A rumor spreading model based on two propagation channels in social networks

Physica A 524 (2019) 342–353

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Physica A journal homepage: www.elsevier.com/locate/physa

A rumor spreading model based on two propagation channels in social networks ∗

Pingqi Jia a,b , Chao Wang a,b , Gaoyu Zhang c , , Jianfeng Ma a,b a

School of Cyber Engineering, Xidian University, Xi’an 710071, China Shanxi Key Laboratory of Network and System Security, Xidian University, Xi’an 710071, China c School of Information Management, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China b

highlights • • • •

An improved SIR model based on two propagation channels for rumors is proposed. Dynamics of the model including stability and bifurcation point are investigated. Global stability of an endemic equilibrium is obtained by geometric approach. Numerical simulations are presented to illustrate the theoretical results.

article

info

Article history: Received 13 July 2018 Available online 25 April 2019 Keywords: Social network Rumor spreading SIR epidemic model Stability analysis

a b s t r a c t Social networks have become one kind of the most important information media in the world, and the study of the phenomena of rumor propagation in social networks is helpful to understand the intrinsic laws of propagation behavior. We distinguish two propagation channels, that is, point to point propagation and group propagation, of rumor spreading on social networks. Thus we propose an improved SIR model and establish the corresponding mean-field equation. By using the differential dynamics method and the next generation matrix theory, the equilibrium point and the basic regeneration number R0 are calculated. Moreover, the geometric method is used to prove the asymptotic stability of the model at the equilibrium point and the bifurcation phenomenon at R0 =1. Finally, numerical simulations are carried out to verify the correctness of the theoretical results, and the influence of the rumor spreading mechanisms on the propagation process is analyzed. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The rapid development of network technologies and the wide application of mobile devices have promoted the growth of social networks, such as Facebook, Twitter, Wechat and Sina Weibo. Social networks have gone beyond the traditional media and become the most popular way to publish and disseminate information [1]. As a kind of special information, rumors emerge and spread all along the whole human history. In recent years, the rapid development of social network has provided new spreading ways and characteristics for rumors, and made it possible to spread faster and wider. Due to the similarity between rumor propagation and disease transmission, epidemic models are often used as the basis for the research of rumor propagation [2–8]. Zhao et al. [9–11] proposed a SIR model with forgetting mechanism and memory mechanism, and analyzed the application of the model in this new media age. The simulation result on ∗ Corresponding author. E-mail address: [email protected] (G. Zhang). https://doi.org/10.1016/j.physa.2019.04.163 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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LiveJournal showed that the forgetting rate has an effect on rumor propagation. Liu et al. [12] proposed an immune mechanism with stochastic delayed, established a SIR model with temporary immune mechanism. They analyzed the stability of the model, and concluded that the temporary immune mechanism would promote the spread of rumors. Wang et al. [13] presented a SIR model by considering the trust mechanism between the ignorant nodes and the spreader nodes. The introduction of trust mechanism not only reduces the influence of rumors, but also delays the spreading of rumors. Xia et al. [14] proposed a modified SEIR model with hesitating mechanism by considering the attractiveness and fuzziness of the content of rumors. It was verified that the spreading of rumors is positively related to the fuzziness of rumors. By analyzing the influence of independent propagators on the process of information propagation, Ma et al. [15] introduced the independent spreaders to the SIR model, and the simulation results demonstrated that the emergence of independent propagators is positively correlated with the propagation range of rumors. All of these researches have explored the spreading mechanisms of rumors on social networks and tried to give explanations to the phenomena of information spreading. But none of these researches considered the difference between the channels through which rumors spread. Different spreading channels lead to different transmission efficiencies, which affect the spreading speed and the spread range of rumors. In this work, we divide the spreading channels into two categories: group propagation and point-to-point propagation. The spreading mechanism of rumors is explored and an improved SIR model is established. Then the equilibrium points of the model and the basic reproduction number are calculated, and the asymptotic stabilities of the equilibrium points are proved by both theoretical analysis and numerical simulation. The organization of this paper is as follows. In Section 2, we establish an improved rumor spreading model, and formulate the mean-field equations with the different propagation channels of rumors taken into account. In Section 3, we calculate the equilibrium points and the basic reproduction number of the model, and prove the asymptotic stabilities of the equilibrium points theoretically. In Section 4, numerical simulations are performed to support the theoretical analysis. Finally, we make a brief conclusion in Section 5. 2. Model establishment Over social networks, users can communicate with their friends one-to-one, and they can also complete a one-to-many spreading of information in user groups, or dynamically disseminate information by updating their status [16]. Although the forms are different, the channels of information spreading on social networks are similar, and can be divided into two categories: group propagation and point-to-point propagation. Rumors can also be spread in both channels. SIR model is widely used and can reflect the most basic characteristics of rumor propagation. The SIR model divides the states of the user nodes into three groups: the susceptible state S, the infected state I and the recovered state R [17–21]. In the spreading process of a rumor, the susceptible nodes have not received the rumor, the infected nodes have received and spread the rumor, and the recovered nodes have received the rumor but never spread. In our model, if an infected node spreads a rumor by group propagation, all its neighbors will receive the rumor. Otherwise, if the infected node spreads a rumor by point-to-point propagation, only one of its neighbors will receive the rumor at a time. A node which is susceptible can be transformed into the infection state through either of these two channels. The spreading rules are described as follows. (1) After contacting with an infected node and receiving a rumor through group propagation, a susceptible node will become an infection node with a probability of λ1 or change into the recovered state with a probability of θ . Otherwise, the susceptible node gets the rumor through the point-to-point propagation, and then it will become an infection node with a probability of λ2 , or change into the recovered state with a probability of θ . (2) In each time step, an infected node chooses group propagation with a probability of γ1 or selects point-to-point propagation with a probability of γ2 to spread the rumor. Meanwhile, the containment mechanism [22] makes an infected node become a recovered node with a probability of α . (3) A recovered node changes into the susceptible state with a probability of β as a result of the temporary immune mechanism [23]. We denote the average degree of the network by ⟨k⟩, the birth rate by Λ, and the death rate by u. The SIR model for the rumor spreading process is shown in Fig. 1. The sum of the individuals in the social network is assumed to be constant during the spreading of a rumor. We describe the density of susceptible nodes, infected nodes and recovered nodes at moment t by I(t), S(t) and R(t). At the initial moment t = 0, there are at least one infected node and no recovered nodes in the network, and the remaining nodes are all susceptible nodes. If the sum of network nodes is much larger than one, we have

