numbers about parameters that correspond to measures adopted by government should be discussed. On the other hand, when or , the sensitivity of the final scale of the spreader about parameters should be studied.
Now, we carry out the sensitivity analysis under different cases.
Case 1 ().
Consider
Observing Figure 1, is linear function of , , and . is the concave function with the rest of parameters, where the influences of , and are greater on . Observing the values of ordinate axis, and have the biggest influence on ; that is, the releasing amount of messages is the most important. With regard to and , the submerged rate of message issued by the government has bigger effect than . So, for message by spreader, we should control the distribution of message. Once the message is issued, the deleting of message has a small effect on controlling rumor spreading. For the government, in order to prevent the rumor spread, the quantity and the survival time of message are important factors. From and , the reliability of government and the cognizance ability of the public are equally important and is more sensitive with the reliability of government. With regard to , the concavity of curve is the biggest. When the minority of people are published, the effect has been big on . From Figures 1(h), 1(i), and 1(j), it is easy to know that has the biggest effect on . The effect of on is the smallest.
When , as time goes on, the rumor will eventually disappear. In this case, what we should focus on is the final scale of the spreader. Next, we discuss the influences of parameters on the final scale.
From Figures 2(a) and 2(b), we can see that has a bigger influence on the final scale than , which means that the reliability of government is more important. Comparing Figures 2(c) and 2(d), the change of the final scale caused by the is bigger than , which implies that the effect of release of message is more obvious than punishment from government. From Figures 2(e), 2(f), and 2(g), we know that when , the influence of is smaller. For , , and , the influence of is the biggest and is followed by and .
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Case 2 (). The basic reproduction number of the whole system is . We can know that the basic reproduction number does not change. For such reason, we should focus on the final scale of the spreader, which is showed in Figure 3.
When , then has a bigger influence on the final scale than , which is different from Case 1. Observing Figure 2(c), the effect of is smaller. The changes of the final scales of the spreader are very little under the changes of and . The effect of is larger than and .
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
7. Discussion
Applying the dynamical system, this paper describes the government measures by the parameters , , , , , , , and . More specifically, indicates the reliability of government, indicates the ability of cognizance of the public, indicates the punishment rate of the government, reflects the management strength of government for the internet, and show the amount of messages released by authority, is the transmission rate between humans directly, is the transmission rate of media to human, and is the transmission rate of the government to human by issuing presentation.
According to the above dynamical analysis and sensitivity analysis, we can know that and have the greatest influence on the rumor spread. The effects of and are almost big. When , the influence of is larger which means that the reliability of government is more important when the government issues message beforehand. When , the influence of is larger which means that the cognizance ability of the public is more important when the government releases the message according to the number of the spreaders. The effects of , , and explain that monitoring the internet to prevent the diffusion of rumor is more important than the deleting message in media that has appeared. Moreover, extending the survival time of government message is also necessary. The relationship of in terms of shows that when government punishes the minority of people, the effect is obvious. However, with the increase of , the effect is weakened. is a concave function with . From Figure 1(j), we can know that the influence of is larger than and .
In [26], an ISRW model was presented and its dynamical behaviors were well investigated. Reference [26] was mainly based on the Jacobian matrix and obtained the final size of rumor. However, this paper is based on spectral radius and focuses on the effects of different measures. The obtained results well will enrich the findings in rumor spreading.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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Copyright © 2014 XiaXia Zhao and JianZhong Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.