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A mathematical model described the propagation of information including rumor and truth presented and its properties investigated. We explored exists of the equilibria, local stability and global asymptotical stability, and obtained the propagation threshold of rumor spreading. Numerical simulation is shown to demonstrate our results. Uncertainty and sensitivity analysis shows the importance of the parameters in our model.

Rumors, as daily events have been occurred. With the development of new media such as Internet, mobile phone has broadened the public access to information, optimized the circulation of information, but also exacerbated the spread of rumors. In emergency communication, financial markets fluctuation and all kinds of contagions, rumors play an important role [

The dynamic behavior of the rumor spread has a great similarity with the spread of infectious diseases. Epidemic models have been developed by many researchers [

The remainder of this paper is organized as follows. In Section 2, we derived the model considering the influence of rumors, after which a detailed steady-state analysis is carried out in Section 3, and the threshold of the rumor propagation is obtained. In Section 4, numerical simulation, uncertainty and sensitivity analysis of the model are presented. Finally, Section 5 summarizes this work.

We assume that there are two kinds of spreaders in the whole population which called spreader with rumor and spreader with truth, respectively. As shown in

removed with probability λ + and λ − . The spreaders ( S + \ S − ) spontaneously become the removed at a rate ( γ + \ γ − ) for losing interesting.

According to the dynamic interact as mentioned above, we established the S2IR rumor spreading model based on the above assumptions. The model is described as follows:

I ˙ = Λ − β − S − I − β + S + I − μ 1 I S ˙ + = β + S + I − λ + S − S + − γ + S + − μ 2 S + S ˙ − = β − S − I − λ − S − S + − γ − S − − μ 3 S − R ˙ = λ + S − S + + λ − S − S + − γ − + γ + S + + γ − S − − μ 4 R (1)

The parameters of the model descript as follows:

Λ , the constant recruitment rate of the population;

μ 1 , μ 2 , μ 3 , the natural fade away rate to recover of the ignorants, spreaders with rumor and spreaders with truth respectively;

β + , β − , the force of infection from ignorants to spreaders of rumor, spreaders of truth respectively;

λ + , the fade away rate to recover when a spreader of rumor disseminate the rumor to a spreader of truth;

λ − , the fade away rate to recover when a spreader of truth disseminate the truth to a spreader of rumor;

γ + , γ − , the forgetting rate of spreader of rumor and spreader of truth, respectively.

The initial condition of (1) is given as I ( 0 ) > 0 , S + ( 0 ) ≥ 0 , S − ( 0 ) ≥ 0 , R ( 0 ) ≥ 0 , for the fourth equation of system of (1) is not comprised in the first three equation. Then we can consider the simplified model as follows:

I ˙ = Λ − β − S − I − β + S + I − μ 1 I S ˙ + = β + S + I − λ + S − S + − γ + S + − μ 2 S + S ˙ − = β − S − I − λ − S − S + − γ − S − − μ 3 S − (2)

The initial condition of (2) is given as I ( 0 ) > 0 , S + ( 0 ) ≥ 0 , S − ( 0 ) ≥ 0 . Note μ = m a x { μ 1 , μ 2 , μ 3 } Then we can easily verify that the system of (2) is mathematically well posed in the positive invariant region G = { ( I , S + , S − ) | I > 0 , S + ≥ 0 , S − ≥ 0 , I + S + + S − ≤ Λ / μ } . And all the solutions with I ( 0 ) > 0 , S + ( 0 ) ≥ 0 , S − ( 0 ) ≥ 0 approaches or enter the region G, so it is sufficient to consider solutions in G.

Noting that

R 0 = max { R 01 , R 02 } (3)

where R 01 = Λ β + μ 1 ( γ + + μ 2 ) , R 02 = Λ β − μ 1 ( γ − + μ 3 ) , and

R * = min { R 1 * , R 2 * } (4)

where R 1 * = Λ β + λ − + β + ( μ 2 + γ + ) ( μ 3 + γ − ) β − ( μ 2 + γ + ) 2 + λ − μ 1 ( μ 2 + γ + ) , R 2 * = Λ β − λ + + β − ( μ 3 + γ − ) ( μ 2 + γ + ) β + ( μ 3 + γ − ) 2 + λ + μ 1 ( μ 3 + γ − ) .

