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A dynamic alcohol consumption model with awareness programs and time delay is formulated and analyzed. The aim of this model is to capture the effects of awareness programs and time delay in controlling the alcohol problems. We introduce awareness programs by media in the model as a separate class with growth rate of the cumulative density of them being proportional to the number of mortalities induced by heavy drinking. Susceptible population will isolate themselves and avoid contact with the heavy drinkers or become aware of risk of heavy drinking and decline to drink due to such programs. In particular, we incorporate time delay because the nonconsumer population will take a period of time to become an alcohol consumer. We find that the model has two equilibria: one without alcohol problems and one where alcohol problems are endemic in population. The model analysis shows that though awareness programs cannot eradicate alcohol problems, they are effective measures in controlling the alcohol problems. Further, we conclude that the time delay in alcohol consumption habit which develops in susceptible population may result in Hopf bifurcation by increasing the value of time delay. Some numerical simulation results are also given to support our theoretical predictions.

Alcoholism, also known as alcohol dependence, is a function of social epidemic, environmental contexts, individuals’ preferences, and family history. Alcohol consumption has been identified as a major contributor to the global burden of chronic disease, injury, and economic cost [

The spread of health risk behavior within a community can be viewed as a diffusion process with its own incidence rate. In this situation, the social interaction is considered to be the key factor in spreading the behavior which can result in adverse health effects. For this reason, alcoholism can be viewed as a treatable contagious disease. Mathematical model is a predictive tool which can mimic the process of infectious diseases and provide useful measures to analyze the spread and control of infectious diseases [

A rational step is to make people aware of the alcohol problems through the media. Media (e.g., Radio, Newspapers, Billboards, TV, and Internet), being the prime source of information, can not only influence the individuals’ behavior but also increase the governmental health care involvement to control the spread of heavy drinking. These behavioral responses can change the transmission patterns and declination to drink. In view of this, there is a need to incorporate the effect of awareness programs through the media in the mathematical models. In recent years, many mathematical models have been used for studying the impact of awareness programs by media on epidemic outbreaks (see [

Recently, Huo and Wang [

It is well known that delay differential equations exhibit much more complicated dynamics than ordinary differential equations since the time delay may affect the stability of the system and even lead to instability, oscillation, or bifurcation phenomena [

In this paper, motivated by the above works, we present a nonlinear alcoholism model with awareness programs and time delay. First, we assume that the growth rate of the cumulative density of awareness programs is proportional to the number of mortalities induced by heavy drinking. Hence, the awareness about drinking will alert the susceptible individuals so that they isolate themselves and decline to drink or drink moderately (small intake of alcohol may be beneficial to health). Second, we assume that the heavy drinkers can recover from heavy alcohol drinking due to counselling, health reasons, treatment, prohibition, tax hiked on alcohol beverages, and so forth. A fraction of recovered population will join the unaware susceptible population whereas the rest will join the aware population. In particular, we incorporate time delay

The remainder of this paper is organized as follows. In Section

The total population in the model is

Considering homogenous mixing, the transfer diagram is shown in Figure

The transfer diagram of system (

The transfer diagram leads to the following dynamic alcohol consumption model with awareness programs and time delay:

All the above parameters are assumed to be positive constants. We suppose that the initial condition for system (

Using the fact that

From (

It is obvious that system (

The basic reproduction number of the alcohol consumption model can be easily obtained by the next generation matrix method formulated in [

System (

For equilibrium

It is apparent that the existence of equilibrium

In this section, we will investigate the stability of the equilibria of system (

First, we investigate the stability under the condition of

For

The alcohol-free equilibrium

The alcohol-free equilibrium

Furthermore, equilibrium

When

(i) For the alcohol-free equilibrium

(ii) If

The characteristic equation of system (

Notice that

Moreover, we know that

Using the approach in Chitnis et al. [

The effect of increasing

Trajectories of populations and awareness program changes for different values of

In this subsection, we will be concerned with the effect of time delay on the stability of the alcohol-present equilibrium

Now, we will derive the conditions for the stability of

If the coefficients

If

Next, we consider the case that the coefficients

(H1) Equation (

Suppose that (H1) holds. Without loss of generality, we assume that (

Define

In order to investigate whether the Hopf bifurcation occurs or not as time delay

(H2)

Assume that (H2) holds; the following result is true.

Assume that (H2) holds; then the transversality condition

Differentiating the two sides of (

From the above discussion and Lemma

If

With the help of

First, we can see that the value of

Second, the effect of awareness programs on the alcohol consumption behavior is demonstrated in Figure

Moreover, according to the above given parameter values, the critical value of time delay

The alcohol equilibrium

This corresponding phase plot of system (

Hopf bifurcations occur with

Periodic orbit of system (

The goal of this paper is to analyze the impacts of awareness programs and time delay on the alcohol consumption behavior. A vast body of literature (see [

At the same time, the condition of transcritical bifurcation is also given when

Not only awareness programs but also time delay is introduced in our model. It can help us to reduce the economic burden of disease caused by alcohol abuse. It is also beneficial to the social problems such as traffic accident and violent crime. Whether in theory or in the practical sense, these issues are very interesting and need further studies.

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

This work is supported by the NNSF of China (11461041), the NSF of Gansu Province (148RJZA024), and the Development Program for HongLiu Outstanding Young Teachers in Lanzhou University of Technology.