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In this study, we proposed a deterministic mathematical model that attempts to explain the propagation of a rumor using epidemiological models approach. The population is divided into four classes which consist of ignorant individuals,
*I*(t), spreaders targeting community through media,
*M*(t), spreaders targeting community through verbal communication,
*G*(t) and stiflers,
R(t). We explored existence of the equilibria and analyzed its stability. It was established that rumour-free equilibrium E
_{0 }is locally asymptotically stable if R
_{0}<1; meaning rumor can seize spreading in a population, and unstable if R
_{0}>1 leads to new rumor spreading in the population. Numerical simulations of the dynamic model are carried out on the system to confirm the analytical results. We see that the dynamics of rumor spreading show similar behavior to that found in the dynamics of infectious diseases except that the spread depends on the classes of spreader.

The propagation of rumors is a typical form of social interaction that exerts a powerful influence on human affairs. Rumors represent unproven expositions about or interpretations of news, events, or problems that are of public interest. Because rumors are unconfirmed information, it is hard to determine whether they are true or false [

In recent years, online social media is growing rapidly. Social media provides a convenient communication scheme for people. Meanwhile, the scheme enables unreliable sources to spread large amounts of unverified information among people [

At the beginning, mathematical models for the rumors were considered merely speculative and imprecise, but for the fact that rumor spreading is now seen like the transmission of disease [

Rumor can be viewed as an infection of the mind [

Different possible behaviors in the dynamics of rumor spreading were studied by [

Rumor propagation through different types of mathematical models was investigated by [

system.

The dynamics of rumor model in complex heterogonous networks was analyzed by [

Following the classical assumption on the variables described in

1) Recruitment into the ignorant compartment is in constant rate,

2) There is no movement between the classes of spreaders,

3) Ignorant individual become spreader after interaction with spreader through media or spreader through verbal communication,

Variable and parameter | Description |
---|---|

I ( t ) M ( t ) G ( t ) R ( t ) θ β 1 β 2 α 1 α 2 σ π μ | Ignorant Spreader through media Spreader through verbal communication Communication Stifler Probability of ignorant to become stifler Effective contact rate between the ignorant and the spreaders through media Effective contact rate between the ignorant and the spreaders through verbal communication Rate at which spreaders through media become stiflers Rate at which spreaders through verbal communication become stiflers Proportion of I that interact with M and G and choose not to spread Recruitment rate into the ignorant population Death rate |

4) The probability of an ignorant to become a spreader through media or verbally are equal,

5) The spreading process are mutually exclusive event,

6) Spreaders through media and verbal communications become stifler at the rate α 1 and α 2 respectivelly,

7) Proportion of ignorant will interact with the spreaders at the rate σ and become stifler.

From the model diagram (

Thus, the differential equations for the deterministic model are as follow:

d I d t = π − μ I − θ 2 β 1 I M − θ 2 β 2 I G − ( M + G ) σ I (1.1)

d M d t = θ 2 β 1 I M − μ M − α 1 M (1.2)

d G d t = θ 2 β 2 I G − μ G − α 2 G (1.3)

d R d t = ( M + G ) σ I + α 1 M + α 2 G − μ R (1.4)

N = I + M + G + R (1.5)

Since the model (1.1) to (1.4) monitors human populations, all the variables and the associated parameters are non-negative at all time. It is important to show that the model variables of the model remain non negative for all non-negative initial conditions.

Lemma 1: The region D = { ( I , M , G , R ) ∈ R + 4 : I + M + G + R ≤ π μ } is positively invariant and attract all solutions in R + 4 .

Proof:

Adding all the equations from (1.1) to (1.4), gives the rate of change of the total human population

d N d t = π − μ N

Since d N d t = π − μ N whenever N ( t ) > π μ , then d N d t < 0 , implying d N d t is bounded by π − μ N .

Thus, a standard comparison theorem by [

N ( t ) ≤ N ( 0 ) e − μ t + π μ ( 1 − e − μ t )

In particular, N ( t ) ≤ π μ if N ( 0 ) ≤ π μ . Thus, R is positively invariant (i.e. all solution in D remain in D for all time). Furthermore, if N ( t ) > π μ then either the solution enters R in finite time or N ( t ) approaches π μ and the spreader variables M and G approaches zero. Hence, D is attracting (i.e. all solution in R + 4 eventually approach, enter or stay in D). Therefore, the model is epidemiologically and mathematically well posed since all the variables remain non-negative for all t ≥ 0 . Hence it is sufficient to study the dynamics of the system (1.1) to (1.4) in D.