Letting the left-hand side of the differential equations of model (2) equal to zero yields the following equations

0 = Λ − β − S − I − β + S + I − μ 1 I 0 = β + S + I − λ + S − S + − γ + S + − μ 2 S + 0 = β − S − I − λ − S − S + − γ − S − − μ 3 S − (5)

Obviously, the system exists a trivial equilibrium E 0 ( I 0 ,0,0 ) where I 0 = Λ / μ 1 . If R 01 > 1 , let S − = 0 , we easily get the rumor-free equilibrium E 1 ( I 1 , S + 1 ,0 ) where I 1 = ( γ + + μ 2 ) / β + , and

S + 1 = Λ / ( γ + + μ 2 ) − μ 1 / β + = μ 1 β + ( R 01 − 1 ) . Similarly, If R 02 > 1 , we get the counter-free equilibrium E 2 ( I 2 ,0, S − 2 ) where I 2 = ( γ − + μ 3 ) / β − , and S − 2 = Λ / ( γ − + μ 3 ) − μ 1 / β − = μ 1 β − ( R 02 − 1 ) .

In the following, we shall study the existence of the positive equilibrium E * ( I * , S + * , S − * ) of system (1). From the second and the third equation of (5), we have S + * = ( β − I * − ( μ 3 + γ − ) ) / λ − , S − * = ( β + I * − ( μ 2 + γ + ) ) / λ + .

Substituting them in the first equation of (5), we have

F ( I ) = A ( I * ) 2 + B I * − C (6)

where A = β + β − ( λ + + λ − ) , B = ( λ + + λ − ) μ 1 − β + λ + ( μ 3 + γ − ) − β − λ − ( μ 2 + γ + ) , and C = Λ λ + λ − . Obviously, A > 0 , C > 0 then exist a unique positive solution I = I * , yield F ( I * ) = 0 . For the S + * > 0 and S − * > 0 , we have F ( ( μ 2 + γ + ) / β + ) < 0 and F ( ( μ 3 + γ − ) / β − ) < 0

From (6) we get that R 1 * > 1 , R 2 * > 1 , namely R * > 0 . Thus, we obtain the following theorem.

Theorem 1. System (2) has the following equilibrium:

1) if R 0 < 1 , system (2) has only the trivial equilibrium E 0 ;

2) if R 01 > 1 , R 02 < 1 , system (2) has a trivial equilibrium E 0 and a truth-free equilibrium E 1 ;

3) if R 01 < 1 , R 02 > 1 , system (2) has a trivial equilibrium E 0 and a rumor-free equilibrium E 2 ;

4) if R 01 < 1 , R 02 > 1 , system (2) has a trivial equilibrium E 0 , a truth-free equilibrium E 1 and a rumor-free equilibrium E 2 ;

5) if R 01 > 1 , R 02 > 1 , R * > 1 , system (2) has a trivial equilibrium E 0 , a truth-free equilibrium E 1 , a rumor-free equilibrium E 2 and coexist equilibrium E * .

The general Jacobian of (2) is given by

J ( 2.2 ) = | − β + S + − β − S − − μ 1 − β + I − β − I β + S + β + I − λ + S − − γ + − μ 2 − λ + S + β − S − − λ − S − β − I − λ − S + − γ − − μ 3 | (7)

Theorem 2. The boundary equilibrium points of (1) have the following local stability properties:

1) if R 0 < 1 , E 0 ( I 0 ,0,0 ) is a stable node, and if R 0 > 1 , E 0 ( I 0 ,0,0 ) is a saddle node.

2) E 1 ( I 1 , S + 1 ,0 ) is a stable if and only if R 1 * > 1 .

