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- Open Access
Global stability of a SEIR rumor spreading model with demographics on scale-free networks
- Chen Wan^{1},
- Tao Li^{1}Email author and
- Zhicheng Sun^{1}
https://doi.org/10.1186/s13662-017-1315-y
© The Author(s) 2017
Received: 22 April 2017
Accepted: 8 August 2017
Published: 24 August 2017
Abstract
In this paper, a new SEIR (susceptible-exposed-infected-removed) rumor spreading model with demographics on scale-free networks is proposed and investigated. Then the basic reproductive number \(R_{0}\) and equilibria are obtained. The theoretical analysis indicates that the basic reproduction number \(R_{0}\) has no correlation with the degree-dependent immigration. The globally asymptotical stability of rumor-free equilibrium and the permanence of the rumor are proved in detail. By using a novel monotone iterative technique, we strictly prove the global attractivity of the rumor-prevailing equilibrium.
Keywords
- rumor spreading model
- scale-free networks
- hesitate mechanism
- demographics
- stability
1 Introduction
With the development of online social networks, rumor has propagated more quickly and widely, coming within people’s horizons [1–3]. Rumor propagation may have tremendous negative effects on human lives, such as reputation damage, social panic and so on [4–7]. In order to investigate the mechanism of rumor propagation and effectively control the rumor, lots of rumor spreading models have been studied and analyzed in detail. In 1965, Daley and Kendal first proposed the classical DK model to study the rumor propagation [5]. They divided the population into three disjoint categories, namely, those who who never heard the rumor, those knowing and spreading the rumor, and finally those knowing the rumor but never spreading it. From then on, most rumor propagation studies were based on the DK model [8–14].
In the early stages, most rumor spreading models were established on homogeneous networks [15–18]. However, it is well known that a significant characteristic of social networks is their scale-free property. In networks, the nodes stand for individuals and the contacts stand for various interactions among those individuals. Scale-free networks can be characterized by degree distribution which follows a power-law distribution \(P(k)\sim k^{ - \gamma}\) (\(2 < \gamma \le 3\)) [19]. Recently, some scholars have studied a variety of rumor spreading models and found that the heterogeneity of the underlying network had a major influence on the dynamic mechanism of rumor spreading [18, 20–26].
It is noteworthy that the influence of hesitation plays a crucial role in process of rumor spreading. Lately, there were a few researchers who have studied the effects of hesitation. For instance, Xia et al. [27] proposed a novel SEIR rumor spreading model with hesitating mechanism by adding a new exposed group (E) in the classical SIR model. Liu et al. [28] presented a SEIR rumor propagation model on the heterogeneous network. They calculated the basic reproduction number \(R_{0}\) by using the next generation method, and they found that the basic reproduction number \(R_{0}\) depends on the fluctuations of the degree distribution. However, in most of the research work mentioned above, the immigration and emigration are not considered when rumor breaks out. Although references [27, 28] proposed a SEIR model with hesitating mechanism, neither could serve as a strict proof of globally asymptotically stability of rumor-free equilibrium and the permanence of the rumor. In this paper, considering the immigration and emigration rate, we study and analyze a new SEIR model with hesitating mechanism on heterogeneous networks and comprehensively prove the globally asymptotical stability of rumor-free equilibrium and the permanence of rumor in detail.
The rest of this paper is organized as follows. In Section 2, we present a new SEIR spreading model with hesitating mechanism on scale-free networks. In Section 3, the basic reproduction number and the two equilibria of the proposed model are obtained. In Section 4, we analyze the globally asymptotic stability of equilibria. Finally, we conclude the paper in Section 5.
2 Modeling
Here, \(1 / i\) represents the probability that one of the infected neighbors of an individual, with degree i, will contact this individual at the present time step; \(P(i|k)\) is the probability that an individual of degree k is connected to an individual with degree i. In this paper, we focus on degree uncorrelated networks. Thus, \(P(i|k) = iP(i) / \langle k \rangle\), where \(\langle k\rangle = \sum_{i} iP(i)\) is the average degree of the network. For a general function \(f(k)\), it is defined as \(\langle f(k)\rangle = \sum_{i} f(i)P(i)\). The function \(\varphi (k)\) is the infectivity of an individual with degree k.
