In this paper, the SECIR rumor spreading model is formulated and analyzed, in which the social education level and the counterattack mechanism are taken into consideration. The results show that improving education level and increasing the ratio of counter are effective in reducing the risk of rumor propagation and enhanc ing the resistance to rumor propagation.
Rumor is a kind of social phenomenon that an unverified account or explanation of events spreads on a large-scale in a short time through people’s communication [
The standard model of rumor spreading is the Daley-Kendal (DK) model [
In this paper, inspiring of Zan et al. [
The remaining part of the paper is organized as follows. We formulate the propagation mechanism of the SECIR model in a social network, and derive a system of nonlinear ordinary differential equations that describe dynamics of rumor spreading process in Section 2. In Section 3, we analyze the steady-state of the SECIR model. We give some of the conclusions in the last section.
We study the SECIR model in a closed homogeneously mixed population that we differentiate into five distinct classes: the rumor-mongers (spreader, S), those who are spreading the rumor, the non-educated ignorants individuals class (Ignorant, I), the people who never heard the rumor, the educated ignorants individuals class (Educatee, E), the people who never heard the rumor, but have more sophisticated behaviours with the non-educated ignorant individuals when they encountered the spreader, the counterattack class (Counter, C), those who do not agree but refute the rumor, and persuade others to agree with him (refute the rumor), and the stiflers class (Recovered, R), the ones who heard the rumor but have lost interest in disseminating it. For simplicity, we refer to the rumor spreading model as the SECIR model.
According to the MK model, we assume the rumor spreads by directed contact of the spreads with others in the population, and the contacts between rumor-mongers and the rest of the population are governed by the following dynamics (As shown in
・ Whenever a non-educated ignorant contact a spreader, the ignorant will be a relatively large probability ( β 1 ) into the spreader, but also with a smaller probability ( α 1 ) of transition to a stifler;
・ Whenever an educated ignorant contact a spreader, the ignorant will be one of the three class: spreader or stifler analog the non-educated ignorant, and he can be evolved into counterattack, and the change probability is β 2 , α 2 and θ , respectively;
・ Whenever a spreader contact a spreader, a stifler or a counter, the spreader will be into stifler with the probability g, g and η , respectively;
Let I ( t ) , E ( t ) , C ( t ) , C ( t ) , and R ( t ) respectively represent the density of the corresponding compartment in the total population. Namely that we have I ( t ) + E ( t ) + C ( t ) + S ( t ) + R ( t ) = 1 . Note that the resistance to the rumor of the educated ignorants more than the non-educated ignorants, we assume that
β 1 > β 2 , α 1 > α 2 and η > g . (1)
In accordance with the above rules, the mean-field equations of the SECIR model can be described as follows:
d I ( t ) d t = − ( β 1 + α 1 ) k ¯ I ( t ) S ( t ) d E ( t ) d t = − ( β 2 + α 2 + θ ) k ¯ E ( t ) S ( t ) d C ( t ) d t = θ k ¯ E ( t ) S ( t ) d S ( t ) d t = β 1 k ¯ I ( t ) S ( t ) + β 2 k ¯ E ( t ) S ( t ) − g k ¯ S ( t ) ( S ( t ) + R ( t ) ) − η k ¯ S ( t ) C ( t ) d R ( t ) d t = α 1 k ¯ I ( t ) S ( t ) + α 2 k ¯ E ( t ) S ( t ) + g k ¯ S ( t ) ( S ( t ) + R ( t ) ) + η k ¯ S ( t ) C ( t ) (2)
We assume that all in the population are I or E but only one spreader at the beginning of the rumor spreading, and the ratio of E to the sum of I and E is ϵ when t = 0 . Namely, when t = 0 , the initial condition of rumor spreading is given as follows:
I ( 0 ) = N − 1 N ( 1 − ϵ ) , E ( 0 ) = N − 1 N ϵ , S ( 0 ) = 1 N , C ( 0 ) = 0 , R ( 0 ) = 0. (3)
Note that for an ignorant, he/she can be a spreader or stifler, so we have β 1 + α 1 < 1 , and 1 − β 1 − α 1 is the probability that no one tells him/her the rumor. For the same reason, β 2 + α 2 + θ < 1 is considered.