3) E 2 ( I 1 ,0, S − 2 ) is a stable if and only if R 2 * > 1 .

4) E * ( I * , S + * , S − * ) is always unstable if it exists.

Proof. 1) At E 0 , we have

J ( 2.2 ) | E 0 = | − μ 1 − Λ β + μ 1 − Λ β − μ 1 0 Λ β + μ 1 − ( γ + + μ 2 ) 0 0 0 Λ β − μ 1 − ( γ − + μ 3 ) | (8)

The eigenvalues of the Equation (8) are λ 1 = − μ 1 , λ 2 = ( Λ β + − μ 1 ( γ + + μ 2 ) ) / μ 1 = ( γ + + μ 2 ) / ( R 01 − 1 ) , λ 3 = ( Λ β − − μ 1 ( γ − + μ 3 ) ) / μ 1 = ( γ − + μ 3 ) / ( R 02 − 1 ) , and therefore the E 0 is an unstable node if R 0 < 1 .

2) At E 1 , the Jacobian of (2) is

J ( 2.2 ) | E 1 = | − Λ β + γ + + μ 2 − ( γ + + μ 2 ) − β − ( γ + + μ 2 ) β + Λ β + γ + + μ 2 − μ 1 0 − ( Λ β 1 − μ 1 ( λ + + μ 2 ) ) γ + β + ( λ + + μ 2 ) 0 0 ( R 1 * − 1 ) ( β − ( λ + + μ 2 ) + γ − μ 1 ) β + | (9)

The eigenvalues are λ 1 = ( 1 − R 1 * ) ( β + ( γ − + μ 3 ) ) / β 2 , λ 2,3 = − μ 1 ( R 01 ± R 01 2 − ( R 01 − 1 ) ( λ + + μ 2 ) / μ 1 2 , obviously they all have negative real parts and E 1 is local stability if R 1 * > 1 and R 01 > 1 .

In a similar fashion, we can conclude that if R 2 * > 1 and R 02 > 1 . E 2 is local stability.

For the positive equilibrium E * ,

| Λ I * − β + I * − β − I * β − I * − ( λ − + μ 3 ) β + β − 0 − ( β − I * − ( λ − + μ 3 ) ) γ + β − ( β + I * − ( λ + + μ 2 ) ) β − β + − ( β + I * − ( λ + + μ 2 ) ) γ − β + 0 | (10)

The characteristic polynomial of (10) can be calculated as follows is

λ 3 + Λ I * λ 2 + ( β + 2 S + * I * + β − 2 S − * I * − λ + λ − S + * S − * ) λ − f = 0 (11)

which f = ( β + β − ( λ + + λ − ) S + * S − * ) + Λ S + * S − * λ + λ − I * , the eigenvalues satisfied

λ 1 λ 2 λ 3 = f > 0 , It can be obtained by the Routh-Hurwitz criterion: It exists at least one eigenvalue of the polynomial equation has positive real part, then E * is always unstable.,

Next, we will consider the global stability of quiet equilibrium E 0 of system (2).

Theorem 3. The quiet equilibrium E 0 = ( I 0 ,0,0 ) is globally asymptotically stable provided that R 0 < 1 .

Proof. Define the Lyapunov function

V ( I , S + , S − ) = V 1 ( I , S + , S − ) + V 2 ( I , S + , S − ) (12)

where V 1 ( I , S + , S − ) = I − I 0 − I 0 l n I I 0 , V 2 ( I , S + , S − ) = S + + S − , then differentiating V 1 with respect to t along solutions of system (2) gives

d V 1 d t = ( 1 − I 0 I ) ( Λ − β − S − I − β + S + I − μ 1 I ) = ( 1 − I 0 I ) ( − β − S − I − β + S + I ) + μ 1 I 0 ( 1 − I 0 I ) ( 1 − I I 0 ) ≤ ( 1 − I 0 I ) ( − β − S − I − β + S + I ) = − β − S − I − β + S + I + β − S − I 0 + β + S + I 0 (13)

d V 2 d t = β + S + I − λ + S − S + − γ + S + − μ 2 S + + β − S − I − λ − S − S + − γ − S − − μ 3 S − (14)

