A compartment epidemic model with delay is given to discuss the impact of awareness programs on the spread and control of infectious diseases in a given region. It is assumed that there is a constant recruitment rate in the cumulative density of awareness programs, and further it is assumed that awareness programs can influence the susceptible to a limited extent. The system exhibits two equilibria: the disease-free equilibrium is stable if the basic reproduction number is less than unity for any delay and the unique endemic equilibrium exhibits Hopf-bifurcation under certain conditions. Numerical simulations prove the results of analysis and the significance of awareness programs in preventing and controlling the diseases, by investigating the relationship between the proportion of the infective and the dissemination rate and the implementation rate, respectively.

Plenty of evidence shows that awareness programs, which can influence the susceptible to a limited extent due to some objective factors, play an important role in the spread and control of infectious diseases. For example, during the outbreak of SARS, H1N1 influenza pandemic, and HIV epidemic, public media had massive reports on the number of the infections and deaths per day, which had a great impact on the diseases control [

Recently some scholars used mathematical models to discuss the impact of awareness programs on the diseases spreading and controlling in a given region [

Some scholars consider some other factors in awareness program models. Liu and Wang et al. took into account the random perturbation [

But what we regard as unreasonable is that most of the articles assume that the cured infective become unaware of the disease. In fact people have a certain consciousness about the disease once they get sick. Therefore we propose a delayed mathematical model for predicting the future course of any epidemic by considering some of the infect join the aware susceptible after recovery. We hold the opinion that awareness programs can influence the susceptible to a limited extent for some objective factors and then consider the interaction between the susceptible and awareness programs as Holling type-II functional response. In addition we make a constant to represent the density level of media coverage from other regions with the disease because other regions can also effect the region that we consider. In fact the results about global stability of other delayed systems could be further utilized for other related problems [

The rest of this paper is organized as follows. In the next section, a mathematical model with delay has been proposed to capture the dynamics of the effect of awareness programs. It is assumed that diseases spread due to the contact between the susceptible and the infective only. Then we analyze the conditions of the stability of equilibria and the existence of Hopf-bifurcation in Section

In the region under consideration the rate of immigration of the susceptible is

Consider that the cumulative density of awareness programs driven by media in that region at time

In the above model, the constants

Using the fact that

For the analysis of system (

The system (

Disease-free equilibrium

Endemic equilibrium

Define the basic reproduction number

Solving (

From the expression of

In this section we present the local stability of

The disease-free equilibrium

The Jacobian matrix corresponding to the system (

The characteristic equation at

The form of characteristic equation at

When the endemic equilibrium

For the characteristic equation

it is easy to show

The disease-free equilibrium

Similar to the proof of Theorem

We linearize system (

Then the form of the characteristic equation of the system is

To show the Hopf-bifurcation, we need to show that (

If the coefficients in

If

For

Denote

Now we turn to the bifurcation analysis. We use the delay

One has the following transversality condition:

Differentiating with respect to

The endemic equilibrium

To check the feasibility of our analysis in Section

Schematic model flow diagram.

The stability of the disease-free equilibrium

Then we choose the following set of parameter values which satisfy the condition in Theorem

The stability of the endemic equilibria

The stability of the endemic equilibria

In the following we let

When

A comparison between the oscillations in

Comparison between variations of

In this paper, a nonlinear mathematical model with delay and awareness programs driven by media has been proposed and analyzed. It is assumed that pathogens are transmitted via direct contact between the susceptible and the infective. The model exhibits two equilibria, and the disease-free equilibrium has been shown to be stable for basic reproduction number

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

This paper is supported by a Project of the National Sciences Foundation of China (10901145, 11301491, and 11331009) and the National Sciences Foundation of Shanxi Province (2012011002-1).