Open Access Library Journal

How to cite this paper: Wan, C., Li, T., Wang, Y.M. and Liu, X.D. (2016) Rumor Spreading of a SICS Model on Complex Social Networks with Counter Mechanism. Open Access Library Journal, 3: e2885. http://dx.doi.org/10.4236/oalib.1102885

Rumor Spreading of a SICS Model on Complex Social Networks with Counter Mechanism Chen Wan, Tao Li*, Yuanmei Wang, Xiongding Liu School of Electronics and Information, Yangtze University, Jingzhou, China

Received 7 July 2016; accepted 22 July 2016; published 26 July 2016

Copyright © 2016 by authors and OALib. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract The rumor spreading has been widely studied by scholars. However, there exist some people who will persuade infected individuals to resist and counterattack the rumor propagation in our social life. In this paper, a new SICS (susceptible-infected-counter-susceptible) rumor spreading model with counter mechanism on complex social networks is presented. Using the mean-field theory the spreading dynamics of the rumor is studied in detail. We obtain the basic reproductive number ρ and equilibriums. The basic reproductive number is correlated to the network topology and the influence of the counter mechanism. When ρ < 1 , the rumor-free equilibrium is globally asymp-totically stable, and when ρ > 1 , the positive equilibrium is permanent. Some interesting patterns of rumor spreading involved with counter force have been revealed. Finally, numerical simula-tions have been given to demonstrate the effectiveness of the theoretical analysis.

Keywords Rumor Spreading Model, Complex Social Networks, Counter Mechanism, Stability, Permanence Subject Areas: Network Modeling and Simulation

1. Introduction Nowadays, more and more SNS (Social Networking Services) networks are emerging in our social life, such as Facebook, WeChat, LinkedIn and so on, which are seemingly like cobwebs to connect people from different places. With the rapid increase of the number of SNS users, rumor will be quickly into people’s horizons. Each coin has its two sides, as the rumors spread on the impact of our social lives. Sometimes, the rumor spreading

*Corresponding author.

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may play a positive role, for instance, we can let more people to concern about something and take pertinent precaution measures by utilizing the rapid and efficient characteristic of rumor spreading [1] [2]. However, most rumors induce public panic, social disarray and severe economic loss, etc. [3] [4]. Therefore, it is very important to investigate the mechanism of rumor spreading and how to effectively control the rumor.

Rumor can be viewed as an “infection of the mind”, and its spreading shows an interesting similarity to the epidemic spreading [5]-[9]. Daley and Kendal [5] first proposed the classic DK model of rumor spreading. Since then, most of the studies are based on DK model [10]-[17]. In order to overcome the weaknesses of DK model, more and more researchers consider the topological characteristics of underlying networks that they have started to study the problems of rumor spreading on complex networks [15]-[20]. Nekovee and Moreno et al. [16] de-rived a conclusion that scale-free social networks were prone to the spreading of rumors. In Ref. [17], the au-thors found that the degree distribution influenced directly the final rumor size. Recently, researchers [18]-[20] started to take full into account of the role of human behaviors and different mechanisms in the rumor spreading. Zhao et al. [18] presented a novel model by introducing the forget mechanism. Wang et al. [19] presented a novel SIR model by introducing the trust mechanism between the ignorant nodes and the spreader nodes. Han et al. [20] presented a novel model based on the heat energy theory to analyze the mechanisms of rumor propaga-tion on social networks.

However, most of the previous models didn’t consider that people may not agree with the rumor and counte-rattack it strongly. Based on some realistic perspectives, different people may have different views to the rumor on social networks. Some people may be in conflict with their beliefs when they hear rumor. They will persuade infected individuals to resist and counterattack the rumor propagation. In order to study this phenomenon, we present a SICS (susceptible-infected-counter-susceptible) rumor spreading model with counter mechanism on complex social networks to explain it. Obviously, the counter mechanism can change the contacts among people, i.e. network topology structure. Within the counter mechanism of the SICS model, when an infected individual contacts a counter individual, it may become a counter individual with a certain probability.