{

S(0) ≈ 1 I(0) ≈ 0 R(0) = 0.

(1)

The transition probabilities between states are all in the range of [0, 1], i.e., 0 ≤ Λ, u, λ1 , λ2 , α, γ1 , γ2 , β, θ ≤ 1.

(2)

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Fig. 1. Schematic diagram of the transformation of a User’s State.

According to the above rules, the mean-field equation of the model can be written as,

⎧• ⎪ ⎪ ⎨S(t) = Λ − (λ1 γ1 + λ2 γ2 /⟨k⟩ + θ (γ1 + γ2 /⟨k⟩))S(t)I(t) − uS(t) + β R(t) • I(t) = (λ1 γ1 + λ2 γ2 /⟨k⟩)S(t)I(t) − (u + α )I(t) ⎪ ⎪ ⎩• R(t) = α I(t) + θ (γ1 + γ2 /⟨k⟩)S(t)I(t) − (u + β )R(t).

(3)

The initial state is described by Eqs. (1) and (2).

3. Stability analysis 3.1. Positive and boundedness of the model Theorem 3.1. Since there are S(0), I(0), R(0) ≥ 0, the solution (S(t), I(t), R(t)) of the model is non-negative for t ≥ 0. Proof. Assume t1 = sup{t ≥ 0: S ≥ 0, I ≥ 0, R ≥ 0 ∈ [0, t ]}, then we have t1 ≥ 0. It follows from the first equation of (3) as ds dt

= Λ − (m1 + m2 )I(t) · S(t) − uS(t) + β R(t) ≥ Λ − (m1 + m2 )I(t) · S(t).

(4)

It can be written as ds dt

+ (m1 + m2 )I(t) · S(t) ≥ Λ,

(5)

where

{

m1 = λ1 γ1 + λ2 γ2 /⟨k⟩ m2 = θ (γ1 + γ2 /⟨k⟩).

