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Information plays an important role in modern society. In this paper, we presented a mathematical model of information spreading with isolation. It was found that such a model has rich dynamics including Hopf bifurcation. The results showed that, for a wide range of parameters, there is a bistable phenomenon in the process of information spreading and thus the information cannot be well controlled. Moreover, the model has a limit cycle which implies that the information exhibits periodic outbreak which is consistent with the observations in the real world.

In recent years, information plays a more and more important role in the real world. For a decision maker, if he does not have reasonable information, he will make a wrong decision which may cause bad consequences. However, long time series of information are hard to obtain and thus it may provide useful information by constructing mathematical models. As a result, modeling the process of information spreading has been one of the central themes in the field of science [

Information can be divided into many types. For some useful information such as advanced culture and useful knowledge, it is encouraged to develop both widely and deeply. However, there is also some bad information and the typical one is rumor [

There is some work on information spreading by using mathematical models. Based on SIS models, Leskovec et al. presented a cascading information model and found how blogs behaved and how information propagated through the blogosphere [

In this paper, we aim to present a mathematical model to describe the information spreading with isolation. What is more, we want to reveal its dynamical behavior by both mathematical analysis and numerical results. The paper is organized as follows. In Section

To well describe the model, we firstly give four main assumptions, which are as follows.

There are three kinds of individuals: the population who has no information

There are birth and death events and all the newborn has no information.

The spread rate of information is

Since information spreading has stages, isolation term may have different forms at different stages of information spreading. We choose isolation term having the following form:

Plot of

For the small number of

On the basis of the above assumptions, we have the following model:

The first step in analyzing the model (

System (

Denote

In order to obtain positive solutions of (

It should be noted that

As a result, we can conclude that system (

We can also give an explicit criterion of

If (

Note that

In that case, we have that

We fix

The sizes of population

In the following, we discuss the case when

Note that

The discussions above yield the following results.

(I) For the case

(II) For the case

In the following, we aim to investigate the global stability of the equilibrium by considering two cases: (i)

For the first case, we can obtain the Jacobian matrix of system (

Let

It is easy to obtain that

Take the Dulac function

As a result, we have the following conclusions.

The equilibrium

When

In Figure

Phase plane of

In Figure

Phase plane of

For the second case

Since dynamical behaviors of system (

If

By calculations, one can find that

If

In Figure

Phase plane of

We assume that

In Figure

Phase plane of

Information has a great effect on social network, which is a double-edged sword. In this paper, we investigated an information spreading model with isolation. For the dynamical model, we obtained the global stability of the equilibria and bifurcation behaviors, based on both mathematical analysis and numerical results. In a word, information spreading model can have rich dynamics which may provide some new insights for policy decisions on information.

It should be noted that our results are based on the assumption of a homogeneously mixing population. That is to say, we assume that contacts between different individuals are in the same form. This assumption ignores individual differences and thus is not reasonable [

The authors declare that there is no conflict of interests regarding the publication of this paper.