The rest of this paper is organized as follows. In Section 2, we present a SICS rumor spreading model and de-rive the corresponding mean-field equations to describe the dynamics of the model. In Section 3, the basic re-productive number obtained at first. Then we analyze the globally asymptotic stability of rumor-free equilibrium and the permanence of the rumor in detail. Simulation results of the proposed model are shown in Section 4. Fi-nally, we conclude the paper in Section 5.

2. Model Formulation As mentioned earlier, we present a SICS rumor spreading model. The population is divided into three classes: susceptible individuals who have ambiguous attitude about the rumor; infected individuals who believe and spread it actively; counter individuals who reject the rumor, refute the rumor and persuade neighbors don’t be-lieve in it. Taking into account the heterogeneity induced by the presence of vertices with different connectivi-ties, let ( ) ( ) ( ), ,k k kS t I t C t be the densities of susceptible, infected and counter individuals of connectivity k at time t, respectively.

The SICS model has the flow diagram given in Figure 1. In the course of rumor spreading, a susceptible indi-vidual is infected with probability 1β if it is connected to an infected individual. When a counter individual

Figure 1. The flow diagram of the SICS model.

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contacts an infected individual, the counter individual can persuade infected individual to resist and counterat-tack the rumor, so the infected individual becomes a counter node with probability 2β . A susceptible individual transform into a counter individual with probability α . Due to some own reason, an infected individual turns into a counter individual with probability 1γ . However, some counter individuals, due to loss of counterattack ability, join the susceptible individuals again, i.e., moving back to susceptible state, with probability 2γ . We assume that the immigration rate and emigration rate are both constant l in the spreading process of rumor. All recruitment is into the susceptible class.

Thus, the dynamic mean-field reaction rate equations can be written as

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

2 1 1

1 1 2 1

2 2 1 2

dd

dd

dd

kk k k k

kk k k k

kk k k k k

S tC t S t k t S t S t

tI t

k t S t k t I t I t I tt

C tS t k t I t I t C t C

l

l

t

l

tl

γ β

β β γ

β

α

α γ γ

2

= + − − Θ −

= Θ − Θ − −

= + Θ + − −

(1)

The probability ( )1 tΘ describes a link pointing to an infected individual,

( ) ( ) ( )( ) ( ) ( )1

1kkkk

k

kP k I tt kP k I t

sP s kΘ = =∑ ∑∑

(2)

the probability ( )2 tΘ describes a link pointing to a counter individual which satisfies the relation

( ) ( ) ( )( ) ( ) ( )2

1kkkk

s

kP k C tt kP k C t

sP s kΘ = =∑ ∑∑

(3)

where k is the average degree within the network. And ( ) ( ) ( )kkI t P k I t= ∑ is the density of infected in-

dividuals in the whole network, ( ) ( ) ( )kkC t P k C t= ∑ is the density of counter individuals in the whole net-

work, ( )P k is the connectivity distribution.

3. Stability Analysis In this section, we present an analytic solution to the deterministic equations describing the dynamic of the (SICS) rumor spreading process.

Theorem 1. Let. ( )( ) ( )

21 2 2

1 2

ll kl

kβ γ βρ

γ γαα

+ −= ⋅

+ + +. There always exists a rumor-free equilibrium

20

2 2

, 0,ll l

E γαγ

ααγ

+ + + + +

and when 1ρ > , then system (1) has a positive equilibrium solution

( ), ,k k kE S I C∞ ∞ ∞+ .