(6)

Now integrating both sides of the above inequality and using the theory of differential inequality, we get t

∫ S(t1 ) exp{

t1

∫ (m1 + m2 )I(u)du} − S(0) ≥

t

∫

Λ(m1 + m2 )I(u)du}dx.

exp{

0

0

(7)

0

So that,

∫ t S(t1 ) ≥ S(0) exp{ Λ(m1 + m2 )Idu}−1 + 0 ∫ t1 ∫ t ∫ t Λ{ Λ(m1 + m2 )Idu}dx · exp{ Λ(m1 + m2 )Idu}−1 ≥ 0. 0

0

(8)

0

Hence, S(t) is non-negative for t ≥ 0. In a similar way it can be show that I(t) ≥ 0, R(t) ≥ 0 for t ≥ 0. Theorem 3.2. The solutions of the model are uniformly bounded. Proof. Suppose N(t) = S(t) + I(t) + R(t) holds for t ≥ 0, then we get dN dt

=

dS dt

+

dI dt

+

dR dt

= Λ − uN(t).

(9)

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It is easy to see that N(t) =

Λ u

+ N(0)e−ut .

(10)

When t tends to positive infinity, we obtain N = Λ . So all solutions of the model are confined in the region u R = {(S(t), I(t), R(t)) ∈ R3+ ∪ {0}: N(t) = Λ }. u As from Theorems 3.1 and 3.2, we can see that all solutions of the model are positive and uniformly bounded, and the considered region for the model is

Ω = {(S(t), I(t), R(t)) ∈ R3+ ∪ {0}: S(t) + I(t) + R(t) ≤

Λ u

, t ≥ 0}.

(11)

3.2. Equilibrium points When the time-dependent ratios of S(t), I(t), R(t) become constant, the system reaches the equilibrium state which can be describe as,

⎧• ⎪ ⎪ ⎨S(t) = Λ − (λ1 γ1 + λ2 γ2 /⟨k⟩ + θ (γ1 + γ2 /⟨k⟩))S(t)I(t) − uS(t) + β R(t) = 0 • I(t) = (λ1 γ1 + λ2 γ2 /⟨k⟩)S(t)I(t) − (u + α )I(t) = 0 ⎪ ⎪ ⎩• R(t) = α I(t) + θ (γ1 + γ2 /⟨k⟩)S(t)I(t) − (u + β )R(t) = 0. From Eq. (12) we can get the equilibrium points E0 = (S0 , 0, 0), where S0 =

Λ u

(12)

ˆ where , and E1 = (Sˆ , Iˆ, R),

⎧ u+α ⎪ ⎪ Sˆ = ⎪ ⎪ m1 ⎪ { }−1 ⎪ ⎪ ⎨ − Λ + u · Sˆ α + m2 · Sˆ m2 · Sˆ + u + α Iˆ = · − β α+β β ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ⎪ ⎪Rˆ = α + m2 · S · Iˆ. ⎩ u+β

(13)

So the system has two equilibrium points. When the system reaches the rumor-free equilibrium point (DEF) E0 = (S0 , 0, 0), the rumor disappears in the system and all the individuals are susceptible. While at the rumor endemic ˆ the rumor is disseminated stably and the ratios of the three types of individuals are equilibrium point E1 = (Sˆ , Iˆ, R), kept at a dynamic equilibrium. Since all system parameters are non-negative, the equilibrium point E0 always exists, and the other equilibrium point E1 is accessible under a specific set of conditions. 3.3. Basic reproduction number In general, the basic reproduction number R0 can be defined as ‘‘the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual’’ [24]. In our model, we calculate R0 by the next generation matrix method [25]. The model is written as follows dx dt

= φ (x) − ψ (x),

(14)

where x = (I(t), R(t), S(t))′ and

( φ (x) =

m1 · Λ 0 , 0

( ψ (x) =

)

(15)

(u + α ) · I −α · I − m2 · S · I + (u + β ) · R . −Λ + (m1 + m2 ) · S · I + uS − β R

)

(16)

, 0, 0) are given by The Jacobian matrices of φ and ψ evaluated at the rumor-free equilibrium E0 = ( Λ u

( ) ⎧ F 0 ⎪ ⎪ ⎨J(φ |E0 ) = 0 0 ( ) ⎪ ⎪ ⎩J(ψ |E0 ) = V 0 J1

J2

(17)

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where

⎧ m1 · Λ ⎪ ⎪ ⎪ F = ⎪ ⎪ u ⎪ ⎪ ⎪ ⎪ ⎪ V =u+α ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ m2 · Λ ⎪ ⎨ ⎜ −α + u ⎟ J1 = ⎝ ⎠ ⎪ (m1 + m2 ) · Λ ⎪ ⎪ ⎪ ⎪ ⎪ u ⎞ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪J2 = ⎝u + β 0⎠ . ⎪ ⎪ ⎩ −β u

(18)