In this section, we discuss the existence and uniqueness of Rumor Free Equilibrium (RFE) of the model and its analysis. The model Equations (1.1) to (1.4) has an RFE given by

E o = ( I * , M * , G * , R * ) = ( π μ , 0 , 0 , 0 ) .

The local stability of RFE given will be investigated using the next generation matrix method. We calculate the next generation matrix for the system of the question (1.1) to (1.4) by enumerating the number of ways that: 1) new spreaders arise 2) number of ways that individuals can move but only one way to create a spreader. So, let

F = rate of appearance of new spreaders into the compartment and,

V = rate of transfer into (out) of compartment

F = ( θ 2 β 1 I * 0 0 θ 2 β 2 I * )

V = ( μ + α 1 0 0 μ + α 2 )

Hence the NGM with large domain is two dimensional and is given by F V − 1

F V − 1 = ( θ β 1 π 2 μ ( μ + α 1 ) 0 0 θ β 2 π 2 μ ( μ + α 2 ) ) (1.6)

The dominant eigenvalue of (1.6) is equal to R o , therefore we evaluate the characteristic equation of (1.6) by using | ( F V − 1 ) − λ I | = 0 , which gives

R o = R o m + R o v

R o = π θ [ β 1 ( μ + α 2 ) + β 2 ( μ + α 1 ) ] 2 μ ( μ + α 1 ) ( μ + α 2 )

where

R o m = θ β 1 π 2 μ ( μ + α 1 )

R o v = θ β 2 π 2 μ ( μ + α 2 )

The jacobian of (1.1) to (1.4) at the equilibrium point E o = ( π μ , 0 , 0 , 0 ) is

= ( − μ − ( θ β 1 + σ μ ) π 2 μ − ( θ β 2 + σ μ ) π 2 μ 0 0 π θ β 1 2 μ − μ − α 1 0 0 0 0 π θ β 2 2 μ − μ − α 2 0 0 α 1 + σ π μ α 2 + σ π μ − μ ) (1.7)

Now we try to calculate the eignvalues of (1.7) by finding the characteristic equation using the formula | J E − λ I | = 0

| ( − μ − λ ) − ( θ β 1 + σ μ ) π 2 μ − ( θ β 2 + σ μ ) π 2 μ 0 0 ( π θ β 1 2 μ − μ − α 1 − λ ) 0 0 0 0 ( π θ β 2 2 μ − μ − α 2 − λ ) 0 0 α 1 + σ π μ α 2 + σ π μ ( − μ − λ ) | = 0 (1.8)

Solving (1.8), we have

λ 1 = − μ

λ 2 = ( R o M − 1 ) ( μ + α 1 )

λ 3 = ( R o V − 1 ) ( μ + α 2 )

λ 4 = − μ

Theorem 1: The rumor-free equilibrium of the model equation (1.1) and (1.4) given by E o , is locally asymptotically stable if R o < 1 and unstable if R o > 1 . Thus this theorem 1 implies that for any given rumor in a population, it can be eliminated when R o < 1 .

Proof

Having λ 1 and λ 4 to be negative, we also see that λ 2 and λ 3 are both negative too when R o m < 1 and R o v < 1 . Since all the eigenvalues of (1.8) have negative real parts when R o m < 1 and R o v < 1 , we conclude that rumor-free equilibrium is locally asymptotically stable.

Stability Analysis of the Rumor Endemic Equilibrium

When rumor persists in a population (i.e. at least M ≠ 0 or G ≠ 0 ), the model question (1.1) to (1.4) has two equilibrium points denoted by

E 1 = ( I Λ , M Λ , G Λ , R Λ ) and E 2 = ( I Λ Λ , M Λ Λ , G Λ Λ , R Λ Λ )

called rumor verbal-endemic equilibrium and rumor media-endemic equilibrium point. For the existence and uniqueness of E 1 and E 2 their coordinate has to satisfy the following I Λ > 0 , M Λ > 0 G Λ > 0 , R Λ > 0 and I Λ Λ > 0 , M Λ Λ > 0 , G Λ Λ > 0 , R Λ Λ > 0 respectively.