3 The basic reproduction number and equilibria
In this section, we reveal some properties of the solutions and obtain the equilibria of system (2.2).
Theorem 1
Define the basic reproduction number \(R_{0} = \frac{\beta h}{(\beta + \mu )(\delta + \mu ) - \beta h\delta m}\frac{ \langle \varphi (k)\lambda (k) \rangle}{ \langle k \rangle} \), then there always exists a rumor-free equilibrium \(E_{0}(\eta_{k},0,0,0)\). And if \(R_{0} > 1\), system (2.2) has a unique rumor-prevailing equilibrium \(E_{ +} (S_{k}^{\infty},E_{k}^{\infty},I_{k}^{\infty},R_{k}^{\infty} )\).
Proof
System (2.2) admits a unique rumor equilibrium \(E_{ +} (S_{k}^{\infty},E_{k}^{\infty},I_{k}^{\infty},R_{k}^{\infty} )\) satisfying equation (3.1) if and only if \(R_{0} > 1\). The proof is completed. □
Remark 1
The basic reproductive number \(R_{0}\) is obtained by equation (3.4), which depends on some model parameters and the fluctuations of the degree distribution. Interestingly, the basic reproductive number \(R_{0}\) has no correlation with the degree-dependent immigration \(b(k)\). According to the form of \(R_{0}\), we see that increase of the emigration rate μ will make \(R_{0}\) decrease. If \(b(k) = 0\) and \(\mu = 0\), then system (2.2) become the network-based SEIR model without demographics, and \(R_{0} = \frac{h}{\delta (1 - hm)}\frac{ \langle \varphi (k)\lambda (k) \rangle}{ \langle k \rangle} \), which is in consistence with reference [28].
4 Discussion
4.1 The stability of the rumor-free equilibrium
Theorem 2
The rumor-free equilibrium \(E_{0}\) of SEIR system (2.2) is locally asymptotically stable if \(R_{0} < 1\), and it is unstable if \(R_{0} > 1\).
Proof
Theorem 3
The rumor-free equilibrium \(E_{0}\) of SEIR system (2.2) is globally asymptotically stable if \(R_{0} < 1\).
Proof
When \(R_{0} < 1\), we can easily find that \(V(t) \le 0\) for all \(V(t) \ge 0\), and that \(V(t) = 0\) only if \(\Theta (t) = 0\), i.e., \(I_{k}(t) = 0\). Thus, by the LaSalle invariance principle [29], this implies the rumor-free equilibrium \(E_{0}\) of system (2.2) is globally attractive. Therefore, when \(R_{0} < 1\), the rumor-free equilibrium \(E_{0}\) of SEIR system (2.2) is globally asymptotically stable. The proof is completed. □
4.2 The global attractivity of the rumor-prevailing equilibrium
In this section, the permanent of rumor and the global attractivity of the rumor-prevailing equilibrium are discussed.
Theorem 4
When \(R_{0} > 1\), the rumor is permanent on the network, i.e., there exists a positive constant \(\varsigma > 0\), such that \(\lim \inf I(t)_{t \to \infty} = \lim \inf_{t \to \infty} \sum_{k} P(k)I_{k}(t) > \varsigma\).
Proof
In the following, we will explain that system (2.2) is uniformly persistent with respect to \((X_{0},\partial X_{0})\).