Hence, R0 of the model defined by the spectral radius (FV −1 ) is given by R0 =

m1 · Λ

(19)

u · (u + α )

When the system reaches stability, the rumor can be disseminated stably and continuously if R0 > 1 holds, while rumor disappears if R0 < 1, and R0 = 1 is a critical condition for rumor spreading. 3.4. Local stability Theorem 3.3. If R0 < 1, the DEF of the system is local asymptotically stable, otherwise unstable for R0 > 1. Proof. At E0 the Jacobian matrix J(E0 ) of the system is given by

( J(E0 ) =

−u 0 0

−(m1 + m2 ) · S0 m1 · S0 − (u + α ) m2 · S0

β

)

0

−(u + β )

.

(20)

The characteristics equation associated to J(E0 ) can be given as

|λE − J(E0 )| = (λ + u)(λ + u + β )(λ + u + α − m1 · S0 ) = 0.

(21)

According to the Routh–Hurwitz stability criterion [26], if R0 < 1, all the eigenvalues of the matrix J(E0 ) are negative, so E0 is local asymptotically stable. If R0 > 1, the matrix J(E0 ) has a positive eigenvalue, so E0 is unstable. Theorem 3.4. The rumor endemic equilibrium E1 of the system is locally asymptotically stable if R0 > 1. Proof. The Jacobian matrix of the model at E1 is given by

⎛ −u − (m1 + m2 )Iˆ J(E1 ) = ⎝ m1 · Iˆ m2 · Iˆ

−(m1 + m2 ) · Sˆ

β

⎞ ⎠.

0

0

α + m2 · Sˆ

−(u + β )

(22)

The characteristics equation of J(E1 ) is

⏐ ⏐λ + u + (m1 + m2 ) · Iˆ (m1 + m2 ) · Sˆ ⏐ |λE − J(E1 )| = ⏐⏐ −m1 · Iˆ λ ⏐ −m2 · Iˆ −α − m2 · Sˆ = λ3 + a1 λ2 + a2 λ + a3 = 0

⏐ ⏐ ⏐ ⏐, 0 ⏐ λ + u + β⏐ −β

(23)

where

⎧ ⎨a1 = 2 · u + β + m1 · Iˆ + m2 · Iˆ ˆ . a = m21 · Iˆ · Sˆ + m1 · m2 · Iˆ · Sˆ + m2 · u · Iˆ + (u + β ) · (u + m1 · I) ⎩ 2 ˆ ˆ ˆ ˆ ˆ m3 = m2 · I · S · (u + β ) − m1 · I · α · β + m1 · m2 · u · I · S

(24)

When R0 > 1, Iˆ > 0 holds, we obtain m1 , m2 , m3 > 0. According to the Vieta theorem, we can get

{ |J(E1 )| = λ1 · λ2 · λ3 = −m1 < 0 tr(E1 ) = λ1 + λ2 + λ3 = −m3 < 0 tr(E1 ) = λ1 · λ2 + λ2 · λ3 + λ3 · λ2 = m2 > 0.

(25)

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Assume λ1 , λ2 > 0, λ3 < 0, then we get λ1 + λ2 < −λ3 . So that

−(λ1 λ3 + λ2 λ3 ) > λ1 λ2 ⇒ λ1 λ2 + λ1 λ3 + λ2 λ3 < 0.

(26)

It contradicts what we already know, so the hypothesis does not hold, but λ1 , λ2 , λ3 < 0 holds, or all the eigenvalues of matrix J(E0 ) are negative. According to the Routh–Hurwitz stability criterion, E1 is locally asymptotically stable. 3.5. Bifurcation When R0 < 1, there is only one equilibrium E0 and it is locally asymptotically stable. When R0 > 1, there are two equilibriums E0 and E1 . The DEF E0 is unstable and the rumor endemic equilibrium E1 is locally asymptotically stable. So we have the theorem as follows. Theorem 3.5. The proposed system has a transcritical bifurcation at R0 = 1. m ·Λ

Proof. R0 = 1 ⇔ u·(u1+α ) = 1. It can be seen from Theorem 3.3 that when R0 = 1, the eigenvalues of matrix J(E0 ) are λ1 = −u, λ1 = −u − β, λ3 = 0. That is, λ1 and λ2 are negative, λ3 is zero. So when R0 = 1, E0 is a non-hyperbolic equilibrium point. Now denote by w ⃗ = (w1 , w2 , w3 )T a right eigenvector associated with the zero eigenvalue λ3 = 0, and it follows J(E0 ) · w ⃗ = 0, that is

{ −uw1 − (m1 + m2 ) · S0 · w2 + β2 · w3 = 0 (m1 · S0 − u − α ) · w2 = 0 m2 · S0 · w2 − (u + β ) · w3 = 0.