Proof. To get the equilibrium solution ( ), ,k k kE S I C∞ ∞ ∞+ , we need to make the right side of system (1) equal to

zero. Then the equilibrium ( ), ,k k kE S I C∞ ∞ ∞+ should satisfy

2 1 1

1 1 2 2 1

2 2 1 2

0

0

0

k k k k

k k k k

k k k k k

C S k S S

k S k I I I

S k I I C C

l l

l

l

γ β

β

γ

α

β γ

β γα

∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ ∞ ∞ ∞

+ − − Θ − = Θ − Θ − − = + Θ + − − =

(4)

where ( )11

kk

kP k Ik

∞ ∞Θ = ∑ , ( )21

kk

kP k Ck

∞ ∞Θ = ∑ , one has

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( )

( ) ( )( )

1

1 1 1

1 2

k k

k k

kS I

k

k k kC I

k

l

l

l

β γ

β

β β γ β γ

β

α

γ

∞2 2∞ ∞

∞1 1

∞ ∞ ∞1 2 2 2 2∞ ∞

∞1

Θ + + =

Θ

Θ Θ + + Θ + += Θ +

(5)

According to the following normalization condition for all k:

1k k kS I C∞ ∞ ∞+ + = .

We can obtain:

( )( ) ( ) ( )

1 2

1 2 1 2 2 1 2k

kI

kl

l l lk kβ γ

β β γ γ αγ β γ

∞1∞

∞ ∞ ∞1 2 2

Θ +=

Θ Θ + + + + Θ + + + + (6)

( ) ( )( ) ( ) ( )

1 1 1

1 2 1 2 2 1 2k

lk k kC

l lk k lk

β β γ β γ

β β γ γ β γ γ

α

α

∞ ∞ ∞1 2 2 2 2∞

∞ ∞ ∞1 2 2

Θ Θ + + Θ + +=

Θ Θ + + + + Θ + + + + (7)

Inserting Equation (6) into Equation (2), we obtain the following equation

( ) ( )( ) ( ) ( )

2

1 2

1 2 1 2 2 1 2

kk P k

k kl

l l lk kβ γ

β β γ γ β γ α γ

∞1∞

1 ∞ ∞ ∞1 2 2

Θ +Θ = ⋅

Θ Θ + + + + Θ + + + +

∑ (8)

Inserting Equation (7) into Equation (3), we obtain the following equation

( ) ( ) ( )( ) ( ) ( )

2

1 1 1

1 2 1 2 2 1 2

kl kk P k k k

k k k l l lk

αβ β γ β γ

β β γ γ β γ γα

∞ ∞ ∞1 2 2 2 2∞

2 ∞ ∞ ∞1 2 2

Θ Θ + + Θ + +Θ = ⋅

Θ Θ + + + + Θ + + + +

∑ (9)

Equation (9) divided by Equation (8), we obtain the following equation

( )( ) ( )

1 1 1

1 2 2 1

kk

lk kl

γ β γβ γ

ααβ β

∞1∞

2 ∞1

Θ + +Θ =

+ − Θ + (10)

Inserting Equation (10) into Equation (8), we can obtain

( ) ( )( )( )

( ) ( ) ( )( )

2

1 2

2 1 1 11 1 2 2 1

1 2 2 1

.k l

ll l

k P kf

k

kl

kk k

β γ

β γ β γγ β γ γ β

β γ β β

αα

α

∞1∞ ∞

1 1∞1 ∞ ∞

1 1∞1

Θ +Θ = ⋅ Θ

Θ + + + + Θ + + + + Θ + − Θ +

∑ (11)

Obviously, 0∞1Θ = is a solution of Equation (11), i.e., ( ) 0f ∞

1Θ = . To ensure Equation (11) have a non-trivial solution, i.e. 1∞

10 < Θ ≤ , the following conditions must be satisfied

( ) ( )0

d1 and 1

d

ff

∞1

∞1 ∞

1∞1

Θ =

Θ> Θ ≤

Θ.

