 Research
 Open Access
Dynamic analysis of a spatial diffusion rumor propagation model with delay
 Chunru Li^{1, 2}Email author and
 Zujun Ma^{3}
https://doi.org/10.1186/s1366201506558
© Li and Ma 2015
 Received: 18 May 2015
 Accepted: 30 September 2015
 Published: 27 November 2015
Abstract
In this paper, we study the dynamics of a delayed reactiondiffusion rumor model with government control. By using the theory of partial functional differential equations, a Hopf bifurcation of the proposed system with delay as the bifurcation parameter is investigated. It reveals that the discrete time delay has a destabilizing effect in the rumor dynamics, and the phenomenon of Hopf bifurcation occurs as the delay increases through a certain threshold. Then by numerical simulations the impact of government control is explored. It is found that government control has strong effects on the dynamics of the model.
Keywords
 rumor
 spread
 delay
 diffusion
MSC
 34C23
 34D23
1 Introduction
Rumor is the kind of social phenomenon that a similar remark spreads on a large scale in a short time through chains of communication [1]. Compared with the way of rumor propagation by word of mouth in the old days, nowadays because of the appearance of the radio, television, newspapers, and mobile phone and so on, rumor appears and becomes widespread.
It is well known that the spreading of harmful rumors can deeply endanger a society. 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 [2]. In order to reduce and avoid the dangers of the rumor propagation in online social networks, it is necessary to adequately understand the dynamic characteristics of rumor propagation. Rumor propagation is very similar to the diffusion of a virus, thus, most of the existing models of rumor propagation are derived from the models of infectious diseases [3–11]. The most popular model for information or rumor spreading, introduced by Daley and Kendall [12, 13], see also [14, 15], is conceptually similar to the SIR. This is a susceptibleinfectiverecovered model for epidemiology. Agents are divided into three classes: ignorants, spreaders, and stiflers, i.e., those who have lost interest in diffusing the information or rumor. Their role is exactly the same as the susceptible, infective, and recovered agents of the SIR model, respectively. Epidemiological models have since been repeatedly used for describing information spread, such as topic flow in blog space, and word of mouth in product marketing.
The models mentioned above have concentrated only on the temporal dimension without diffusion. Recently, Wang et al. [16] proposed a diffusive logistic (DL) model with spatialtemporal diffusion terms to study the information propagation process in online social networks. The authors described the spatial distance by using a new concept: friendship hops, and abstractly divided the information diffusion process in online social networks into two separate processes: growth process and social process. In [17], Wang et al. further proposed a partial differential equation (PDE) based on a linear diffusive model to understand the information diffusion process over both temporal and spatial dimensions. Combined with the actual observations in the Digg data set, they proved the performance of the proposed linear diffusive model. To our knowledge, the study of a PDE rumor propagation model is still at the preliminary stage and there are many problems to be researched. Therefore, these spatialtemporal models will provide a new insight to research of the rumor propagation in online social networks.
It is worth noting that most works mentioned above on rumor propagation modeling assume that there is no time delay over rumor spreading. In fact, similarly to epidemic models [18, 19], as regards the rumor spreading process we should consider that there exists an incubation period before an influenced ignorant user has the ability to spread rumors. Consequently, delay needs to be considered.
In this paper, our objective is to propose a novel rumor propagation model with more realistic significance in theory and further analyze the dynamic characteristic of this model in mathematics.
The structure of this paper is arranged as follows. In Section 2, the modeling approach is described explicitly. In Section 3, we consider the existence of equilibrium points of system (1), which is studied. In Section 4, we study the local stability and the existence of a Hopf bifurcation through the study of associated characteristic equations. In Section 5, we prove the global asymptotical stability of the interior equilibrium. In Section 6, some numerical simulations are given to support our theoretical predictions. Finally, this paper ends with a brief conclusion.
2 The model
This section describes a delayed spatialtemporal rumor propagation model. Our goal is to create a realistic model which can provide wide insight into predicting and controlling rumor prevalence in online social networks.
 (i)
We consider the ignorant users and stifle users usually to have logistic growth with a carrying capacity as well as an intrinsic growth rate.
 (ii)
In online social networks, when an ignorant user is infected by spreading users, there is a spreading incubation period during which the infectious agents develop on networks, and it is only after that time that the infected user becomes himself infectious. Therefore, defining a delay for the spreading incubation period is more appropriate.
 (iii)
Usually, when a rumor spreading in online social networks the government will take effective actions to control and remove the spreading users.
3 Existence of equilibrium points
In this section, we will find all possible nonnegative equilibria.
 (1)
The trivial point \(E_{0} (0,0,0)^{T}\).
 (2)
The boundary equilibrium \(E_{1} (0,0,\frac{c\gamma}{d})^{T}\), as \(c>\gamma\), representing the state corresponding to the extinction of ignorants and spreaders.
 (3)
The boundary equilibrium \(E_{2} (\frac{a\gamma}{b},0,\frac{c\gamma }{d})^{T}\), as \(c>\gamma\) and \(a>\gamma\), representing the state corresponding to the extinction of the spreaders.
 (4)
The interior equilibrium \(E^{*} (S^{\ast},I^{\ast},R^{\ast})^{T}\).
For simplicity, we denote \(\Delta=A_{2}^{2}4A_{1}A_{3}\). The following results are obvious.
Lemma 3.1
 (a)
If \(\Delta>0\) and \(A_{3}<0\), then (4) has a unique positive root \(I^{*}=\frac{A_{2}+\sqrt{\Delta}}{2A_{1}}\).
 (b)
If \(\Delta=0\) and \(A_{2}<0\), then (4) has a unique positive root \(I^{*}=\frac{A_{2}}{2A_{1}}\).
 (c)
If \(\Delta>0\), \(A_{3}>0\), and \(A_{2}<0\), then (4) has two positive roots \(I^{*}=\frac{A_{2}\pm\sqrt{\Delta}}{2A_{1}}\).
As follows from Lemma 3.1, system (1) will have at least one positive steady state \(E^{*}(S^{*}, I^{*}, R^{*})\), where \(S^{*}=\frac{1}{\beta }(\gamma+\eta)(1+\alpha I^{*})\) and \(R^{*}=\frac{c\gamma+\sqrt{(c\gamma )^{2}+4d\eta I^{*}}}{2d}\).
4 Local stability and Hopf bifurcation
In this section, we will discuss the local stability and Hopf bifurcation of system (1) by analyzing the corresponding characteristic equations.
 (H_{1}):