(27)

So

w ⃗ =

(

β − (m1 + m2 ) ·

u+β m2

u+β

,

u

m2 · S0

)T ,

1

.

(28)

Furthermore, the left eigenvector η ⃗ = (η1 , η2 , η3 ) satisfies η⃗ · J(E0 ) = 0, that is,

{ − uη 1 = 0 βη1 − (u + β ) · η3 = 0 −(m1 + m2 ) · S0 · η1 − (m1 · S0 − u − α ) · η2 + m2 · S0 · η3 = 0.

(29)

So

( η⃗ = 0, Denote m =

)T

1,

0

∑3

k,i,j=1

.

(30)

fk ηk wi wj ∂w∂ ·∂w (P0 , θ ∗ ) and n = 2

i

m = η2 · m1 · w1 · w2 = m1 ·

n = η2 · w2 · S0 =

u+β m2 · S0

j

u+β m2 · S0

·

β−

m1 m2

∑3

k,i=1

fk ηk wi ∂w∂ ·∂θ (P0 , θ ∗ ), then there are 2 i

· (u + β ) − (u + β ) u

< 0,

· S0 > 0.

(31)

(32)

By applying Hopf bifurcation theorem [27], we have verified that the system exhibits a transcritical bifurcation at R0 = 1. 3.6. Global stability When R0 < 1 holds, the rumor-free equilibrium point E0 is globally asymptotically stable. When R0 > 1 and u > ω∗ , the rumor endemic equilibrium point E1 is globally asymptotically stable in int(Ω ). Theorem 3.6. The rumor-free equilibrium point E0 of the system is globally asymptotically stable when R0 < 1. Proof. Let w = (S(t)), v = (I(t), R(t)), E0 = (w0 , 0), where w0 = (S0 ) = dw dt dw dt

=

dS dt

Λ u

, then we have

= F (w, v ) = Λ − (m1 + m2 ) · S(t) · I(t) − u · S(t) + β · R(t).

(33)

When S(t) is equal to S0 , we can obtain F (w, 0) = 0 and ddtw = Λ − u · w . As t tends to positive infinity, there are → 0, w → Λu , w → w0 . Hence, w = w0 is globally asymptotically stable.

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Now G(w, v ) = B

( B=

(

I(t) R(t)

)

ˆ , v ) holds, where − G(u

m1 · S0 − (u + α ) α + m2 · S0

)

0

−(u + β )

,

(34)

and

ˆ , v) = G(u

(

)

m1 I(S0 − S(t)) . m2 I(S0 − S(t))

(35)

ˆ , v ) ≥ 0. Hence, by applying Lemma 4.3 of Ref. [28], the rumor-free equilibrium point Clearly, B is an M-matrix and G(u E0 is globally asymptotically stable if R0 < 1. Lemma 1. When R0 > 1, the system is uniformly persistent [28]. Proof. Taking into account that the rumor-free equilibrium E0 is unstable if R0 > 1 and E0 ∈ ∂ Ω , we can find that the system is uniformly persistent for R0 > 1 according to Theorem 4.3 of Ref. [28]. The result implies the existence of a compact absorbing set in int(Ω ) and, consequently, the geometric approach [29] can be used. Thus, another theorem follows. Theorem 3.7. The unique rumor endemic equilibrium point E1 of the system is globally asymptotically stable in int(Ω ), when R0 > 1 and u > w ∗ . Proof. The second additive compound matrix J [2] (E1 ) is given by

⎛ −u − (m1 + m2 )Iˆ J [2] (E1 ) = ⎝ α + m2 Sˆ −m2 Iˆ

As Q = Q (S(t), I(t), R(t)) = diag( I(t) , S(t)

Qg =

dQ dx

= diag(

˙ S(t) I(t)

−

⎞ −β −(m1 + m2 )Sˆ ⎠ . −(u + β )

0 −2u − β − (m1 + m2 )Iˆ m1 Iˆ

S(t) I 2 (t)

S(t) I(t)

˙ , · I(t)

(36)

, S(t) ), there is I(t)

˙ S(t) I(t)

−

S(t) I 2 (t)

˙ , · I(t)

˙ S(t) I(t)

−

S(t) I 2 (t)

˙ . · I(t))

(37)

It is easy to check that Qg · Q −1 = diag(

˙ S(t) I(t)

−

˙ ˙ I(t) S(t) ,

I(t) I(t)

−

˙ ˙ I(t) S(t) ,

I(t) I(t)

−

˙ I(t) I(t)

).

