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R E S E A R C H

Open Access

Global stability of a

SEIR

rumor

spreading model with demographics

on scale-free networks

Chen Wan, Tao Li

*

and Zhicheng Sun

*Correspondence:

yuanyzq@163.com School of Electronics and Information, Yangtze University, Jingzhou, 434023, P.R. China

Abstract

In this paper, a newSEIR(susceptible-exposed-infected-removed) rumor spreading model with demographics on scale-free networks is proposed and investigated. Then the basic reproductive numberR0and equilibria are obtained. The theoretical analysis indicates that the basic reproduction numberR0has 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 [–]. Rumor propagation may have tremendous negative effects on human lives, such as reputation damage, social panic and so on [–]. 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 , Daley and Kendal first proposed the classical DK model to study the rumor propagation []. 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 [–].

In the early stages, most rumor spreading models were established on homogeneous networks [–]. However, it is well known that a significant characteristic of social net-works is their scale-free property. In netnet-works, 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 distributionP(k)∼k–γ

( <γ ≤) []. 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 [, –].

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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, Xiaet al. [] proposed a novelSEIRrumor spreading model with hesitating mechanism by adding a new exposed group (E) in the classicalSIRmodel. Liuet al. [] presented aSEIRrumor propagation model on the heterogeneous network. They calcu-lated the basic reproduction numberRby using the next generation method, and they

found that the basic reproduction numberR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 [, ] proposed aSEIRmodel 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 ana-lyze a newSEIRmodel with hesitating mechanism on heterogeneous networks and com-prehensively 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 , we present a newSEIR spread-ing model with hesitatspread-ing mechanism on scale-free networks. In Section , the basic repro-duction number and the two equilibria of the proposed model are obtained. In Section , we analyze the globally asymptotic stability of equilibria. Finally, we conclude the paper in Section .

2 Modeling

Consider the whole population as a relevant online network. TheSEIRrumor spreading model is based on dividing the whole population into four groups, namely: the susceptible, referring to those who have never contacted with the rumor, denoted byS; the exposed, referring to those who have been infected, in a hesitate state not spreading the rumor, denoted byE; the infected, referring to those who have accepted and spread the rumor, denoted byI; the recovered, referring to those who know the rumor but have ceased to spread it, denoted byR. During the period of rumor spreading, we suppose that the indi-viduals with the same number of contacts are dynamically equivalent and belong to the same group in this paper. LetSk(t),Ek(t),Ik(t) andRk(t) be the densities of the above-mentioned nodes with the connectivity degreekat timet. Then the aggregate number of population at time isN(t), and the density of the whole population with degreek satis-fies

Nk(t) =Sk(t) +Ek(t) +Ik(t) +Rk(t). (.)

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Figure 1 Transfer diagram forSEIRrumor propagation model.

Based on the above hypotheses and notation, the dynamic mean-field reaction rate equa-tions described by

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

dSk(t)

dt =b(k) –λ(k)(t)Sk(t) –μSk(t), dEk(t)

dt =λ(k)(t)Sk(t) –βEk(t) +δmIk(t) –μEk(t), dIk(t)

dt =βhEk(t) –δIk(t) –μIk(t), dRk(t)

dt =β( –h)Ek(t) +δ( –m)Ik(t) –μRk(t),

(.)

whereλ(k) >  is the degree of acceptability ofkfor individuals for the rumor, and the probability(t) denotes a link to an infected individual, satisfying

(t) =  k

i

ϕ(i)

i P(i|k) Ii(t)

Ni(t)

. (.)

Here, /irepresents the probability that one of the infected neighbors of an individual, with degreei, will contact this individual at the present time step;P(i|k) is the probability that an individual of degreekis connected to an individual with degreei. In this paper, we focus on degree uncorrelated networks. Thus,P(i|k) =iP(i)/k, wherek=iiP(i) is the average degree of the network. For a general functionf(k), it is defined asf(k)=

if(i)P(i). The functionϕ(k) is the infectivity of an individual with degreek.

Adding the four equations of system (.), we have dNk(t)

dt =b(k) –μNk(t). Then we can

obtainNk(t) =bμ(k)( –e–μt) +Nk()e–μt, whereNk() represents the initial density of the

whole population with degreek. Hence,lim supt→∞Nk(t) =b(k)/μ, thenNk(t) =Sk(t) +

Ek(t) +Ik(t) +Rk(t)≤b(k)/μfor allt> . In order to have a population of constant size, we suppose thatSk(t) +Ek(t) +Ik(t) +Rk(t) =Nk(t) =ηk, whereηk=b(k)/μ. Thus, we have

(t) =  k

k=

ϕ(k) ηk

P(k)Ik(t). (.)