We then postulate that the number of individuals in the I class that have heard about the rumor is the same as the number of individuals in the E class that have heard about the rumor. This simply translates to
β 1 + α 1 = β 2 + α 2 + θ = M (4)
From the first and the second equation of (2), we have
d E ( t ) d I ( t ) = E ( t ) I ( t ) (5)
Solve the above differential equation with the initial conditions (3), we obtain
E ( t ) I ( t ) = E ( 0 ) I ( 0 ) = ϵ 1 − ϵ (6)
From the first and the third Equation (2), we obtain
d C ( t ) d t = − θ α 2 + β 2 + θ d E ( t ) d t
With the initial conditions (3), we can derive the relational expression between C ( t ) and E ( t ) by separation of variable,
C ( t ) = θ α 2 + β 2 + θ ( ϵ − E ( t ) ) (7)
From (6) and (7), we have
S ( t ) + R ( t ) = 1 − ( I ( t ) + C ( t ) + E ( t ) ) = 1 − 1 − ϵ ϵ E ( t ) − θ α 2 + β 2 + θ ( ϵ − E ( t ) ) − E ( t ) = ( 1 − θ ϵ α 2 + β 2 + θ ) − ( 1 ϵ − θ α 2 + β 2 + θ ) E ( t ) (8)
Note that τ = g ϵ + ( η − g ) θ α 2 + β 2 + θ and β = 1 − ϵ ϵ β 1 + β 2 , then we have
d S ( t ) d t = β 1 k ¯ I ( t ) S ( t ) + β 2 k ¯ E ( t ) S ( t ) − g k ¯ S ( t ) ( S ( t ) + R ( t ) ) − η k ¯ S ( t ) C ( t ) = β 1 k ¯ 1 − ϵ ϵ E ( t ) S ( t ) + β 2 k ¯ E ( t ) S ( t ) − η k ¯ S ( t ) θ α 2 + β 2 + θ ( ϵ − E ( t ) ) − g k ¯ S ( t ) ( ( 1 − θ ϵ α 2 + β 2 + θ ) − ( 1 ϵ − θ α 2 + β 2 + θ ) E ( t ) ) = ( 1 − ϵ ϵ β 1 + β 2 + g ϵ + ( η − g ) θ α 2 + β 2 + θ ) k ¯ E ( t ) S ( t ) − ( g + θ ( η − g ) ϵ α 2 + β 2 + θ ) k ¯ S ( t ) = ( β + τ ) k ¯ E ( t ) S ( t ) − ϵ τ k ¯ S ( t ) (9)
From the second equation of (2), therefore (9) becomes
d S ( t ) d t = ϵ τ E ( t ) − ( β + τ ) M d E ( t ) d t
Solving the differential equations above by the method of separation of variables, we have
S ( t ) = ϵ τ M ( ln E ( t ) − ln ϵ ) − β + τ M ( E ( t ) − ϵ ) (10)
By the second equation of (2), it is easy to see that d E ( t ) d t < 0 and E ( t ) are monotonically decreasing and continuous function. Let d S ( t ) d t = 0 , we obtain E ( t ) = ϵ τ β + τ . It is easy to see that, the peak value of spreader is
S max = ϵ β M + ϵ τ M ln ( τ β + τ ) (11)
d R ( t ) d t = α 1 k ¯ I ( t ) S ( t ) + α 2 k ¯ E ( t ) S ( t ) + g k ¯ S ( t ) ( S ( t ) + R ( t ) ) + η k ¯ S ( t ) C ( t ) = α 1 k ¯ 1 − ϵ ϵ E ( t ) S ( t ) + α 2 k ¯ E ( t ) S ( t ) + η k ¯ S ( t ) θ α 2 + β 2 + θ ( ϵ − E ( t ) ) + g k ¯ S ( t ) ( ( 1 − θ ϵ α 2 + β 2 + θ ) − ( 1 ϵ − θ α 2 + β 2 + θ ) E ( t ) ) = ( 1 − ϵ ϵ α 1 + α 2 − g ϵ − ( η − g ) θ α 2 + β 2 + θ ) k ¯ E ( t ) S ( t ) + ( g + θ ( η − g ) ϵ α 2 + β 2 + θ ) k ¯ S ( t ) = ( α − τ ) k ¯ E ( t ) S ( t ) + ϵ τ k ¯ S ( t ) (12)