(38)

And consequently QJ [2] Q −1 = J [2] .

(39)

Let B = Qg Q

−1

[2]

+ QJ Q

−1

( =

B11 B21

B12 B22

)

.

(40)

We can calculate the element of the matrix as

⎧ ˙ ˙ ⎪ I(t) S(t) ⎪ ⎪ − − u − (m1 + m2 ) · Iˆ B11 = ⎪ ⎪ ⎪ S(t) I(t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B12 = (0, −β) ⎪ ⎪ ⎪ ⎨ ( )T B21 = α + m2 Sˆ , −m2 · Iˆ ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ˙ ˙ S(t) I(t) ⎪ ⎪ ⎪ ⎜ − − 2 · u − β − (m1 + m2 ) · Iˆ ⎪ ⎪ ⎜ I(t) ⎪ B22 = ⎜ S(t) ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩ m1 · Iˆ

(41)

⎞ −(m1 + m2 ) · Sˆ ˙ S(t) S(t)

−

˙ I(t) I(t)

− (u + β )

⎟ ⎟ ⎟ ⎠

Now consider the norm in R3 as ∥(x, y, z)∥ = max {∥x∥ , ∥y∥ + ∥z ∥}, where (x, y, z) denotes the vector in R3 , and denote the Lozinskii measure [30] with respect to this norm as γ , then we have

γ (B) ≤ sup {q1 , q2 } ≡ sup {γ1 (B11 ) + ∥B12 ∥1 , γ1 (B22 ) + ∥B21 ∥1 } ,

(42)

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where ∥B12 ∥1 , ∥B21 ∥1 are matrix norms with respect to the L1 vector norm, and γ1 denotes the Lozinskii measure with respect to the L1 vector norm. In Eq. (42), there are

⎧ ˙ ˙ I(t) S(t) ⎪ ⎪ ⎪ − − u − (m1 + m2 ) · Iˆ γ1 (B11 ) = ⎪ ⎪ S(t) I(t) ⎪ ⎪ { } ⎨ ˙ ˙ S(t) I(t) γ1 (B22 ) = − − (u + β ) + max −u − m2 Iˆ, −(m1 + m2 )Sˆ . S(t) I(t) ⎪ ⎪ ⎪ ∥ { ∥ = max 0 , B ⎪ 12 1 ⎪ { |−β|} = β } ⎪ ⎪ ⎩∥B21 ∥1 = max α + m2 Sˆ , m2 Iˆ

(43)

Then,

⎧ ˙ S(t) ⎪ ⎪ ⎪ q1 = − u − (m1 + m2 ) · Iˆ + β ⎪ ⎪ S(t) ⎪ ⎨ { } { } ˙ ˙ S(t) I(t) q2 = − − (u + β ) + max −u − m2 Iˆ, −(m1 + m2 )Sˆ + max α + m2 Sˆ , m2 Iˆ . ⎪ S(t) I(t) ⎪ ⎪ ⏐ ⏐ ⎪ ˙ S(t) ⎪ ⏐ ⏐ ⎪ ⎩ = − (2u + β ) + ⏐α + m2 Sˆ − m2 Iˆ⏐

(44)

γ (B) ≤ sup {q1 , q2 } ⏐} ⏐ { ˙ S(t) ⏐ ⏐ = − u + max β − (m1 + m2 ) · Iˆ, −(u + β ) + ⏐α + m2 Sˆ − m2 Iˆ⏐ .

(45)

S(t)

So

S(t)

⏐ ⏐

{

⏐} ⏐

Now denote w ∗ = max β − (m1 + m2 ) · Iˆ, −(u + β ) + ⏐α + m2 Sˆ − m2 Iˆ⏐ , we can get 1 t

t

∫

γ (B)ds < 0

1 t

log

S(t) S(0)

− (u − w∗ ).