Furthermore,

S(t) =

k

P(k)Sk(t), E(t) =

k

P(k)Ek(t),

I(t) =

k

P(k)Ik(t), and R(t) =

k

P(k)Rk(t)

are the global average densities of the four rumor groups, respectively. From a practical perspective, we only need to consider the case ofP(k) >  fork= , , . . . . The initial con-ditions for system (.) satisfy

≤Sk(),Ek(),Ik(),Rk()≤ηk,

Sk() +Ek() +Ik() +Rk() =ηk, () > .

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3 The basic reproduction number and equilibria

In this section, we reveal some properties of the solutions and obtain the equilibria of system (.).

Theorem  Define the basic reproduction number R=(β+μ)(δ+μ)–ββh hδm ϕ(k)λ(k)

k ,then there always exists a rumor-free equilibrium E(ηk, , , ).And if R> , system(.) has a unique rumor-prevailing equilibrium E+(Sk ,Ek∞,Ik∞,Rk ).

Proof We can easily see that the rumor-free equilibriumE(ηk, , , ) of system (.) is

always existent. To obtain the equilibrium solutionE+(Sk∞,Ek∞,Ik∞,Rk ), we let the right

side of system (.) be equal to zero. Thus, we have

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

b(k) –λ(k)∞SkμSk∞= ,

λ(k)∞SkβEk∞+δmIk∞–μEk∞= , βhEkδIk∞–μIk = ,

β( –h)Ek∞+δ( –m)Ik∞–μRk = ,

where∞=kk=ϕ(ηk) k P(k)I

k . One has

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Ek =(δ+μ)βh Ik∞,

Sk∞=(β+μ)(δ+μ)–ββhλ(k)hδmIk∞,

Rk∞=(δ+μ)(–hh)+μhδ(–m)Ik∞.

(.)

According toSk +Ek +Ik∞+Rk =ηkfor allk, we have

Ik∞= μβhλ(k)

η k

μ(β+μ)(δ+μ) –μβhδm+λ(k)∞[(β+μ)(δ+μ) –mβhδ]. (.)

Inserting equation (.) into equation (.), we can obtain the self-consistency equation:

∞=  k

k=

ϕ(k) ηk

×P(k) μβhλ(k)

η k

μ(β+μ)(δ+μ) –μβhδm+λ(k)∞[(β+μ)(δ+μ) –mβhδ]

f∞ . (.)

Obviously,∞=  is a solution of (.), thenSk∞=ηkandEk∞=Ik∞=Rk = , which is a

rumor-free equilibrium of system (.). In order to ensure equation (.) has a nontrivial solution,i.e.,  <∞≤, the following conditions must be fulfilled:

df(∞)

d

∞=

>  and f()≤.

Thus, we can obtain

βh

(β+μ)(δ+μ) –βhδm

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Let the base reproduction number as follows:

Remark The basic reproductive numberRis obtained by equation (.), which depends

on some model parameters and the fluctuations of the degree distribution. Interestingly, the basic reproductive numberRhas no correlation with the degree-dependent

immi-grationb(k). According to the form ofR, we see that increase of the emigration rateμ

will makeRdecrease. Ifb(k) =  andμ= , then system (.) become the network-based SEIRmodel without demographics, andR=δ(–hhm)ϕ(kk)λ(k), which is in consistence with

reference [].

4 Discussion

4.1 The stability of the rumor-free equilibrium

Theorem  The rumor-free equilibrium Eof SEIR system(.)is locally asymptotically stable if R< ,and it is unstable if R> .

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

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where

λ(k)ϕ(k)=λ()ϕ()P() +λ()ϕ()P() +· · ·+λ(kmax)ϕ(kmax)P(kmax).

The stability ofEis only dependent on

(z+β+μ)(z+δ+μ) –βhδmβhλ(k)ϕ(k)

k = . (.)

Then we have

z+ (β+δ+ μ)z+ (β+μ)(δ+μ) –βhδmβhλ(k)ϕ(k)

k = . (.)