We can obtain the basic reproductive number

( )( ) ( )

21 2 2

1 2

1l

l lkk

β γ βρ

γ γαα

+ −= ⋅ >

+ + + (12)

So, a nontrivial solution exists if and only if 1ρ > . Substitute the nontrivial solution of (11) into (6), we can get kI ∞ . By (5) and (6), we can easily obtain

0 1,0 1,0 1k k kS I C∞ ∞ ∞< < < < < < .

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Therefore, the positive equilibrium ( ), ,k k kE S I C∞ ∞ ∞+ is well-defined. Hence, when 1ρ > , one and only one

positive equilibrium ( ), ,k k kE S I C∞ ∞ ∞+ of system (1) exists. This completes the proof.

Remark. The basic reproductive number is obtained by Equation (12), which depends on the fluctuations of the degree distribution and the influence of counter mechanism. The 2β can affect the basic reproductive number.

Theorem 2. If 1ρ < , the rumor-free equilibrium 0E of the system (1) is globally asymptotically stable. Proof. We rewrite the system (1) as

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 21

21 2

d1

d

d1

d

kk k k k k k

k k

kk k k k k k

k

I t k kI t C t kP k I t I t kP k C t I tt k k

C t kI t C t I t kP k C t I t C tt k

l

l

β β γ

β γα γ

= − − − − +

= − − + + − +

∑ ∑

∑ (13)

The Jacobian matrix of system (13) at 2

0,l

ααγ

+ +

is a max max2 2k k× as follows

max

max

max max max max

1 12 13 1

21 2 23 2

1 2 3

k

k

k k k k

A B B B

B A B BJ

B B B A

=

,

where

( )( ) ( )

( )( )

2

1 2 2 12

2

2 1 22

0

j

j P jk

Aj P j

l ll

llk

αα

α

β γ β γγ

β γ γαγ

α α

+ − − − + + =

− + − − − + +

,

( )( ) ( )

( )( )

1 2 22

22

0

0ij

ijP jk

BijP j

l

k

l

l

β γ βγ

βγ

αα

αα

+ − + + =

+ +

.

By mathematical induction method, the characteristic equation can be calculated as follows

( )( )( )( )

( )( )( )( )

max

max

121 1 2 2

2 12

21 2 2

2 12

0

k

k ll l

l

ll l

l

k

k

k

k

αα

α

αα

α

β γ βλ γ λ γ

γ

β γ βλ γ λ γ

γ

−

− + − + + + + + + +

+ − × + + + + + = + +

−

−

where

( ) ( ) ( )2 2 2max max1 2 2k P P k P k= + + + .

The stability of 0E is only dependent on

( ) ( )( )( )

21 2 2

2 12

0kk

ll l

lβ γ β α

αα

λ γ λ γγ

+ − + + + + + = + +

− .

Note that

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( )( ) ( )

21 2 2

1 2

ll kl

kβ γ βρ

γ γαα

+ −= ⋅

+ + +.

So, we have obtained

( ) ( ) ( )( )2 1 1 0l lαλ γ λ γ ρ+ + + + + − = .

When 1ρ < , all real-valued eigenvalues are negative. Hence, 0E is locally asymptotically stable if 1ρ < .

Now we will prove that 20

2 2

, 0,ll l

E γαγ

ααγ

+ + + + +

is globally attractive. From the second equation of

system (1) we can get

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( )

11 1 2 1

1

1 2 1 11

1 1 11 1

2 21 2

1 12 2

21 2

2

d 1d

1

1 1

n

k k k kk

n

kk

n n

k kk k

tkP k k t S t k t I t I t I t

t k

kP k k S t k l tk

kP k k S t k

l

P k C t k

ll l

l

l tk k

k kl t

k k

kl

β β γ

β β γ

β β γ

β γ β γγ γ

β γ β

αα α

ααγ

2=

2=

2= =

2

2

Θ= Θ − Θ − −

< − Θ − + Θ

= − − + Θ + < − − + Θ

+ + + +

+ −=

+ +

∑

∑

∑ ∑

( ) ( )

( ) ( ) ( )

1 1

1 11

l t

tl

kγ

γ ρ

− + Θ

= + − Θ

Now we consider the comparison equation with the condition ( ) ( )1 0 0ϕΘ = as follows

( ) ( ) ( ) ( )1d

1d

lt

tt

ϕγ ρ ϕ= + − ,

integrating from 0 to t yields

( ) ( ) ( )( )1 10 e l tt γ ρϕ ϕ + −= .