where α = 1 − ϵ ϵ α 1 + α 2 . From the second equation of (2), therefore (12) becomes
d R ( t ) d t = ( τ − α α 2 + β 2 + θ − ϵ τ α 2 + β 2 + θ 1 E ( t ) ) d E ( t ) d t (13)
Solving the above differential equations, we get
R ( t ) = α − τ α 2 + β 2 + θ ( ϵ − E ( t ) ) + ϵ τ α 2 + β 2 + θ ( ln ϵ − ln E ( t ) ) (14)
It is easy know that S ∞ = 0 , let t → ∞ , then (8) becomes
R ∞ = ( 1 ϵ − θ α 2 + β 2 + θ ) ( ϵ − E ∞ ) (15)
Let t → ∞ , Substituting (15) into (14), it becomes
( 1 ϵ − α − τ + θ α 2 + β 2 + θ ) ( ϵ − E ∞ ) = ϵ τ α 2 + β 2 + θ ( ln ϵ − ln E ∞ ) (16)
α + β = 1 − ϵ ϵ ( β 1 + α 1 ) + ( β 2 + α 2 ) = 1 − ϵ ϵ ( β 2 + α 2 ) + ( β 2 + α 2 ) = 1 ϵ ( β 2 + α 2 + θ ) − θ (17)
Solve from (17), we get ϵ = β 2 + α 2 + θ β + α + θ , substitute into (16), we have
( ϵ − E ∞ ) = ϵ τ β + τ ( ln ϵ − ln E ∞ ) (18)
Let A ( t ) = I ( t ) + E ( t ) = 1 − ϵ ϵ E ( t ) + E ( t ) = E ( t ) / ϵ , from (18) we get the final size
A ∞ = τ β + τ ln A ∞ + 1 (19)
Theorem 1. For 0 < σ < 1 , the equation x = σ ln x + 1 has two solutions, x = 1 and a nontrivial solution x 1 , where 0 < x 1 < σ .
Proof. Obviously x = 1 is a solution of x = σ ln x + 1 .
Let f ( x ) = x − σ ln x − 1 , and take the derivative of f ( x ) with respect to x: f ′ ( x ) = 1 − σ / x , f ″ ( x ) = σ / x 2 > 0 .
Let f ′ ( x ) = 0 , we obtain the unique minimum point x = σ , and the function f ( x ) is a convex function, we have f ( σ ) = σ − σ ln σ − 1 < σ + σ ( 1 / σ − 1 ) − 1 = 0 , and f ( 0 + ) = ∞ . According to the Mean Value Theorem, f ( x ) have a nontrivial solution x 1 , where 0 < x 1 < σ . ,
Theorem 2. If the parameters are satisfied (1), (3) and (4), we have
(1) S max is decrease with ϵ , if the other parameters keep constant.
(2) S max is increase with β 1 , β 2 but decrease with α 1 , α 2 , if the other parameters keep constant.
(3) S max is decrease with g , η , if the other parameters keep constant.
Proof. (1) From (11), differential with ϵ , we have
d S max d ϵ = − β 1 − β 2 M + ϵ τ M ( ln τ β + τ ) ϵ + ( ϵ τ ) ′ ϵ M ln τ β + τ = − β 1 − β 2 M + ϵ τ M ( − β g + τ β 1 M ϵ ( β + τ ) ) + ( η − g ) θ M 2 ln τ β + τ
≤ − β 1 − β 2 M + ϵ τ M ( − β g + τ β 1 M ϵ ( β + τ ) ) + ( η − g ) θ M 2 ( τ β + τ − 1 ) = − ( ( 1 − ϵ ) β 1 + β 2 ϵ ) ( β 1 − β 2 ) M ( β + τ ) ϵ (20)
Therefore, d S max d ϵ < 0 , which implies that S max decreases as ϵ increases.