Equating (1.1) to (1.4) to zero, we obtained

π − μ I − θ 2 β 1 I M − θ 2 β 2 I G − ( M + G ) = 0 (1.9)

θ 2 β 1 I M − μ M − α 1 M = 0 (1.10)

θ 2 β 2 I G − μ G − α 2 G = 0 (1.11)

( M + G ) σ I + α 1 M + α 2 G − μ R = 0 (1.12)

From (1.10), we have

I Λ = 2 ( μ + α 1 ) θ β 1 (1.13)

M Λ = 0 (1.14)

Substituting (1.13) and (1.14) in (1.9), we have

G Λ = π θ β 1 − 2 μ ( μ + α 1 ) β 2 θ ( μ + α 1 ) + 2 σ ( μ + α 1 ) (1.15)

Using (1.13), (1.14) and (1.15) in (1.12), we obtained

R Λ = ( 2 σ ( μ + α 1 ) + α 1 θ β 1 θ β 1 μ ) ( π θ β 1 − 2 μ ( μ + α 1 ) β 2 θ ( μ + α 1 ) + 2 σ μ ( μ + α 1 ) ) (1.6)

Thus E 1 = ( I Λ , M Λ , G Λ , R Λ )

Local Stability of rumor verbal-endemic equilibrium

We used the Jacobian Stability approach to prove the stability of the rumor verbal-endemic equilibrium.

The Jacobian of (1.9) to (1.12) at the equilibrium point E 1 = ( I Λ , M Λ , G Λ , R Λ ) is

( − μ − θ 2 β 1 M Λ − θ 2 β 2 G Λ − ( M Λ + G Λ ) σ − θ 2 β 1 I Λ − σ I Λ − θ 2 β 2 I Λ − σ I Λ 0 θ 2 β 1 M Λ θ 2 β 1 I Λ − μ − α 1 0 0 θ 2 β 2 G Λ 0 θ 2 β 2 I Λ − μ − α 2 0 ( M Λ + G Λ ) σ α 1 + σ I Λ α 2 + σ I Λ − μ ) (1.17)

If we evaluate (1.17) at E 1 and find the eigenvalue using characteristic equation | J ( E 1 ) − λ I | = 0 , we will obtain

λ 1 = − μ − ( θ β 2 + 2 σ 2 ) ( π θ β 1 − 2 μ ( μ + α 1 ) θ β 2 ( μ + α 1 ) + 2 σ σ ( μ + α 1 ) )

λ 2 = 0

λ 3 = β 2 ( μ + α 1 ) β 1 − μ − α 2

λ 4 = − μ

From the above questions, it is observed clearly that only λ 1 < 0 and λ 4 < 0 , therefore we conclude that rumor verbal-endemic equilibrium is unstable since some of the eigenvalue are greater than zero. Furthermore, once a rumor spreads, the truth is at risk of being distorted in the public sphere, therefore it must spread first before it decline naturally or by counter rumor.

Rumor media-endemic equilibrium

To obtain the rumor media endemic equilibrium we use the same approach as in rumor verbal-endemic equilibrium.

From (1.11), we have

I Λ Λ = 2 ( μ + α 1 ) θ β 2 (1.18)

G Λ Λ = 0 (1.19)

Substituting (1.18) and (1.19) in (1.9), we have

M Λ Λ = π θ β 2 − 2 μ ( μ + α 2 ) β 1 θ ( μ + α 2 ) + 2 σ ( μ + α 2 ) (1.20)

Using (1.18), (1.19) and (1.20) in (1.12), we obtained

R Λ Λ = ( 2 σ ( μ + α 1 ) + α 1 θ β 1 θ β 1 μ ) ( π θ β 1 − 2 μ ( μ + α 1 ) β 2 θ ( μ + α 1 ) + 2 σ μ ( μ + α 1 ) ) (1.21)

Thus E 2 = ( I Λ Λ , M Λ Λ , G Λ Λ , R Λ Λ )

Local Stability of Rumor Media-endemic Equilibrium

We also use the Jacobin stability approach to prove the stability of the rumor media endemic. The Jacobian of (1.9) to (1.12) at equilibrium point E 2 = ( I Λ Λ , M Λ Λ , G Λ Λ , R Λ Λ ) is

