Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

Abstract

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.

1. Introduction

Emergencies usually cause serious negative impacts on people’s work and life in a variety of ways. For example, the event itself may lead to financial loss or personal injuries, and also at the same time the rumor may begin to prevail and it leads to panic feelings or irrational behavior. Rumor is usually defined as the unconfirmed elaboration or annotation of the public interesting things, events, or issues that spread through various channels [1–3]. When emergencies occur, people often fail to reflect rumor’s seriousness and take active measures to deal with the related issues in time. Rumor especially the exaggerated rumor, as a product of emergencies, may bring helplessness and distress to the public, hamper the regulation of public opinions, and simultaneously destroy social harmony and stability. For instance, during the 2011 earthquake and Fukushima nuclear leak in Japan, because of the propagation of the rumors about the water pollution, many people in China blindly rush to buy salts, which greatly affects the regular social order [4]. Thus, how to effectively control and prevent rumor propagation in emergencies has become one of the most valuable research topics.

Accurate analysis of the dynamic behavior of rumor propagation is significant for controlling and preventing rumor propagation in emergencies. Using mathematical models to analyze the dynamic characteristics about rumor propagation has now become a compelling topic [5–9]. Rumor propagation is very similar to the diffusion of the virus; thus, most of the existing models of rumor propagation are derived from the models of infectious diseases. The classical models (i.e., the DK model) for rumor propagation were introduced by Daley and Kendall [10]. They mainly divided people into the following three classes: ignorants (those who cannot identify the rumors), spreaders (those who spread rumors), and stiflers (those who do not believe the rumors again). Thenceforth, most of the researchers applied or improved DK model to study rumor propagation. Maki and Thompson applied a mathematical model to study rumor propagation, and they mainly focused on the theoretical analysis [11]. Z.-L. Zhang and Z.-Q. Zhang [12] supposed that the emergency state and the spreading rate of rumors were continuous functions of time. Then, they established a rumor propagation model in emergencies based on three-molecule models. Their results showed that rumor spreading at certain rate had positive effects on situation stability and especially had delaying effects on the rapid proliferation of an emergency. Based on the analysis of [12], Zhao [13] considered the impact of authorities’ media and rumor propagation on the evolution of emergencies, and they presented a new rumor propagation model with the impact of authorities’ media. Huo [14] studied a rumor transmission model with constant immigration and incubation. Their analysis results meet the actual circumstances better and more externally, and they are easier to understand. Zhao and Wang [15], based on epidemic-like models, established a ISRW dynamical model considering the medium to describe the rumor propagation in emergencies. They discussed the stability of the rumor propagation model and suggested that the government should enhance the management of Internet. Recently, Zhang et al. [16] considered an 8-state ICSAR rumor propagation model with official rumor refutation. They believed that high spread probability and low block value could lead to rumor refutation. However, this model did not consider the impact of delays on rumor propagation.

In fact, delays always exist in rumor propagation. For example, when rumor susceptible people are infected by rumor spreading people, there is a time during which the infected people can have the ability to spread rumor. On the other hand, when rumor spreading people are educated by our government, there also exists a delayed time after which the rumor spreading people can really recover and will not believe the rumor. Therefore, delay is an important factor in rumor propagation.

Inspired by the abovementioned work, this paper deals with a novel delayed rumor propagation model with media report, which is of the following form: with initial conditions where is the density of the rumor susceptible people at time , is the density of the rumor infected people at time , is the density of the rumor recovered people at time , and the coefficients , , , , are positive constants. represents the natural growth rate of the rumor susceptible people, represents the impact of media report, represents the impact of our government on the rumor infected people, is the infected rate, and describes a rate that people remove from the system since they lose interest in rumors [17]. is a delay, which implies that when rumor infected people are educated by our government, there exists a delayed time after which the rumor spreading people can really recover. , , are the initial densities of the rumor susceptible people, the rumor infected people, and the rumor recovered people, respectively.

Accordingly, the remaining structure of this paper is arranged as follows. In Section 2, we are concerned with local stability and Hopf bifurcation of the equilibrium points for our model. In Section 3, the properties of Hopf bifurcation, formula determining the direction of Hopf bifurcation, and the stability of spatially homogeneous bifurcating periodic solutions are investigated. In Section 4, numerical simulations are presented to demonstrate our theoretical results. Finally, conclusions are drawn in Section 5.