Then, we have

d V d t = d V 1 d t + d V 2 d t ≤ β + S + I 0 − λ + S − S + − ( γ + + μ 2 ) S + + β − S − I 0 − λ − S − S + − ( γ − − μ 3 ) S − = ( γ + + μ 2 ) ( R 01 − 1 ) S + + ( γ − + μ 3 ) ( R 02 − 1 ) S − − ( λ + + λ − ) S − S + ≤ ( γ + + μ 2 ) ( R 0 − 1 ) S + + ( γ − + μ 3 ) ( R 0 − 1 ) S − − ( λ + + λ − ) S − S + (15)

Therefore, when R 0 < 1 , d V d t ≤ 0 , and the equality holds only for I = I 0 ,

S + = 0 and S − = 0 , According to the LaSalle invariant principle, we have l i m t → ∞ I ( t ) = I 0 , l i m t → ∞ S + ( t ) = 0 and l i m t → ∞ S − ( t ) = 0 . Thus, the proof is completed.,

We have formulated a S2IR rumor-spread model. We have established some threshold conditions for the steady state. To illustrate the analytical results, we do some numerical simulations. From

I ˙ = Λ − β − S − I − β + S + I − 0.3 I S ˙ + = β + S + I − 0.38 S − S + − 0.4 S + − 0.3 S + S ˙ − = β − S − I − 0.1 S − S + − 0.4 S − − 0.3 S − (16)

Let Λ = 0.2 , β + = 0.705 , β − = 0.725 , then R 0 = 0.6905 < 1 . By theorem 1.(a), theorem 2.(a) and theorem 3, system (16) has only the trivial equilibrium E 0 ( 0.6667,0,0 ) and it is globally asymptotically stable (see

Let Λ = 0.312 , β + = 0.856 , β − = 0.656 . We have R 01 = 1.272 > 1 , R 02 = 0.974 < 1 , R 1 * = 1.303 > 1 , R 2 * = 0.7996 < 1 , It presents the system (16) has have two equilibria, E 0 ( 1.04,0,0 ) and E 1 ( 0.8178,0.095,0 ) where E 0 is unstable but E 1 is stable according to theorem 1.(b) and theorem 2.(a)-2.(b) (see

two equilibria, E 0 ( 1.077,0,0 ) and E 2 ( 0.819,0,0.11 ) where E 0 is unstable but E 2 is stable according to theorem 1.c and theorem 2.a, 2.c (see

Let Λ = 2.1 , β + = 0.6 , β − = 0.9 , then R 0 = 9 > 0 , R 1 * = 0.9091 < 1 , R 2 * = 3.101 > 1 . By theorem 1.(d), system (16) has three equilibria E 0 ( 7,0,0 ) , E 1 ( 1.167,2.5,0 ) and E 2 ( 0.778,0,2.667 ) and E 0 , E 1 is unstable, but E 2 is stable (see

R 0 = 10.13 > 0 , R * = R 1 * = 1.566 > 1 , R 2 * = 2.474 > 1 . By theorem 1.(e), system (16) has three equilibria E 0 ( 9.33,0,0 ) , E 1 ( 0.921,3.605,0 ) , E 2 ( 0.972,0,3.583 ) and E * ( 1.244,1.956,0.646 ) , which E 0 and E * are not stable but E 1 and

E 2 are bistable. The phase portrait is represented in

In this part, uncertainty and sensitivity analysis, based on Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) scheme, will explore the dependence of R 0 , R 1 * and R 2 * due to the change of input parameters in the estimation of the uncertainty. Each input parameters is sampling 1000 times. A uniform distribution function was used and the variation ranges of all parameters are given in

