Dynamical Model about Rumor Spreading with Medium

. Rumor is a kind of social remark, that is untrue, and not be confirmed, and spreads on a large scale in a short time. Usually, it can induce a cloud of pressure, anxiety, and panic. Traditionally, it is propagated by word of mouth. Nowadays, with the emergence of the internet, rumors can be spread by instant messengers, emails, or publishing. With this new pattern of spreading, an ISRW dynamical model considering the medium as a subclass is established. Beside the dynamical analysis of the model, we mainly explore the mechanism of spreading of individuals-to-individuals and medium-to-individual. By numerical simulation, we find that if we want to control the rumor spreading, it will not only need to control the rate of change of the spreader subclass, but also need to control the change of the information about rumor in medium which has larger influence. Moreover, to control the effusion of rumor is more important than deleting existing information about rumor. On the one hand, government should enhance the management of internet. On the other hand, relevant legal institutions for punishing the rumor creator and spreader on internet who can be tracked should be established. Using this way, involved authorities can propose efficient measures to control the rumor spreading to keep the stabilization of society and development of economy.


Introduction
Rumor is a kind of social phenomenon that a remark spreads on a large scale in a short time through chains of communication and runs through the whole evolutionary history of mankind [1].Narrowly speaking, rumor is defined as untrue information, but has not been confirmed.Usually, it is dispersed by some people in order to achieve the specific purpose: slandering others, manufacturing momentum, diverting attention, causing panic, and so on [2][3][4].Most rumors induce panic psychology or economic loss in the accompanying unexpected events.Emergencies cause serious negative impacts on people's life in several ways: not only the event itself might lead to financial loss or personal injuries, but also the rumor might lead to panic feelings and irrational behavior [5].For example, the nuclear leakage in Japan caused an alt-buying frenzy in China.With the rumors spreading, this frenzy swept and caused social panic in just a few days and the preternatural rising of the salt price, which has a negative effect on society and economy [6].Recently, there appeared a hearsay that beef in Suzhou city was infected bacillus anthracis; but after a few days, the rumor cannot stand and is confirmed.
The spreading of rumor is in many ways similar to the spreading of epidemic infection, by the spreader or infectious to notify or infect the susceptible.After notified or infected, that the spreader can become the stifler is similar as the infectious can recover after a few time.Thus many epidemic models have been used to describe the spread of information in the form of rumors [6][7][8][9][10][11].A classical model for the spread of rumor was introduced by Daley and Kendall [12] who first proposed the basic DK model of rumor spreading.Maki and Thompson [13] focused on the analysis of the rumor spreading model, which was based on mathematical theory and developed the MK model.The DK and MK models have been used extensively for quantitative studies of rumor spreading [14][15][16].The literatures [5,17,18] give a detailed introduction for the models about rumor spreading.
Traditionally, rumors are propagated by word of mouth.In this propagation mode, the spreading speed is slower and can produce a small effect on society stability.Nowadays, with the emergence of the internet, rumors can spread rapidly by instant messengers, emails, or publishing bloggings that provide faster velocity of transmission and result in widespread of information [6].Under this case, it is more difficult to control the rumor spreading.The paper about infective vector can be seen in [19,20].With this new pattern of rumor spreading, we establish an  dynamical model including spreading between individuals and medium-toindividuals to describe more accurately the actual pattern of transmission, which has not been studied in previous papers.
Then the mechanism under spreading between individuals and by medium can be investigated by resorting to the model.Furthermore, we also give the main influence factors of transmission to government that can propose efficient measures to keep the stabilization of society and development of economy.