(2) Since that τ = g ϵ + ( η − g ) θ α 2 + β 2 + θ and β = 1 − ϵ ϵ β 1 + β 2 , we have τ ′ β i = 0 , τ ′ α i = 0 , i = 1 , 2 , β ′ β i > 0 , β ′ α i < 0 . Taking the derivative of (11), with respect to β i , we have d S max d β i = ϵ M β ′ β i − ϵ τ M β ′ β i β + τ = ϵ M β β + ϵ β ′ β i > 0 . It is easy to see that S max increases as β i ( i = 1 , 2 ) increases.
By the same way, we can proof that S max is decrease with α 1 , α 2 .
(3) Similarly (2), we have β ′ η = 0 , β ′ g = 0 , τ ′ x > 0 , ( x = η or g ) , then
d S max d x = ϵ M ln τ β + τ τ ′ x + ϵ τ M ( τ ′ x τ − τ ′ x β + τ ) = ( ϵ M ln τ β + τ + ϵ β M ( β + τ ) ) τ ′ x < ( ϵ M ( τ β + τ − 1 ) + ϵ β M ( β + τ ) ) τ ′ x = 0 (21)
which means that S max is decrease with g , η The proof is complete. ,
Theorem 3. If the parameters are satisfied (1), (3) and (4), we have
(1) the other parameters keep constant, the final state A ∞ is increased with ϵ .
(2) the other parameters keep constant, the final state A ∞ is decreased with β 1 , β 2 but decrease with α 1 , α 2 .
(3) the other parameters keep constant, the final state A ∞ is increased with g , η .
Proof. Let σ = τ β + τ , from (19) we have
d A ∞ d χ = d σ d χ ln A ∞ + σ 1 A ∞ d A ∞ d χ = A ∞ ln A ∞ A ∞ − σ d σ d χ (22)
From Theorem 1, 0 < A ∞ < σ , it is obviously that A ∞ ln A ∞ A ∞ − σ > 0 , so, if we know that the plus-minus sign of d σ d χ , then the plus-minus sign of d A ∞ d χ can be determined, and furthermore, the monotonicity between A ∞ and parameters χ can be proved.
(1)
d σ d ϵ = τ ′ ϵ β − τ β ′ ϵ ( β + τ ) 2 = β 1 τ − g β ϵ 2 ( β + τ ) 2 = ( β 1 − β 2 ) g + ( η − g ) θ M β 1 ϵ 2 ( β + τ ) 2 (23)
so d σ d ϵ > 0 , that is, σ increases as ϵ increases, and also implies that A ∞ increases as ϵ increases.
(2)
d σ d β i = − τ β ′ β i ( β + τ ) 2 < 0 , ( i = 1 , 2 ) (24)
similarly, we can get d σ d α i > 0 . and so A ∞ d β i < 0 , A ∞ d α i < 0 , where i = 1 , 2 , which proves (2).
(3)
d σ d x = τ ′ x β ( β + τ ) 2 , x ∈ { η , g } (25)
then, d σ d x > 0 , it means that A ∞ increases as η ( g ) increases. The proof is complete.
We assume N = 10 5 , the average degree of network k ¯ = 10 , and the initial condition of the model follows equation of (2).
always decreases, and eventually evolve to a stable value.
Figures 3-5 show how the densities of spreaders change over time for different system parameters include ϵ , α 1 , β 1 , α 2 , β 2 , g and η , and the change is consistent with theorem 2. It is interest that the higher parameter β 1 , β 2 , the earlier the outbreak, the larger the peak of the outbreak, but the shorter the outbreak period (
In this paper, considering the social education level and the counterattack mechanism,
we analyze the dynamics of rumor propagation. The results of simulations show that improving education level and increasing the ratio of counter are effective in reducing the risk of rumor propagation and enhancing the resistance to rumor propagation.
The research has been supported by The National Natural Science Foundation of China (11561004) and The 12th Five-year Education Scientific Planning Project of Jiangxi Province (15ZD3LYB031).
The authors declare no conflicts of interest regarding the publication of this paper.
Liu, Y.J., Zeng, C.M. and Luo, Y.Q. (2019) Steady-State Analysis of SECIR Rumor Spreading Model in Complex Networks. Applied Mathematics, 10, 75-86. https://doi.org/10.4236/am.2019.103007