(46)

∫t

That is limt →∞ sup(S(0),I(0),R(0))∈int(Ω ) 1t 0 γ (B)ds ≤ 0 if u > w ∗ . Hence, when R0 > 1 and u > w ∗ , the rumor endemic equilibrium E1 is globally asymptotically stable in int(Ω ). 4. Numerical simulation results In this section, we use numerical simulations to analyze the effects of rumor spreading mechanisms and verify the correctness of the above theoretical analysis. The initial conditions are given by I(0) = 0.1, S(0) = 0.9, R(0) = 0. Because that the variation period of user numbers is much longer than the rumor spreading cycle, the total number of users remains approximately constant. So we set Λ = u. Figs. 2 and 3 illustrate the change of ratios of various individuals with time at different values of R0 . For R0 valued 0.62 and 0.91 in Fig. 2(a) and (b), respectively, the asymptotically stable points of the system are both (1,0,0). In both cases S(t) approach unity, and I(t) and R(t) approach zero. But with the increase of R0 , the maximum I(t) becomes higher, and the time that the network arrives at the stable state becomes longer. The result in Fig. 2 shows that when R0 <1, a rumor will disappear in a stable social network. Fig. 4 shows that the system exhibits a bifurcation at R0 = 1 and it is consistent with the proof given above. When R0 > 1, with the increase of R0 , I(t) at the stable state increases, that is, I(t) at the rumor endemic equilibrium point increases gradually and the increasing rate decreases gradually. Fig. 5 shows that the change of I(t) with different values of β . Fig. 5(a) shows that β has weak effect on the maximum I(t) and the network convergence time when R0 < 1. In Fig. 5(b), we find that β has great effect on the maximum I(t), but weak effect on the network convergence time. Meanwhile, I(t) is proportional to β when R0 > 1 and β > 0. But when β = 0, there is no rumor endemic equilibrium point. So β keeps the existence of the rumor endemic equilibrium point. And when the system is asymptotically stable at rumor endemic equilibrium point, I(t) will increase with β . Fig. 6 shows that the change of I(t) with time at different values of θ . Fig. 6(a) shows that θ has weak effect on the maximum I(t) and the network convergence time when R0 < 1. In Fig. 6(b), we find that θ has great effect on the maximum I(t) and I(t) at the stable state. But it has weak effect on the network convergence time. When θ is bigger, the peak I(t) is higher and I(t) at the stable state is higher. Fig. 7 shows that the change of I(t) with time at different γ1 or γ2 . Fig. 7(a) shows that with the increase of γ1 , the network changes from the stable state of the rumor-free equilibrium to the stable state of the rumor endemic equilibrium, the maximum I(t) and the network convergence time also increase. Fig. 7(b) shows that with the increase of γ2 , the system changes from the stable state of the rumor-free equilibrium to the stable state of the rumor endemic equilibrium, the maximum I(t) and the network convergence time also increase.

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Fig. 2. Dynamic of the model when R0 <1.

Fig. 3. Dynamic of the model when R0 > 1.

Fig. 4. System branches at R0 = 1.

Fig. 8 shows that the change of I(t) with γ1 and γ2 when the model reaches stable states. It is demonstrated that the group propagation infection rate γ1 has greater influence on the maximum I(t) compared with the point-to-point propagation infection rate γ2 . At each time step, an infected node can choose one or two channels to spread rumor or do not spread rumor. It seems that γ2 > γ1 . But because of the large base of group propagation, the total group spread rate is greater than that of the point-to-point propagation. So group propagation has a greater effect on the maximum I(t) than that of the point-to-point propagation. It reflects that in social networks, group propagation has a greater impact on the rumor spreading process compared with point-to-point propagation.

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Fig. 5. Effect of β on I(t).

Fig. 6. Densities of I(t) become smaller with increasing θ .

Fig. 7. Effect of γ1 or γ2 on I(t).

5. Conclusion This paper explores the mechanisms of rumor spreading on social networks and establishes an improved SIR model. Compared with the traditional SIR model, the proposed model divides the rumor spreading channels into two categories, group propagation and point-to-point propagation. The equilibrium points and the basic reproductive number are calculated, and the asymptotical stabilities of the network equilibriums are analyzed theoretically. It is proved that when R0 < 1 a rumor in social network will disappear at last, and the social network will become globally asymptotically stable at the equilibrium point E0 . When R0 > 1, a rumor will keep spreading all the time, and the social network will become globally asymptotically stable at the equilibrium

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Fig. 8. Effect of communication channels on rumor spreading (λ1 = 0.5, λ2 = 0.8, Λ = 0.05, u = 0.05, α = 0.1, β = 0.3, θ = 0.1, ⟨k⟩ = 2).

