Dynamical Behaviors of Rumor Spreading Model with Control Measures

and Applied Analysis 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4


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][2][3][4][5][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][10][11][12][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][15][16][17][18][19].Some are built on the basis of the random theory [20][21][22][23].
With rapid development of today's society, besides propagation by word of mouth, rumors also can be spread by public homepage, SMS, e-mails, 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 medium-to-individuals 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 The initial quantity of messages by spreader measures [28,29].In order to control the rumor spreading, we can focus on the credibility of the authorities' media [30][31][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.

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: (1)
Proof.(a) Define a Lyapunov function  =  +  0  ≥ 0. ( When  0 < 1, the Lyapunov function satisfies It is easy to know that L = 0 only hold when  = 0.As a result, the disease-free equilibrium point  0 is the only fixed point of the system.By applying the Lyapunov-LaSalle asymptotic stability theorem [36,37], the disease-free quilibrium point  0 is globally asymptotically stable.(b) The Jacobian matrix at the  0 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  0 > 1,( +  −  0 ) −  0 < 0; that is, there must exist a positive root.That means that  0 is unstable.
With regard to the positive equilibrium  * = ( * ,  * ,  * ,  * ), it should satisfy By calculating the equations, we have where The analysis about  1 ,  2 , and  3 is more complex and we list the result in Table 2.

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: (10) (10) What this paper mainly discusses is the effect of measures carried out by authority.At first, we can assume that the  Table 2: Equilibria and stability of system (1).

Cases
Conditions Positive equilibria No positive equilibrium  2 < 0,  1 > 0,  3 > 0, Δ = 0 Unique positive equilibrium  2 < 0,  1 > 0,  3 > 0, Δ > 0 Two positive equilibria   The submerged rate of messages by government (0) 0 Individual The initial quantity of messages by government 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.
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 () =   ,  ∈ N + .
Case 2 (() =   ,  ∈ N + ).In Case 2, the diseasefree equilibrium  0 = (/, 0, 0, 0, 0).Similarly, the basic reproduction number is  02 = ( −1 ) =  0 /( +  + ) +  0 /( +  + ), which does not relate to the parameters , .From the expression of  02 , we can know that  02 is the same as  0 .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.

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  01 < 1 or  0 < 1, 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,  01 is linear function of , , and . 01 is the concave function with the rest of parameters, where the influences of , , and  are greater on  01 .Observing the values of ordinate axis,  and  have the biggest influence on  01 ; 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  01 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  01 .From Figures 1(h), 1(i), and 1(j), it is easy to know that  has the biggest effect on  01 .The effect of  on  01 is the smallest.
When  01 < 1, 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  01 < 1, the influence of  is smaller.For ,  1 , and  2 , the influence of  is the biggest and is followed by  2 and  1 .
Case 2 (() = ).The basic reproduction number of the whole system is  0 =  0 /( +  + ) +  0 /( +  + ).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  1 .The effect of  is larger than  2 and .

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  01 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. 01 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.

Figure 1 :
Figure 1: The sensitivity of  01 in terms of parameters.

Table 1 :
Description of parameters in the system (1).