An epidemic model with media is proposed to describe the spread of infectious diseases in a given region. A piecewise continuous transmission rate is introduced to describe that the media has its effect when the number of the infected exceeds a certain critical level. Furthermore, it is assumed that the impact of the media on the contact transmission is described by an exponential function. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity. On the other hand, when the basic reproduction number is greater than unity, a unique endemic equilibrium exists, which is also globally asymptotically stable. Our analysis implies that media coverage plays an important role in controlling the spread of the disease.

When an infectious disease outbreaks, the spread and the control of the disease will be reported by the media including television programs, newspapers, and online social networks. Many examples are the massive reports and daily updates in the public media on the number of the infections and deaths, which had important impacts on the diseases control [

Recently, such impact on disease spreading and controlling has been investigated by mathematical modeling approach [

More recently nonsmooth media functions [

In fact, during infectious disease outbreaks, individuals may reduce their activities after receiving information about the risk of infection. For example, people will reduce the time that they go out, students will not attend school, and so on, and such information on the ongoing epidemics may impact the dynamics itself. In fact, at the initial stages of the prevalence of disease, most people and public mass media are unaware of the disease; thus, the individuals will not do any protective measures. Only when the number of infectious individuals reaches and exceeds a certain level, the individuals will take precautionary measures against the diseases. Based on the above facts, in this paper we also focus on the incidence rate. Here we introduce

The rest of this paper is organized as follows: in the next section, a mathematical model is proposed in order to reveal the effect of media; then the existence of equilibria is given in Section

In this model the population is divided into three types: the susceptible, the infective and the recovered. Let

In this model we assume that the impact of media is described by an exponential decreasing factor as

For the solutions of system (

For convenience, let the vector

Let

It follows from [

The disease-free equilibria of the subsystem

Let

One can verify that

Let

When

In this section we present the locally and globally asymptotical stability of the equilibria of subsystem

The disease-free equilibrium

The characteristic equation corresponding to the subsystem

The disease-free equilibrium

To establish the global stability of the disease-free equilibrium of subsystem

In fact, when

The unique endemic equilibrium

When the endemic equilibrium

When

Let a Dulac function

When

To check the analysis in Section

When

The stability of the endemic equilibria

Global stability of

Global stability of

In the following we consider the impact of media coverage on the spread of diseases by drawing the curves between

The curves of

Number of the infected individuals

Number of the susceptible individuals

In this paper, a nonlinear mathematical model has been proposed and analyzed to describe the impact of media coverage on the transmission dynamics of infectious diseases. Considering the fact that public mass media usually do not work when the number of the infected individuals is generally small at the initial stages of prevalence of the disease, and by incorporating a piecewise continuous transmission rate to represent that media coverage has its effects only when the number of the infected individuals exceeds a certain level. The disease-free equilibrium has been shown to be globally asymptotically stable when the basic reproduction number

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

The project is supported by the National Sciences Foundation of China (11571324) and the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province.