According to equation (.), if R < , we can easily get (β +μ)(δ +μ) – βhδm

βhλ(kk)ϕ(k)> ,i.e.,z< . Hence,Eis locally asymptotically stable ifR<  and

unsta-ble ifR> . The proof is completed.

Theorem  The rumor-free equilibrium Eof SEIR system(.)is globally asymptotically stable if R< .

Proof First, we define a Lyapunov functionV(t) as follows:

V(t) =

k

ϕ(k) ηk

P(k)Ek(t) +(β+μ) βh Ik(t)

. (.)

Then, according to a calculation of the derivative ofV(t) along the solution of system (.), we have

V(t) =

k

ϕ(k)P(k) ηk

Ek(t) +(β+μ) βh Ik(t)

=

k

ϕ(k) ηk

P(k)

λ(k)(t)Sk(t) – (β+μ)Ek(t) +δmIk(t)

+(β+μ) βh

βhEk(t) – (δ+μ)Ik(t)

k

ϕ(k) ηk

P(k)

λ(k)(tk+

δmβh– (β+μ)(δ+μ) βh Ik(t)

=(t)

k

ϕ(k)P(k)λ(k) +δmβh– (β+μ)(δ+μ) βh

k

ϕ(k) ηk

P(k)Ik(t)

=(t)ϕ(k)λ(k)+δmβh– (β+μ)(δ+μ) βh k(t)

=(t)  βh

βhϕ(k)λ(k)+δmβh– (β+μ)(δ+μ)k

=(t)k[(β+μ)(δ+μ) –βhδm]

βh (R– ).

WhenR< , we can easily find thatV(t)≤ for allV(t)≥, and thatV(t) =  only

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rumor-free equilibriumEof system (.) is globally attractive. Therefore, whenR< ,

the rumor-free equilibriumEofSEIRsystem (.) 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  When R> ,the rumor is permanent on the network,i.e.,there exists a posi-tive constantς> ,such thatlim infI(t)t→∞=lim inft→∞kP(k)Ik(t) >ς.

Proof We desire to use the condition stated in Theorem . in []. Define

X=(S,E,I,R, . . . ,Skmax,Ekmax,Ikmax,Rkmax) :

Sk,Ek,Ik,Rk≥ andSk+Ek+Ik+Rk=η,k= , . . . ,kmax ,

X=

!

(S,E,I,R, . . . ,Skmax,Ekmax,Ikmax,Rkmax)∈X:

k

P(k)Ik> 

"

,

∂X=X\X.

In the following, we will explain that system (.) is uniformly persistent with respect to (X,∂X).

Clearly, X is positively and bounded with respect to system (.). Assume that () =

k

k= ϕ(k)

ηk P(k)Ik() > , then we have Ik() >  for some k. Thus, I() =

k=P(k)Ik() > . ForI(t) =

kP(k)Ik(t)≥–(δ+μ)

kP(k)Ik(t) = –(δ+μ)I(t), we have I(t)≥I()e–(δ+μ)t> . Therefore,X

is also positively invariant. Furthermore, there exists

a compact setB, in which all solutions of system (.) initiated inXultimately enter and remain forever after. The compactness condition (C.) of Theorem . in reference [] is easily verified for this setB.

Denote

M∂=

S(),E(),I(),R(), . . . ,Skmax(),Ekmax(),Ikmax(),Rkmax() :

S(t),E(t),I(t),R(t), . . . ,Skmax(t),Ekmax(t),Ikmax(t),Rkmax(t) ∈∂X,t≥ ,

and

=#ωS(),E(),I(),R(), . . . ,Skmax(),Ekmax(),Ikmax(),Rkmax() :

S(),E(),I(),R(), . . . ,Skmax(),Ekmax(),Ikmax(),Rkmax() ∈X ,

whereω(S(),E(),I(),R(), . . . ,Skmax(),Ekmax(),Ikmax(),Rkmax()) is the omega limit

set of the solutions of system (.) starting in (S(),E(),I(),R(), . . . ,Skmax(),Ekmax(),

Ikmax(),Rkmax()). Restricting system (.) onM∂, we can obtain

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

dSk(t)

dt =b(k) –μSk(t), dEk(t)

dt = –(β+μ)Ek(t), dIk(t)

dt = –(δ+μ)Ik(t), dRk(t)

dt =β( –h)Ek(t) –μRk(t).