point E1 if u > w ∗ . Simulation results verify the theoretical analysis. At R0 = 1, the system will exhibit the phenomena of transcritical bifurcation with some other threshold parametric conditions. The proposed model can be applied to those rumors which mainly spread through social networks, and it is helpful for better understanding of the spreading behavior of rumors, and provides guidelines for practical applications. The next step we will improve this model by considering the propagation of rumor over social networks with different non-linear incident rates. Moreover, the research on the effective control strategy is underway. Acknowledgments This work was supported in part by the National Key Research and Development Program of China under Grant 2016YFB0801100, in part by the Key Program of NSFC, China under Grant U1405255, in part by the Shaanxi Science & Technology Coordination & Innovation Project under Grant 2016KTZDGY05-06, in part by the Fundamental Research Funds for the Central Universities under Grant BDZ011402, in part by the National High Technology Research and Development Program (863 Program) under Grant 2015AA016007 and 2015AA017203, in part by the 13th Five-Year Plan Project of National Education Science under Grant DGA160245. References [1] D. Liu, X. Chen, Rumor propagation in online social networks like Twitter–a simulation study, in: Multimedia Information Networking and Security (MINES), 2011 Third International Conference on, IEEE, 2011, pp. 278–282. [2] L. Zhao, Q. Wang, Complete global stability for an SIR epidemic model with delay distributed or discrete, Nonlinear Analysis: Real World Applications. Elsevier (2010) 55–59. [3] S. Jana, S.K. Nandi, Complex dynamics of an SIR epidemic model with saturated incidence rate and treatment, Acta Biotheoret. (2016) 65–84. [4] M.A. Khan, Y. Kham, Complex dynamics of an SEIR epidemic model with saturated incidence rate and treatment, Physica A (2018) 210–227. [5] L. Wang, L. Zhao, Siraru rumor spreading model in complex networks, Physica A (2014) 43–55. [6] J. Ma, Z. Tian, Rumor spreading in online social networks by considering the bipolar social reinforcement, Physica A (2016) 108–115. [7] X. Wei, G. Xu, Global stability of endemic equilibrium of an epidemic model with birth and death on complex networks, Physica A 7 (2017) 8–84. [8] J.D.H. Guillen, A.M. del Rey, Study of the stability of a SEIRS model for computer worm propagation, Physica A (2017) 411–421. [9] L. Zhao, J. Wang, SIHR Rumor spreading model in social networks, Physica A (2012) 2444–2453. [10] L. Zhao, H. Cui, SIR Rumor spreading model in the new media age, Physica A (2013) 995–1003. [11] L. Zhao, Q. Wang, Rumor spreading model with consideration of forgetting mechanism: A case of online blogging livejournal, Physica A (2011) 2619–2625. [12] Q. Liu, Q. Chen, The threshold of a stochastic delayed SIR epidemic model with temporary immunity, Physica A (2016) 115–125. [13] Y. Wang, X. Yang, Rumor spreading model with trust mechanism in complex social networks, Commun. Theor. Phys. (2013). [14] L. Xia, G. Jiang, Rumor spreading model considering hesitating mechanism in complex social networks, Physica A (2015) 295–303. [15] K. Ma, W. Li, Information spreading in complex networks with participation of independent spreaders, Physica A (2018) 21–27. [16] Z. Zhang, C. Liu, Dynamics of information diffusion and its applications on complex networks, Physica A (2016) 1–34. [17] Z. Qian, S. Tang, The independent spreaders involved SIR rumor model in complex networks, Physica A (2015) 95–102. [18] M. Nekovee, Y. Moreno, Theory of rumor spreading in complex social networks, Physica A (2007) 457–470. [19] C. Ji, D. Jiang, Multi-group SIR epidemic model with stochastic perturbation, Physica A (2011) 1747–1762. [20] Y. Zhao, D. Jiang, The threshold of a stochastic SIRS epidemic model with saturated incidence, Appl. Math. Lett. (2014) 90–93. [21] Z. Jin, M. Haque, Pulse vaccination in the periodic infection rate SIR epidemic model, Int. J. Biomath. (2008). [22] G. Zhu, X. Fu, Spreading dynamics and global stability of a generalized epidemic model on complex heterogeneous networks, Appl. Math. Model. 580 (2012) 8–5817. [23] M.L. Taylor, T.W. Carr, An SIR epidemic model with partial temporary immunity modeled with delay, J. Math. Biol. (2009) 840–880. [24] P. Driessche, James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Physica A (2002) 29–48.

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