Descriptions | Symbols | Values (range) | Reference |
---|---|---|---|

The constant recruitment rate of the population | Λ | 1.5 (0.2, 2.8) | Assumed |

The natural fade away rate of the ignorants | μ 1 | 0.5 (0.1, 0.9) | [ |

The natural fade away rate of spreaders with rumor | μ 2 | 0.5 (0.1, 0.9) | [ |

The natural fade away rate of spreaders with truth | μ 3 | 0.5 (0.1, 0.9) | [ |

The force of infection from ignorants to rumor-monger | β + | 0.8 (0.6, 1) | [ |

The force of infection from ignorants to truth-spreader | β − | 0.8 (0.6, 1) | [ |

The fade away rate of rumor-mongers | λ + | 0.2 (0.01, 0.39) | Assumed |

The fade away rate of truth-spreaders | λ − | 0.2 (0.01, 0.39) | Assumed |

The forgetting rate of spreaders of rumor | γ + | 0.3 (0.1, 0.5) | [ |

The forgetting rate of spreaders of truth | γ − | 0.3 (0.1, 0.5) | [ |

Parameter | R_{0} | R 1 * | R 2 * | |||
---|---|---|---|---|---|---|

PRCC | p-value | PRCC | p-value | PRCC | p-value | |

Λ | 0.0867 | 0.0064 | 0.2224 | 0 | 0.3448 | 0 |

μ 1 | −0.0379 | 0.2348 | −0.0601 | 0.0592 | −0.1306 | 0 |

μ 2 | 0.0186 | 0.5602 | −0.7155 | 0 | 0.4062 | 0 |

μ 3 | −0.0751 | 0.0183 | 0.3442 | 0 | −0.7685 | 0 |

β + | −0.0378 | 0.2349 | 0.5305 | 0 | −0.5457 | 0 |

β − | 0.0721 | 0.0235 | −0.489 | 0 | 0.6155 | 0 |

λ + | 0 | 0 | −0.1302 | 0 | 0.772 | 0 |

λ − | 0 | 0 | 0.7415 | 0 | −0.1256 | 0.0001 |

γ + | 0.038 | 0.2324 | −0.6316 | 0 | 0.3833 | 0 |

γ − | −0.0695 | 0.0291 | 0.3296 | 0 | −0.6966 | 0 |

The PRCC results which illustrate the dependence on each parameters of R 0 , R 1 , R 2 respectively. We considered | PRCC | ≥ 0.4 as indicating that the high correlation between input parameters and output variables, 0.2 ≤ | PRCC | < 0.4 as moderate correlations and | PRCC | < 0.2 as no relations.

The results of the simulation of PRCCs are shown in

Figures 6(a)-(c) shows the LHSs frequency distributions. The mean value for R 0 , R 1 and R 2 are M R 0 = 2.4214 , M R 1 = 1.4797 , M R 2 = 1.4793 respectively. The standard deviation with R 0 is 0.6642 and greater than with R 1 and R 2 (0.3120, 0.3037 respectively) which indicates that the derived frequency distribution for R 0 is dispersed than the distributions for R 1 and R 2 .

Rumor propagation has been investigated through different types of mathematical models. In our study, we consider a rumor propagation model with truth-spreading and determine the threshold which governs the dynamics of the system. On one hand, the propagation of rumor and truth is mutually

exclusive. Therefore, the appropriate increase in the spread of truth is conducive to the elimination of the spread of rumors. On the other hand, the spread of rumor and truth can coexist for a long time under certain conditions. At this point, the improvement of the spread of the truth is also conducive to the spread of rumors in a certain range. This requires the government and related organizations to further develop the corresponding work.

The research has been supported by The Social Science Planning Projects of Jiangxi Province (14XW08) and The 12th Five-year Education Scientific Planning Project of Jiangxi Province (15ZD3LYB031).

Liu, Y.J., Zeng, C.M. and Luo, Y.Q. (2018) Dynamics of a New Rumor Propagation Model with the Spread of Truth. Applied Mathematics, 9, 536-549. https://doi.org/10.4236/am.2018.95038