Since 1ρ < , we obtain ( ) 0tϕ → as 0t → . According to the comparison theorem of functional differential equation, we can get ( ) ( )10 t tϕ≤ Θ ≤ , for all 0t > .

Thus, ( )1 0tΘ → as t → ∞ , which implies 0kI → as t → ∞ , for 1, 2, ,k n= . It follows that the ru-mor-free equilibrium 0E is globally attractive. This completes the proof.

Theorem 3. If 1ρ > , the rumor is permanent on complex social networks, i.e., there exists a 0ξ > , such that

( ) ( ) ( )lim inf lim inf kt t kI t P k I t ξ

→∞ →∞= >∑ .

Proof. We will use the result of Thieme in Theorem 4.6 [21] to prove it. Define

( ) max max max1 1 1 max, , , , , , : , , 0 and 1, 1, ,k k k k k k k k kX S I C S I C S I C S I C k k= ≥ + + = = ,

( ) ( )max max max0 1 1 1, , , , , , : 0k k k k

kX S I C S I C X P k I = ∈ >

∑ ,

0 0\X X X∂ = .

In the following, we will show that (1) is uniformly persistent with respect to ( )0 0,X X∂ .

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Obviously, X is positively invariant with respect to system (1). If (0) 0≥kS , ( ) ( ) 0k P k I k >∑ and

( ) ( ) 0k P k C k >∑ for max1, ,k k= , then ( )0 0kS ≥ , ( ) ( ) 0k P k I k >∑

and ( ) ( ) 0k P k C k >∑ for all

0t > . Since ( ) ( )( ) ( ) ( ) ( )1k kk kP k I t P k I tlγ′ ≥ − +∑ ∑ , ( ) ( )( ) ( ) ( ) ( )2k kk kP k C t P k C tlγ′ ≥ − +∑ ∑ ,

( ) ( )0 0k P k I >∑ and ( ) ( )0 0k P k C >∑ , we have ( ) ( ) ( ) ( ) ( )10 e 0tkk k

lkP k I t P k I γ− +≥ >∑ ∑ ,

( ) ( ) ( ) ( ) ( )20 e 0tkk k

lkP k C t P k C γ− +≥ >∑ ∑ . Thus, 0X is also positively invariant. Furthermore, there exists a

compact set B in which all solutions of (1) initiated in X will enter and remain forever after. The compactness condition (C4.2) in Thieme [21] is easily verified for this set B. Denote

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( )

max max max

max max max

1 1 1

1 1 1 0

0 , 0 , 0 , , 0 , 0 , 0 :

, , , , , , , 0 .

k k k

k k k

M S I C S I C

S t I t C t S t I t C t X t

∂ =

∈ ∂ ≥

Denote

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )( )

max max max

max max max

1 1 1

1 1 1

0 , 0 , 0 , , 0 , 0 , 0 :

0 , 0 , 0 , , 0 , 0 , 0

k k k

k k k

S I C S I C

S I C S I C X

ωΩ =

∈

where ( ) ( ) ( ) ( ) ( ) ( )( )max max max1 1 10 , 0 , 0 , , 0 , 0 , 0k k kS I C S I Cω is the omega limit set of the solutions of system

(1) starting in ( ) ( ) ( ) ( ) ( ) ( )( )max max max1 1 10 , 0 , 0 , , 0 , 0 , 0k k kS I C S I C . Restricting system (1) on M ∂ gives