Clearly, X is positively and bounded with respect to system (2.2). Assume that \(\Theta (0) = \frac{1}{ \langle k \rangle} \sum_{k = 1} \frac{\varphi (k)}{\eta_{k}}P(k)I_{k}(0) > 0\), then we have \(I_{k}(0) > 0\) for some k. Thus, \(I(0) = \sum_{k = 1} P(k)I_{k}(0) > 0\). For \(I'(t) = \sum_{k} P(k)I_{k}'(t) \ge - (\delta + \mu )\sum_{k} P(k)I_{k}(t) = - (\delta + \mu )I(t)\), we have \(I(t) \ge I(0)e^{ - (\delta + \mu )t} > 0\). Therefore, \(X_{0}\) is also positively invariant. Furthermore, there exists a compact set B, in which all solutions of system (2.2) initiated in X ultimately enter and remain forever after. The compactness condition (C4.2) of Theorem 4.6 in reference [30] is easily verified for this set B.
Consequently, \(V(t) \to \infty\) as \(t \to \infty\), which apparently contradicts the boundedness of \(V(t)\). This completes the proof. □
Lemma 1
[32]
If \(a > 0\), \(b > 0\) and \(\frac{dx(t)}{dt} \ge b - ax\), when \(t \ge 0\) and \(x(0) \ge 0\), we can obtain \(\lim \inf_{t \to + \infty} x(t) \ge \frac{b}{a}\). If \(a > 0, b > 0\) and \(\frac{dx(t)}{dt} \le b - ax\), when \(t \ge 0\) and \(x(0) \ge 0\), we can obtain \(\lim \sup_{t \to + \infty} x(t) \le \frac{b}{a}\).
Next, by using a novel monotone iterative technique in reference [33], we discuss the global attractivity of the rumor-prevailing equilibrium.
Theorem 5
Suppose that \(( S_{k}(t),E_{k}(t),I_{k}(t),R_{k}(t) )\) is a solution of system (2.2), satisfying the initial condition equation (2.5). When \(R_{0} > 1\), then \(\lim_{t \to \infty} ( S_{k}(t),E_{k}(t),I_{k}(t),R_{k}(t) ) = ( S_{k}^{\infty},E_{k}^{\infty},I_{k}^{\infty},R_{k}^{\infty} )\), where \(( S_{k}^{\infty},E_{k}^{\infty},I_{k}^{\infty},R_{k}^{\infty} )\) is the unique positive rumor equilibrium of system (2.2) satisfying (3.1) for \(k = 1, 2, \ldots,n\).
Proof
Owing to the existence of sequential limits of equation (4.10), let \(\lim_{t \to \infty} \Omega_{k}^{(h)} = \Omega_{k}\), where \(\Omega_{k}^{(h)} \in \{ X_{k}^{(h)},Y_{k}^{(h)},Z_{k}^{(h)},x_{k}^{(h)},y_{k}^{(h)},z_{k}^{(h)},W_{k}^{(h)},w_{k}^{(h)} \}\) and \(\Omega_{k} \in \{ X_{k},Y_{k},Z_{k},x_{k},y_{k},z_{k},W_{k},w_{k} \}\).
Finally, by substituting \(w = W\) into equation (4.13), in view of equation (3.1) and equation (4.12), it is found that \(X_{k} = S_{k}^{\infty} \), \(Y_{k} = E_{k}^{\infty} \), \(Z_{k} = R_{k}^{\infty} \). This completes the proof. □
5 Conclusions
In this paper, a new SEIR rumor spreading model with demographics on scale-free networks is presented. Through the mean-field theory analysis, we obtained the basic reproduction number \(R_{0}\) and the equilibria. The basic reproduction number \(R_{0}\) determines the existence of the rumor-prevailing equilibrium, and it depends on the topology of the underlying networks and some model parameters. Interestingly, \(R_{0}\) bears no relation to the degree-dependent immigration \(b(k)\). When \(R_{0} < 1\), the rumor-free equilibrium \(E_{0}\) is globally asymptotically stable, i.e., the infected individuals will eventually disappear. When \(R_{0} > 1\), there exists a unique rumor-prevailing \(E_{ +} \), and the rumor is permanent, i.e., the infected individuals will persist and we have convergence to a uniquely prevailing equilibrium level. The study may provide a reliable tactic basis for preventing the rumor spreading.
Declarations
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grant 61672112 and the Project in Hubei province department of education under Grant B2016036.
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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