\(a>\gamma\);
 (H_{2}):

\(c<\gamma\);
 (H_{3}):

\(\gamma+\eta\beta\alpha_{2}>0\).
By a simple calculation, we have the following: (12) and (13) both have a positive root if (H_{1}) holds, (14) has at least a positive one positive root, as \(\tau=0\). Therefore, we obtain the following results.
Theorem 4.1
If (H_{1}) holds, then the boundary equilibrium \(E_{0}\) and \(E_{1}\) are both unstable. As \(\tau=0\), \(E_{2}\) is unstable.
In the following part, we analyze the stability and Hopf bifurcation about the interior equilibrium \(E^{*} (S^{\ast},I^{\ast},R^{\ast})^{T}\).
It is obvious that \(\lambda=0\) is not a root of (15) \(\forall n\in N_{0}\triangleq\{0,1,2,\ldots\}\), as (H_{3}) holds.
Lemma 4.1
If (H_{2}) and (H_{3}) hold, then \(\lambda=0\) is not a root of (15) for \(\forall n \geq0\) and the interior equilibrium \(E^{*}\) of system (1) with \(\tau=0\) is locally asymptotically stable.
Proof
If (H_{2}) and (H_{3}) hold, then \(\lambda_{1}<0\), \(\lambda_{2}\) and \(\lambda_{3}\) have negative real parts. So, system (1) with \(\tau =0\) is locally asymptotically stable. □
 (H_{4}):