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

2

1

2

dd

dd

dd

kk k k

kk k

kk k k

S tC t S t S t

tI t

I t I t

l

tC t

S t C t C

l

t

l

tl

γ α

α

γ

γ

= + − −

= − −

= − −

(14)

It is easy to verify that system (13) has a unique equilibrium 0E in X. Thus 0E is the unique equilibrium of system (1) in M ∂ . It is easy to check that 0E is locally asymptotically stable. This implies that 0E is globally asymptotically stable for (13) is a linear system. Therefore 0EΩ = . And 0E is a covering of Ω , which is isolated and is acyclic (since there exists no solution in M ∂ which links 0E to itself). Finally, the proof will be done if we show 0E is a weak repeller for 0X , i.e.

( ) ( ) ( ) ( ) ( ) ( )( )( )max max max1 1 1 0lim sup dist , , , , , , , 0k k kt

S t I t C t S t I t C t E→∞

>,

where ( ) ( ) ( ) ( ) ( ) ( )( )max max max1 1 1, , , , , ,k k kS t I t C t S t I t C t is an arbitrarily solution with initial value in 0X . By

Leenheer and Smith (2003, Proof of Lemma 3.5, [22]), we need only to prove ( ) 00sW E X = Ø where

( )0sW E is the stable manifold of 0E . Suppose it is not true, then there exists a solution

( ) ( ) ( ) ( ) ( ) ( )( )max max max1 1 1, , , , , ,k k kS t I t C t S t I t C t in 0X , such that

( ) ( ) ( )2

2 2

, 0,k k kS t I tl

tll

Cγαγ

ααγ

+→ → →

+ + + + as t → ∞ . (15)

Since ( )( ) ( )

21 2 2

1 2

ll kl

kβ γ βρ

γ γαα

+ −= ⋅

+ + +, we can choose 0η > such that

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( )

( ) ( )

221 2 2

2

1 2

1l

l k

lkl

l

γαα

β β

γ α

γ ηγ

γ

++ − − + + ⋅ >

+ + +.

For 0η > , by (15) there exists a 0T > such that

( ) ( ) ( )2 2

2 2 2 2

, 0 ,k k kS t I tl l l

tl Cll

α αα α α α

γ γη η η η ηγ γ γ γ

+ +− < < + ≤ < − < < +

+ + + + + + + +.

For all t T≥ and max1, 2, ,k k= . Let

( ) ( ) ( ) ( )2 kk

V t kP k I tlγ= +∑ .

The derivative of V along the solution is given by

( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

( )( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( )

2

2 1 2 1

21 2 2

2

22 2

2 12

21 2 2

2

kk

k kk k

k k kk

kk k

k kk k k

k

V t kP k I t

kP k I t kP k C tkP k k S t k I t I t

k k

kP k kP k I t

k

kP k kP

l

l l

l ll

ll l

l

l

k I t kP

l

k I tk

kP

lk k

k

α

γ

γ β β γ

β γ γ ηγ

β γη γ γ

γα

α

γη

α

β γγ

= +

= + − − + + +

≥ − + + +

− + − + + + +

+ += −

+ +

∑

∑ ∑∑

∑ ∑

∑ ∑ ∑

∑

( )

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

22 2

2 2 12

2 21 2 2 22

2 2

2 1

22

1 2 2 2 12

k

kk

kk

kk

I t

kP k k I t

k

k kP k k I t

k k

l l l ll

l lll l

l l

ll l

P k k I t

kk

kl

lP k

β γ γ η γ γγ

β γ β γγ η ηγ γ

γ γ

γβ γ β η γ γ

α

α

α

α α

α

α αγ α

+ − + + − + + + +

+ + + > − − − + + + + − + + +

+= + − − − + + + + +

∑

∑

∑

( )

( ) ( ) 0

kk

kk

I t

kP k I t

> ≥

∑

∑

Hence ( )V t → ∞ as t → ∞ , which contradicts to the boundedness of ( )V t . This completes the proof.