( − μ − θ 2 β 1 M Λ Λ − θ 2 β 2 G Λ Λ − ( M Λ Λ + G Λ Λ ) σ − θ 2 β 1 I Λ Λ − σ I Λ Λ − θ 2 β 2 I Λ Λ − σ I Λ Λ 0 θ 2 β 1 M Λ Λ θ 2 β 1 I Λ Λ − μ − α 1 0 0 θ 2 β 2 G Λ Λ 0 θ 2 β 2 I Λ Λ − μ − α 2 0 ( M Λ Λ + G Λ Λ ) σ α 1 + σ I Λ Λ α 2 + σ I Λ Λ − μ ) (1.22)

If we evaluate (1.22) at E 2 and find the eigenvalue using characteristic equestion | J ( E 1 ) − λ I | = 0 , we will obtain

λ 1 = − μ − ( θ β 1 + 2 σ 2 ) ( π θ β 2 − 2 ( μ + α 2 ) θ β 1 ( μ + α 2 ) + 2 σ ( μ + α 2 ) )

λ 2 = β 1 ( μ + α 2 ) β 2 − μ − α 1

λ 3 = 0

λ 4 = − μ

Since some of the eigenvalues are not less than zero, we conclude that rumor through media-endemic equilibrium is unstable. Consequently, when rumor occurs in an ignorant population it must spread first before it declines either naturally or by counter rumor.

This section deals with the numerical studies of the developed model and discussion of the simulation result using estimated parameters and adopted some from other models.

Model Simulation

We performed some numerical experiments using ode45 function from MATLAB to study the behavior of the systems on the ignorant, spreaders through media (spreader 1), spreader through verbal communication (spreader 2) and stiflers population. The parameter values used in the simulation of this model are presented in

The results of the simulation of the model (1.1) to (1.4) with parameter value in

Experiment 1: Investigating the impact of contact rates between the ignorant and spreaders.

Experiment 2: Impact of rate at which spreaders become stiflers.

Parameter | Value | Source |
---|---|---|

π | 0.3 | [ |

μ | 0.000005 | Assumed |

θ | 0.02 | [ |

β 1 | 0.8 | [ |

β 2 | 0.5 | [ |

σ | 0.00001 | [ |

α 1 | 0.72 | [ |

α 2 I M G R | 0.5 1000 3 10 2 | [ |

Here we discuss the results obtained in the analytical studies and the numerical experiments.

Analytical Results

The modified model consists of a 4-dimensional system of ordinary differential equations. The Rumour Free Equilibrium was established for the system (1.1) to (1.4). We also obtained the reproduction number of the two spreader comportments using next generation matrix method and we established the stability of the rumor Free Equilibrium and Endemic Equilibrium of the modified model using linearization methods. We observed that all the eigenvalues have negative real parts, implying that the rumor free equilibrium is locally asymptotically stable and unstable otherwise. For the endemic equilibrium point, it is observed that some of the eigenvalues of polynomial (1.17) and (1.22) are positive and we conclude that rumor-endemic equilibrium is unstable (

It is observed that the impact of R o M is the result of the interaction of each parameters and it is easy to decrease R o M by regulating and controlling a few parameters so also R o V . The parameter with the most positive impact on R o M is the rate of contact between the spreaders targeting the community through media and the ignorant. Similarly the most positive impact on R o V is the rate of contact between the spreaders targeting the community through verbal communication and the ignorant. While the parameter that has moderate influence is the recruitment rate. In this study, we proposed two classes of spreaders and we observed that the propagation of rumor through media and verbal communication are mutually exclusive. Therefore, the appropriate increase in the propagation of rumor through media is conducive to the elimination of the propagation of rumor through verbal communication and vice versa (Figures 2 to 7). On the other hand, the propagation of rumor through media and verbal communication coexist for long time under certain condition (

Deterministic model on rumor propagation was analyzed to get insight into its dynamical features and to obtain associated spreaders thresholds. Some of the theoretical findings of the study are as follows:

1) rumor spreading has a life span which depends on the type of class of the spreader [see (Figures 2-4)].

2) the analysis carried out shows that the increase in stiflers population increased the rate at which spreaders convinced.

The model can be used to control rumor propagation by varying the parameters in the model, not only that it can also be modified to manage some other social negative behaviors like smoking, drug abuse, corruption, prostitution, gang, etc.

The authors declare no conflicts of interest regarding the publication of this paper.

Musa, S. and Fori, M. (2019) Mathematical Model of the Dynamics of Rumor Propagation. Journal of Applied Mathematics and Physics, 7, 1289-1303. https://doi.org/10.4236/jamp.2019.76088