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Obviously, system (.) has a unique equilibriumEinX. Thus,Eis the unique

equi-librium of system (.) inM∂. We can easily find thatEis locally asymptotically stable.

For system (.) is a linear system; this indicates thatEis globally asymptotically stable.

Hence={E}. AndEis a covering ofX, which is isolated and acyclic (because there

exists no nontrivial solution inM∂which linksEto itself ). Finally, the proof of theorem

will be completed if it is shown thatEis a weak repeller forX,i.e.,

with initial value in X. In order to use the method of Leenheer and Smith [], we

need only to prove Ws(E

Then we can choose sufficiently smallξ>  such that

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=

Consequently,V(t)→ ∞ast→ ∞, which apparently contradicts the boundedness of

V(t). This completes the proof.

Lemma ([]) If a> ,b> anddxdt(t) ≥bax,when t≥and x()≥,we can obtain

lim inft→+x(t)≥ ba.If a> ,b> and dxdt(t) ≤bax,when t≥and x()≥,we can obtainlim supt→+x(t)≤b

a.

Next, by using a novel monotone iterative technique in reference [], we discuss the global attractivity of the rumor-prevailing equilibrium.

Theorem  Suppose that(Sk(t),Ek(t),Ik(t),Rk(t))is a solution of system(.),satisfying the initial condition equation(.).When R> ,thenlimt→∞(Sk(t),Ek(t),Ik(t),Rk(t)) =

(Sk∞,Ek ,Ik∞,Rk ),where(Sk ,Ek ,Ik ,Rk )is the unique positive rumor equilibrium of sys-tem(.)satisfying(.)for k= , , . . . ,n.

Proof Since the first three equations in system (.) are independent of the fourth one, it suffices to consider the following system:

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where

Xk()= μηk

λ(k)ϑ ε+μ+ ε<ηk.

For(t)≤ki=ϕ(i)P(i) =M, from the second equation of system (.) fort>t, we

can get

dEk(t)

dtλ(k)M

ηkEk(k) –Ik(k) –Rk(k) – (β+μ)Ek(t)

+δmηkEk(k) –Sk(k) –Rk(k)

λ(k)MηkEk(k) – (β+μ)Ek(t) +δm

ηkEk(k)

=ηk

λ(k)M+δmEk(k)λ(k)M+δm+β+μ.

Similarly, for arbitrary given positive constant  <ε<min{/,ε,(λ(k)mM+β+μ)ηm+β+μ)k }, there

exists at>t, such thatEk(t)≤Yk()–εfort>t, where

Yk()= λ(k)Mηk

λ(k)M+δm+β+μ+ ε<ηk.

From the third equation of system (.), we have

dIk(t)

dtβh

ηkIk(t) – (δ+μ)Ik(t) =βhηk– (δ+μ+βh)Ik(t), t>t.

Thus, for arbitrary given positive constant  <ε<min{/,ε,(δ+μ+β(μ+βhhk)}, there exists a t>t, such thatIk(t)≤Z()kεfort>t, where

Z()k = δηk

(δ+μ+βh)+ ε<ηk.

On the other hand, from the first equation of system (.), we can get

dSk(t)

dtb(k) –λ(k)MSk(t) –μSk(t), t>T.

By Lemma , we derive thatlim inft→+∞Sk(t)≥λ(kb)(Mk). Then, for arbitrary given

posi-tive constant  <ε<min{/,ε,[λ(bk()kM)+μ]}, there exists at>t, such thatSk(t)≥x()k +ε,

fort>t, where

x()k = b(k)

λ(k)M+μ– ε> .

It follows that

dEk(t)

dtλ(k)ϑ εx ()

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Hence, for arbitrary given positive constant  <ε<min{/,ε,

λ(k)ϑ εx()kmηk

(β+μ+δm) }, there

ex-ists at>t, such thatEk(t)≥y()k +εfort>t, where

y()k =λ(k)ϑ εx

() k +δmηk

(β+μ+δm) – ε> .

Then

dIk(t) dtβhy

()

k – (δ+μ)Ik(t), t>t.

Hence, for arbitrary given positive constant  <ε<min{/,ε, βhy()k

(δ+μ)}, there exists a t>t, such thatIk(t)≥z()k +εfort>t, where

zk()= βhy

() k

(δ+μ)– ε> .