A Dynamical Model for Rumor Spreading
The populations at any time  we consider are denoted by (), which is classified into three subclasses: the susceptible or ignorant, spreader, and stiflers denoted by (), (), and (), respectively.Besides the rumor spreading from () to (), the susceptible can be notified by medium, such as website news, microblog, Baidu Post Bar, community sites, Renren, and mobile phone.The quantity of rumor in medium is denoted by ().The mathematical model to study the rate of change of the populations is given by system (5).Our assumptions on the dynamical transmission of rumor among humans are demonstrated in the following flowchart (Figure 1).When a susceptible contacts a spreader or sees rumor in medium, the susceptible can get to know about the rumor.Thus, the change over time in the size of the susceptible population () only consists of the output, infected by the rumor −(()()/) − ()().For the incidence rate under direct contact between individuals, we adopt the standard incidence rate.According to the indirect infection between the susceptible individual and the rumor in medium, the probability to be infected for a susceptible individual is proportional to the quantity of rumor in medium.So, the indirect transmission rate is ()().With regard to (), it is very difficult to determine the quantity in medium.In addition, for the susceptible, after notified rumor, they do not completely become the rumor, but can doubt its credibility and become a stifler.Thus, the susceptible notified rumor enters the spreader at a constant rate  ∈ [0,1] per unit time.Consequently, the change over time in the size of the susceptible dairy cattle population () is as follows: The interpretation for the change over time in the size of the spreader population () is as follows.Once infected, the individuals will enter the spreader compartment at a constant rate  ∈ [0, 1] per unit time.However, when two spreaders contact, both of them transmit the rumor, and the spreader can become bored and loses interest in the rumor, and consequently both become stiflers.In this case, the term  1  2 ()/ needs 2 times.During a unite time, the number of the spreader () also can realize truth with the development of the matter.When a spreader contacts a stifler   ( On the one hand, there are (1 − )(()()/) + (1 − )()() from the susceptible department to the stifler department.On the other hand, there are  1  2 ()/, (), and  2 ()()/ from the spreader department to the stifler class.Thus, the change over time in the size of the stifler population () is as follows: For the rumor in medium, they are discharged by the spreader, and the quantity discharged per unite time is assumed as ().However, they have the submerged rate of message ().The quantity of rumor in medium is close to the follows of a person in network, so its unit also can be  1.
individual.Thus, the change over time in the size of the () is as follows: Combining the above equations, the model we study in this paper to govern the temporal evolution of rumor during human population is presented in (5), whose parameters are described in Table (5)
Proof.In order to verify the stability of these points, the linear part of the vector field around them is given by the Jacobian at the equilibrium  * = ( * , 0,  * , 0) as follows: With regard to the above matrix, we know that one zero eigenvalue corresponds to the fact that the order of the dynamical system is two.The other zero eigenvalue is related to the stable center manifold that is the straight line  * + * =  * on the ( * ,  * ) plane.Thus, the stability of the equilibrium depends on the signal of the remainder eigenvalues.The corresponding characteristic polynomial can be written as Using  * +  * =  * and the Hurwitz criterion, we can get when 0 <  * <  1 =  * ( +  2 )/( * +  2  + ),  * is locally asymptotically stable.In fact, applying the twodimensional figures, we can know that when  * is always less than  1 .Here, we give figures about , , , and , seen in Figure 2.
From Figure 2, we can see when  ≤ 0.2, the red curve, representing  1 , is a constant 5000000, and always over the blue line,  * .At  = 0.2, the two curves are very closer, but the gap can be observed by observing the microdiagram round  = 0.2.When  > 0.2, the gap between them becomes larger.At  = 0.2, we have a look at the change of  * and  1 in terms of the parameter .It can be observed that, at  = 0, two curves are the closet, but the red line is still over the blue one.In addition, we also consider the situations of  and  at  = 0.2 and  = 0, which show that the situation at  = 0.1 is the same as  = 0.2, and the  has very small effect on  * and  1 when  and  are too small.This also illustrates that the effect of  and  is the biggest.The situations of other parameters are not given in here for their influences are very little.So, the equilibrium  * is locally asymptotically stable.
Furthermore, by observing the equations in system (5), we find that as  → +∞, () is monotone decreasing and () monotone increasing.The monotonicity of (), () is uncertain, but they are linear.From that, it is impossible to show limit cycle, from which the global stability of  * can be obtained.
We know the monotonicity of (), () is uncertain, depending of the parameter values, which have a certain meaning that will be interpreted later.At the end, the rumor will disappear by undergoing for a short time, and a portion of (0) =  * becomes (∞).Hence, the proportion of all spreader in the process of rumor spreading that had spread rumor to others or made statement about rumor in network should attract the attention of the public, which is called as the final scale of spreader, an important indicator of the severity of rumor spreading.Now, we give the expression for the final scale of spreader, denoted by (∞).