2. Local Stability and Hopf Bifurcation

Because the first two equations in (1) are independent of , we can consider the following reduced model: with initial conditions Obviously, system (3) has the following two kinds of equilibria points:(i)The boundary equilibrium point .(iii)If holds, then there exists the positive equilibrium point

In this part, we will discuss the local stability and Hopf bifurcation of the boundary equilibrium point and the positive equilibrium point with the time delay as the bifurcation parameter.

For further discussion, we firstly make the following assumptions:,,.

2.1. Stability and Hopf Bifurcation of the Boundary Equilibrium Point

By a simple calculation, it is easy to obtain the characteristic equation of system (3) at the boundary equilibrium point as the following form: where .

Lemma 1. If holds, then the boundary equilibrium point of system (3) with is locally asymptotically stable.

Proof. Clearly, (6) always has a negative real root; namely, Therefore, we only consider the form of When , (8) can be rewritten as Obviously, under condition , Thus, the boundary equilibrium point is locally asymptotically stable as .

Now we discuss the effect of the delay on the stability of . Assume that is a root of (8). Then should satisfy the following equations: Taking square on both sides of the equations of (11) and summing them up, we obtain If and hold, then (12) has a positive real root , and Therefore, according to (11), it is easy to show that Furthermore, differentiating both sides of (8) with respect to , we have Based on the above analysis, we obtain the following result.

Theorem 2. If hold, then the boundary equilibrium point of system (3) is locally asymptotically stable when . Moreover, it undergoes a Hopf bifurcation when .

2.2. Stability and Hopf Bifurcation of the Positive Equilibrium Point

For the positive equilibrium point , the characteristic equation of system (3) is as follows

Lemma 3. If holds, then the positive equilibrium point of system (3) with is locally asymptotically stable.

Proof. When , we have Under condition , it is easy to obtain According to the Routh-Hurwitz criteria, all the roots of (17) have negative real parts. Therefore, when , the positive equilibrium point is locally asymptotically stable.

Next we discuss the effect of the delay on the stability of . Assume that is a root of (16). Then should satisfy the following equation: which implies where Let ; then Without loss of generality, define

Lemma 4. If holds, then (20) has a unique positive real root.

Proof. Under condition , a simple calculation shows that Thus, (22) has a unique positive real root; namely, Furthermore, we obtain This completes the proof.

Now, from (19), it is easy to show that

Lemma 5 (see [18]). Let be the root of (16) near satisfying , . Suppose that . Then is a pair of simple purely imaginary roots of (16). Moreover, the following transversality condition holds: Applying Lemma 3–Lemma 5, we have the following result.

Theorem 6. If and hold. Further suppose that satisfies. Then(i)the positive equilibrium point of system (3) is locally asymptotically stable when ;(ii)system (3) undergoes a Hopf bifurcation at when

3. Properties of Hopf Bifurcation

In this section, for the positive equilibrium point we will investigate the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions by using the theory of [19, 20].

Denote by and introduce the new parameter . Normalizing the delay by the time scaling , then system (3) can be rewritten as where for .

The linearized system of (28) at is Based on the discussion in Section 2, we can easily know that for the characteristic equation of (30) has a pair of simple purely imaginary eigenvalues .

Let . By the Riesz representation theorem, there exists a matrix function , , whose elements are of bounded variation such that

In fact, we can choose where , .

Let denote the infinitesimal generator of the semigroup induced by the solutions of (30) and let be the formal adjoint of under the bilinear pairing for , . Then and are a pair of adjoint operators. From the discussion in Section 2, we know that has a pair of simple purely imaginary eigenvalues and they are also eigenvalues of since and are a pair of adjoint operators. Let and be the center spaces, that is, the generalized eigenspaces of and associated with , respectively. Then is the adjoint space of and . In addition, some easy computations give the following result.