Sinceεis a small positive constant, we have  <x()k <Xk()<ηk,  <y()k <Yk()<ηkand

 <z()k <Zk()<ηk. Let

w(j)=  k

k

ϕ(k) ηk

P(k)zjk(t), W(j)=  k

k

ϕ(k) ηk

P(k)Zjk(t), j= ,  . . . .

From the above discussion, we found that

 <w()≤(t)≤W()<M, t>t.

Again, by system (.), we have

dSk(t)

dtb(k) –λ(k)w

()Sk(t) –μSk(t), t>t .

Hence, for arbitrary given positive constant  <ε<min{/,ε}, there exists at>t

such that

Sk(t)≤Xk()min

!

X()kε,

b(k)

λ(k)w()+μ+ε

"

, t>t.

Thus,

dEk(t)

dtλ(k)W ()X()

k +δmZ ()

k – (β+μ)Ek(t), t>t.

Therefore, for arbitrary given positive constant  <ε<min{/,ε}, there exists at>t,

such that

Ek(t)≤Yk()min

!

Yk()–ε,

λ(k)W()X() k +δmZ

() k

(β+μ) +ε

"

.

It follows that

dIk(t) dtβhY

()

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So, for arbitrary given positive constant  <ε<min{/,ε}, there exists at>t, such

that

Ik(t)≤Zk()min

!

Zk()–ε,

βhYk()

(δ+μ)+ε

"

, t>t.

Turning back to system (.), we have

dSk(t)

dtb(k) –λ(k)W

()Sk(t) –μSk(t), t>t .

Hence, for arbitrary given positive constant  <ε<min{/,ε,(λ(k)bW(k)()+μ)}, there

ex-ists at>t, such thatSk(t)≥x()k +εfort>t, where

x()k =max

!

x()k +ε,

b(k)

λ(k)W()+μ– ε

"

.

Accordingly, one obtains

dEk(t)

dtλ(k)w ()x()

k +δmz ()

k – (β+μ)Ek(t), t>t.

Hence, for arbitrary given positive constant  <ε<min{/,ε,

λ(k)w()x()

kmz

()

k

(β+μ) }, there

exists at>t, such thatEk(t)≥y()k +εfort>t, where

y()k =max

!

y()k +ε,

λ(k)w()x() k +δmz

() k

(β+μ) – ε

"

.

Thus,

dIk(t)

dtβhy ()

k – (δ+μ)Ik(t), t>t.

Hence, for arbitrary given positive constant  <ε<min{/,ε, βhy()k

(δ+μ)}, there exists a t>t, such thatIk(t)≥z()k +εfort>t, where

zk()=max

!

z()k +ε,

βhy()k

(δ+μ)– ε

"

.

(13)

and there exists a large positive integer N so that forh≥N

quences of equation (.), we can obtain the following form

Xk=

From equation (.), a direct computation leads to

Zk=

Further, substituting equation (.) intowandW, respectively, we can obtain

 = 

Subtracting the above two equations, a direct computation leads to

(14)

Obviously, it implies thatw=W. So, kkϕ(ηk)

k P(k)(Zkzk) = , which is equivalent to

zk=Zkfor ≤kn. Then, from equation (.) and equation (.), it follows that

lim

t→Sk(t) =Xk=xk, limt→Ek(t) =Yk=yk, t→limIk(t) =Zk=zk.

Finally, by substitutingw=Winto equation (.), in view of equation (.) and equation (.), it is found thatXk=Sk ,Yk=Ek∞,Zk=Rk . This completes the proof.

5 Conclusions

In this paper, a newSEIRrumor spreading model with demographics on scale-free net-works is presented. Through the mean-field theory analysis, we obtained the basic repro-duction numberRand the equilibria. The basic reproduction numberRdetermines the

existence of the rumor-prevailing equilibrium, and it depends on the topology of the un-derlying networks and some model parameters. Interestingly,Rbears no relation to the

degree-dependent immigrationb(k). WhenR< , the rumor-free equilibriumEis

glob-ally asymptoticglob-ally stable, i.e., the infected individuals will eventually disappear. When

R> , there exists a unique rumor-prevailingE+, and the rumor is permanent,i.e., the

infected individuals will persist and we have convergence to a uniquely prevailing equilib-rium level. The study may provide a reliable tactic basis for preventing the rumor spread-ing.

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.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to this work. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 22 April 2017 Accepted: 8 August 2017

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Figure

Figure 1 Transfer diagram for SEIR rumor

References

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