Theorem 2. The final scale of spreader is
Proof.In the initial stages, (0) ≪ (0), so we assume (0) = (0) = 0 and (0) ≈  * , and at the end of rumor spreading, (∞) = (∞) = 0. From the first equation in system (5), we have From the last equation in system (5), it can be obtained Combining (10) and (11) and boundary conditions applied at 0 and  → ∞, the following expression can be obtained: On the other hand, we can have directly another expression about the final scale from the last equation in system (5): Due to the complexity of equations in model ( 5), we cannot give the expression only concluding (∞).Comparing ( 8) with ( 9), Equation ( 8) is simpler and more convenient to use to assess parameters sensitivity.

The Interpretation of Model Parameter Values
In this section, we give the basis of parameter values in Table 1.
Having no real data during the rumor spreading, we assume their values according to individual experience.Firstly, we assume that the average number of acquaintances one person meets per day is  = 30 every day.When a spreader meets a susceptible, the probability that the spreader talks about the rumor and the susceptible is infected by the rumor is  1 = 2/3 every day.After influenced by the rumor, half of the susceptible believe this rumor and begin to spread it, but the other half become the stiflers, not to spread this rumor.Thus,  = 0.5.In other words, a third of people have no response, and a third of people begin to spread the rumor, and the rest, a third of people, become stiflers.So we get  =  1  = 10.When two spreaders contact, both of them discuss the rumor, and they lose interest in it for hearing it again and again.We assume the rate of losing interest is 0.5 every day, so  1 = 30 × 0.5 = 15.When a spreader contacts a stifler, the spreader will transmit the rumor, but the stifler tries to remove it.Assuming the probability that the spreader becomes a stifler is 0.5, so  1 = 30 × 0.5 = 15.As time goes on, the spreader could be aware of the falsity content of the rumor by obtaining involved knowledge.As a result, the spreader also becomes a stifler at a rate of 0.1.The most important and key point is disseminating of the rumor by the medium.Because the quantity of message in medium cannot be measured accurately, we can assume 1% of spreaders will disseminate the rumor into internet, and a person has 500 follower, then the rate that a spreader can disseminate the rumor through the medium is 5.Moreover, due to the update of message on internet, the rumor will be hidden after 12 hours.So the submerged rate of message is 2 every day.With regard to the initial value adopted in this paper, we consider the total number of people is 5000010, where there are 10 spreaders, and (0) = (0) = 0.

Result and Sensitivity Analysis
Taking no media message into account, we know that, when ()/ ≤ 0, () will directly decrease to 0, then the rumor cannot spread to others.When ()/ > 0, () will first increase to its maximum at  = , then decreases to 0. / = 0 gives the threshold to justify whether the rumor spread to others can be obtained, which can be visually seen in Figure 3.

Figure 1 :
Figure 1: Transfer diagram on the dynamical transmission of rumor among humans.

Figure 2 :
Figure 2: The comparison of  * with  1 : (a) in terms of , where  = 0, (b) in terms of , where  = 0.2, (c) in terms of , where  = 0.2 and  = 0, and (d) in terms of , where  = 0.2 and  = 0.The other parameter values and the initial values can be seen inTable 1, unless we say otherwise.