Lemma 7. Let Then is a basis of associated with and is a basis of associated with .
Let and , where for , and for , where

From (33), we can obtain and . Noting that Therefore, we can obtain

Now, we define and construct a new basis for by . Obviously, , the second order identity matrix. In addition, define and for , . Then the center space of linear equation (30) is given by , where and ; here denote the complementary subspace of .

Let be defined by where Then is the infinitesimal generator induced by the solution of (30) and (28) can be rewritten as the following operator differential equation:

Using the decomposition and (43), the solution of (46) can be rewritten as where and with . In particular, the solution of (28) on the center manifold is given by

Setting and noticing that , then (49) can be rewritten as where . Moreover, by [17], satisfies where Let According to (50) we have where

Let . Therefore, we obtain

Since , for appear in , we still need to compute them. It follows easily from (53) that

In addition, by [17], satisfies where with , .

Thus, from (51) and (58)–(60), we can obtain

Noticing that has only two eigenvalues with zero real parts, therefore, (59) has unique solution in given by

From (60), we know that, for ,

Therefore, for ,

By the definition of , we get from (62) that Noting that , ,

Using the definition of and combining (62) and (67), we get From we get From the above expression, we can see easily that By a similar way, we have

Similar to the above, we can obtain

Theorem 8. System (3) has the following Poincaré normal form: where so we can compute the following result: which determine the properties of bifurcating periodic solutions at the critical values ; that is, determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ; determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable), if ; and determines the period of the bifurcating periodic solutions: the periodic increase (decrease), if .

4. Numerical Solutions

Using MATLAB, numerical simulations are provided to substantiate the theoretical results established in the previous sections of this paper.

Example 1. Choose , , , , , , , and in system (1). A simple calculation shows that the boundary equilibrium point . Obviously, the conditions of Theorem 2 satisfy. It is easy to obtain that . According to Theorem 2, the boundary equilibrium point is locally asymptotically stable for , as shown in Figure 1. That is to say, at last there are not any rumor infected people and rumor recovered people except rumor susceptible people, and the rumor propagation is controlled effectively. Furthermore, from Figure 1 we can find that as increasing, system (1) takes more time to converge to the boundary equilibrium point . When , a spatially homogeneous periodic solution emerges from the boundary equilibrium point (see Figure 2). That is, the boundary equilibrium point is unstable, which seriously disrupts the normal social order.

(a)

(b)

(c)

Figure 1 

The boundary equilibrium point is locally asymptotically stable as . (a) The susceptible users; (b) the infected users; (c) the rational users.

(a)

(b)

(c)

Figure 2 

The periodic oscillation derived from Hopf bifurcation around the boundary equilibrium point occurs as . (a) The susceptible users; (b) the infected users; (c) the rational users.

Example 2. Consider system (1) with , , , , , , , and . It is easy to obtain that the positive equilibrium point of system (1) is . Obviously, the conditions of Theorem 6 satisfy. By a simple calculation, we can obtain that . According to Theorem 6, the positive equilibrium point is locally asymptotically stable for , as shown in Figure 3. That is to say, the densities of the rumor susceptible people, the rumor infected people, and the rumor recovered people are controlled in fixed values, which is beneficial in controlling the rumor propagation. When , a spatially homogeneous periodic solution emerges from the positive equilibrium point (see Figure 4), which can lead to social instability and disrupt social harmony.

(a)

(b)

(c)

Figure 3 

The positive equilibrium point is locally asymptotically stable as . (a) The susceptible users; (b) the infected users; (c) the rational users.

(a)

(b)

(c)

Figure 4 

The periodic oscillation derived from Hopf bifurcation around the positive equilibrium point occurs as . (a) The susceptible users; (b) the infected users; (c) the rational users.

5. Conclusion

In this paper, we introduce a delay into rumor propagation in an emergency-affected environment and establish a delayed SIR rumor propagation model. The theoretical analysis and numerical simulation reveal that the discrete delay is responsible for the stability switch of the rumor propagation model, and a Hopf bifurcation occurs as the delay increases through a certain threshold. Furthermore, when the system is stable, as increasing, the system will take more time to converge to the equilibrium state. In future, we will further discuss rumor spreads in space and establish a temporal-spatial rumor propagation model based on the reaction diffusion equation.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant no. 90924012.