4. Numerical Simulations In this section, several numerical simulations are presented to illustrate our analysis. We consider the system (1) on a complex social network with ( ) 3P k k −= , where the parameter η satisfies 3

1 1, 1000nk k nη −

== =∑ .

In Figure 2, the parameters are chosen as 1 2 1 20.1, 0.45, 0.3, 0.5, 0.40.3, ,l γ γα β β= = = = == then the basic reproductive number 0.9462 1ρ = < . We can see that when 1ρ < , kI grows to zero, i.e., the infectious indi-viduals will ultimately disappear.

In Figure 3, we choose 1 2 1 20.1, 0.1, 0.3, 0.5, 0. ,0.3 0, 1l γ γ β βα= = = = == thus the basic reproductive number 6.4073 1ρ = > . We can see that when 1ρ > , the rumor is persist and the infected individuals’ number will converge to a positive constant respectively.

C. Wan et al.

OALibJ | DOI:10.4236/oalib.1102885 9 July 2016 | Volume 3 | e2885

Figure 2. The time series of system (1) with 0.9462 1ρ = < and initial values ( )0 0.1kI = ,

50,100,200,500.k =

Figure 3. The time series of system (1) with 6.4073 1ρ = > and initial values ( )0 0.1kI = ,

50,100,200,500.k =

In Figure 4, numerical simulations show the spread of SICS model on complex social networks with 1 2 10.1, 0.45, 0.0.3, 3, 0.5l α γ γ β= = == = and 2 2 20.3, 0.4, 0.6β β β= = = . The condition of 1ρ < , that dif-

ferent 2β leading to different states. In addition, it is also found that the larger the 2β is, the rumor dies out faster.

In Figure 5, numerical simulations show the spread of SICS model on complex social networks with 1 2 10.1, 0.1, 0.3, 0.0 5.3,l α γ γ β= = == = and 2 2 20.01, 0.04, 0.1β β β= = = . The condition of 1ρ > , that dif-

ferent 2β leading to different states. In addition, it is also found that the larger the 2β is, the positive equili-brium will be lower. The simulations indicate that the numerical results are well consistent with the theoretical analysis.

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

I(t)

I50

I100I200

I500

0 10 20 30 40 50 60 70 80 90 1000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

I(t)

I50

I100I200

I500

C. Wan et al.

OALibJ | DOI:10.4236/oalib.1102885 10 July 2016 | Volume 3 | e2885

Figure 4. The prevalence 300I versus t corresponding to different 2β with identical initial value ( )300 0 0.1I = .

Figure 5. The prevalence 300I versus t corresponding to different 2β with identical initial value ( )300 0 0.1I = .

5. Conclusion In summary, we present a new SICS rumor spreading model with counter mechanism on complex social net-works. By using the mean-field theory, we obtain the basic reproductive number and equilibriums. Theoretical results indicate that the basic reproductive number is significantly dependent on the topology of the underlying networks and the counter mechanism. The basic reproductive number is in direct proportion to 2k k . So, network heterogeneity makes rumor easy to spread. Moreover, we found that the greater 2β can decrease the basic reproductive number ρ , i.e., lower average rumor density and shorter rumor prevalent decay time. The global stability of rumor-free equilibrium and the permanence of rumor are proved in detail. Our theoretical and numerical simulation results give a novel explanation for rumor spreading. This study has valuable guiding sig-nificance in effectively preventing rumor spreading.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

t

I 300

β2=0.3

β2=0.4

β2=0.6

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

I 300

β2=0.01

β2=0.04

β2=0.1

C. Wan et al.

OALibJ | DOI:10.4236/oalib.1102885 11 July 2016 | Volume 3 | e2885

Acknowledgements This work was supported in part by the National Natural Science Foundation of China under Grant 60973012.

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