\(a+2bS^{*}+\beta\alpha_{1}+\gamma>0\),
 (H_{5}):

\(A_{0}B_{0}+A_{0}\beta\alpha_{2}\beta^{2}\alpha_{1}\alpha_{2}<0\),
 (H_{6}):

\(d_{2}A_{0}\beta^{2}\alpha_{1}\alpha_{2}>0\).
Theorem 4.2
If (H_{2})(H_{4}) hold, then all roots of (19) have negative real parts for all \(\tau\geq0\). Furthermore, the interior equilibrium \(E^{*}\) of system (1) is asymptotically stable for all \(\tau\geq0\).
Proof
If (H_{4}) holds, \(A_{n}B_{n}\beta^{2}\alpha_{1}\alpha_{2}+A_{n}\beta\alpha_{2}>0\). These results imply that (19) has no positive roots, and hence the characteristic equation (10) has no purely imaginary roots. Combine with Lemma 4.1, all roots of (10) have negative real parts as \(\tau\geq0\). This completes the proof. □
Remark 1
In Section 5, we will prove that when (H_{2})(H_{4}) hold, then the interior equilibrium is indeed globally asymptotically stable for any \(\tau\geq0\).
Lemma 4.2
If (H_{3}) and (H_{5}) hold, then (20) has a unique positive root, as \(n=0\).
Proof
If (H_{5}) holds, \(A_{0}B_{0}\beta^{2}\alpha_{1}\alpha_{2}+A_{0}\beta\alpha_{2}<0\). Therefore, according to Descartes’ rule of signs [21], (20) has a unique positive root. □
Lemma 4.3
Under the conditions of Lemma 4.2, if (H_{6}) hold, then (20) has no positive real root for any \(n\geq1\).
Proof
That is, (20) has no positive real root for any \(n\geq1\). □
Lemma 4.4
Proof
From the above analysis, we have the following theorem.
5 Global stability
In this section, we prove that when (H_{2})(H_{4}) hold, the interior equilibrium is indeed globally asymptotically stable. To achieve this, we utilize the upperlower solution method in [22, 23].
Lemma 5.1
(See [24])
Theorem 5.1
Assume that (H_{2})(H_{4}) hold, then for any initial value \((S_{0}(x,t), I_{0}(x,t), R_{0}(x,t))>(0,0,0)\), the corresponding nonnegative solution \((S(x,t), I(x,t), R(x,t))\) of system (1) uniformly converges to \(E_{3}(S^{*},I^{*},R^{*})\) as \(t\rightarrow+\infty\). That is, the positive constant equilibrium \(E_{3}(S^{*},I^{*},R^{*})\) is globally asymptotically stable.
Proof
6 Numerical simulation
In this section, we present numerical simulations of some examples to illustrate our theoretical results.
6.1 Stability of the positive steady state for all \(\tau\geq0\)
6.2 Stability and Hopf bifurcation of system (1)
Remark 2
6.3 Effect of the government adjustment power
6.4 The effect of the diffusion
7 Conclusion
In this paper, we introduced delay and diffusion into a rumor model. Through the theoretical analysis and numerical simulation we found that government adjustment power η can affect the system’s stability. These can be found in Section 6.3.
By using PDE stability theory, we take the delay τ as the bifurcation parameter to study the Hopf bifurcation of system (1). Theoretical analysis and numerical simulations show that the discrete delay is responsible for the stability switch of the model and a Hopf bifurcation occurs as the delays increase through a certain threshold (see Section 6.2). When (H_{2})(H_{4}) hold, the interior equilibrium is globally asymptotically stable.
In summary, our study contributes to rumor management in an emergent event by offering an interplay model between rumor spreading and government adjustment. According to the transmission of the rumor, the government should apply TV (the most popular and most believed medium in China) to announce the truth, and the population of the spreaders will be reduced immediately.
Declarations
Acknowledgements
The work is supported by National Natural Science Foundation of China under Grant 90924012. The authors also gratefully acknowledge the helpful comments and suggestions of the reviewers, which have improved the presentation.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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