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In this paper, we propose a rumor transmission model with incubation period considering the fact that incubators may move to stifler class and susceptibles may move to spreader class. The model is formulated with constant recruitment and varying total population. The full system of the model is studied qualitatively producing rumor-free and rumor-existence equilibriums. The existence conditions of the equilibriums are investigated. Moreover, the local and global stability analysis of both equilibriums is examined. Furthermore, numerical simulations are used to support the qualitative analysis. Finally, the impact of different management strategies on the dissipation of rumors is analyzed numerically by varying key parameters in the model.

Rumors disseminate easily these days with expanding social networks. Although rumors are neither true nor false, this does not stop some individuals from spreading it any way before searching for some kind of con- firmation. Transmission of rumors has a major impact on human lives. It may have negative sides such as causing panic and chaos in emergency events or destroying credibility of someone or something. On the other hand, it may create awareness and draw public attention to take action. As a result, great importance lies in studying the dynamics of rumor propagation.

Researchers have applied epidemiological models to study the dynamics of social systems. Daley and kendall are among the earliest researchers to propose a rumor spread model that has some properties in common with epidemic model [

At the beginning of this century, several mathematical models are proposed considering different dynamics of rumor and idea transmission. Thompson et al. [

The above rumor models do not consider the fact that an individual may take time before accepting or rejecting what they hear or read. Some individuals think about rumors for some time before they become spreaders or stiflers. This period is called rumor incubation time. Huo et al. [

We consider a variable population size at any time t and denote it by

The susceptible class

As shown in

Taking the above considerations, the model is described according to the dynamic theory by the following nonlinear system of ordinary differential equations:

Note that

The vector field points into the interior of

To find the equilibriums of system (1), we set the rates in (1) to zero:

The model has a rumor free equilibrium if there are no spreaders and no incubators, that is,

where

The Jacobian matrices of

where

Thus the next generation matrix is

Next, we solve Equations (2)-(5) to find a positive (rumor existence) equilibrium

Substituting for

The non trivial root of the above equation is:

Theorem 1 System (1) has two equilibria: the rumor free equilibrium

Here we investigate the local stability of

Theorem 2 (stability of E_{0}) If

Proof. Linearizing system (1) (by linearization method [

Clearly the eigenvalues of the characteristic equation [

where,

Based on Routh-Hurwitz Criteria [

Simplifying

Therefore, if

Theorem 3 (stability of E^{*}) The rumor existence equilibrium

Proof. Linearizing system (1) at the equilibrium

The eigenvalues of the characteristic equation are:

where,

From (4),

If

First, we explore the global stability of

Hence, the only solution of system (1) in

Theorem 4 (stability of E_{0}) If

Next, we examine the global stability of

Here, we used

Theorem 5 (stability of E^{*}) The rumor existence equilibrium

We summarize the result of this section as follows:

• If

• If

In this section, we illustrate numerical simulations of system (1) to support the qualitative analysis. Furthermore, we examine key parameters that may contribute in controlling the spread of rumors.

Numerical simulations of system (1), with different initial conditions, show that the rumor disappears at an equilibrium level

In reality, rumors prevail if there exist many spreaders. Therefore, It is clear that the parameters:

By varying the transmitting rate

transmission of a rumor becomes difficult. This implies that governments should impose laws with strict punishment as a control measurement to stop the transmission of rumors in order to preserve the stability of a society.

In this paper, we formulated a rumor transmission model with incubation period, constant recruitment and varying total population. The model accommodates for both possibilities: incubators move to stifler class and susceptibles move to spreader class. The full dynamical system of the model is studied qualitatively producing two equilibrium points: rumor-free and rumor-existence. The existence conditions of the equilibriums are investigated. The rumor free equilibrium

exists only if

Salma Al-Tuwairqi,Sarah Al-Sheikh,Reem Al-Amoudi, (2015) Qualitative Analysis of a Rumor Transmission Model with Incubation Mechanism. Open Access Library Journal,02,1-12. doi: 10.4236/oalib.1102040