Modelling Alcoholism as a Contagious Disease : A Mathematical Model with Awareness Programs and Time Delay

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.


Introduction
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 [1][2][3].Alcohol abuse can also lead to a range of negative social effects such as violence and antisocial and criminal behavior.The World Health Organization reports that the harmful use of alcohol causes approximately 3.3 million deaths every year (or 5.9% of all the global deaths), and 5.1% of the global burden of disease is attributable to alcohol consumption [4].Therefore, alcoholism is among the main targeted health risk behaviors due to the high relevance of negative health and social effects.The study of alcoholism has become an important aspect of social epidemic.
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 [5,6].Furthermore, mathematical models for human behaviors related to addictions have been developed from epidemiological models for the spread of infectious diseases, including drinking, smoking, and drug use (see [7][8][9][10] and the references contained therein).In particular, numerous mathematical models for alcohol problems have been formulated and studied recently [11][12][13][14][15][16][17][18][19][20][21].Benedict [11] modelled alcoholism as a contagious disease and studied how "infected" drinking buddies spread problem drinking.Manthey et al. [12] studied campus drinking and suggested that the reproductive numbers are not sufficient to predict whether drinking behavior will persist on campus and that the pattern of recruiting new members plays a significant role in the reduction of campus alcohol problems.Environmental and peer influences combined to create a culture of drinking were studied in [13][14][15][16].In addition, the two-stage models (one stage where people admit having alcohol problem and another stage where people do not admit having alcohol problem) have been developed in [17,18].Bhunu and Mushayabasa [19] studied the cointeraction of alcoholism and smoking in a community.Walters et al. [20] also discussed alcohol problems, and their results showed that an increase in the recovery rate decreased the proportion of binge drinkers in the population.Wang et al. [21] presented a deterministic SATQ-type mathematical model (the model including susceptible, alcoholism, treating, and quitting drinking compartments) for the spread of alcoholism with two control strategies to gain insights into this increasingly concerned-about health and social phenomenon.The optimal control strategies are derived by proposing an objective functional and using Pontryagin's Maximum Principle.
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 [22][23][24][25][26][27][28][29][30][31][32] and the references cited therein).These studies suggested that education and media have a huge impact on controlling the spread of infectious diseases.However, most of these studies assumed that the media reduced the contact rate of susceptible with infective individuals.In addition, these studies assumed that the cumulative density of awareness programs is constant in models.Hence, it is more reasonable to consider that the cumulative density of awareness programs varies.Misra et al. [33] have explicitly incorporated the cumulative density of awareness programs in the modelling process, which is a separate dynamic variable whose growth depends on the size of epidemic.The results showed that awareness programs by media are implemented in accordance with the size of infective population.Considering the effect of susceptible population influenced by media to be limited, Misra et al. [34] have considered interaction between susceptible population and awareness programs as Holling-II functional response function.
Recently, Huo and Wang [35] have studied drinking dynamics and focused on awareness programs and treatment in the modelling process.They have extended the model in [33] via including a treatment class and established some sufficient conditions for the stability of the alcohol-free and alcohol-present equilibria.
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 [34,[36][37][38][39].In most of the above studies, with susceptible individual contacts with heavy drinkers, he/she will become a heavy drinker at once.However, the alcoholism epidemic should consider the effects of the time delay.Since a nondrinker or moderate drinker is infected by heavy drinkers, there is a time delay during which a susceptible individual becomes heavy drinker.Hence, it is more realistic to consider the time delay in the modelling alcoholism process.
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  to describe the time needed for a susceptible individual to become a heavy alcohol user.The goal is to analyze the impact of awareness programs and time delay on the alcohol drinking distribution.We also analyze the effect of time delay on system stability.Our results show that awareness programs are an effective measure in controlling heavy drinking.Our results also show that the system will change its stability with the increase of the value of time delay.
The remainder of this paper is organized as follows.In Section 2, we formulate the delayed mathematical model.The condition for the existence of the equilibria is shown in Section 3. In Section 4, we discuss the stability of equilibria and the existence of Hopf bifurcation.To support our theoretical predictions, some numerical simulations are included in Section 5. A brief conclusion is also given in Section 6.

Model Formulation
The total population in the model is () at time , and it is divided into three types: (), (), and ().() denotes susceptible individuals who do not drink or drink only moderately.() is referred to as heavy drinkers.The class () is aware population who are aware of risk and decline to drink or drink moderately.It is supposed that the cumulative density of awareness programs driven by media at time  is ().The growth rate of () is proportional to the number of mortalities induced by heavy drinking.It is also assumed that heavy drinking only spreads due to the direct contact between susceptible individuals and heavy drinkers.Susceptible population avoid being in contact with the heavy drinkers because the implementation of awareness programs results in awareness about heavy drinking.In view of this, a separate class () is formed.It is assumed that when where Λ is the recruitment rate of susceptible individuals who do not drink or drink only moderately,  is the transmission coefficient of the infection for the susceptible individuals from the heavy drinkers,  is the dissemination rate of awareness programs among susceptible population,  0 limits the effect of awareness programs on susceptible population, known as half saturation point for interactions of Holling-II functional response function,  is the transmission rate of transferring from aware population to unaware susceptible population due to fading of memory or certain factors, and so forth,  is the recovery rate of heavy drinkers,  represents fraction of recovered individuals joining aware class, 1 −  represents fraction of recovered individuals joining unaware susceptible class,  is the natural death rate of the general population in the model,  is the death rate due to heavy alcohol consumption,  is the implementation rate of awareness programs,  is the depletion rate of awareness programs due to ineffectiveness such as social and psychological factors in population, and  is the time lag due to development of alcohol consumption habit.
The basic reproduction number of the alcohol consumption model can be easily obtained by the next generation matrix method formulated in [40], which is given by the following expression: and it acts as a threshold as is shown in the following result.

Stability
Analysis with  = 0. First, we investigate the stability under the condition of  = 0.The stability of two equilibria  0 and  1 of system (2a), (2b), (2c), and (2d) can be given in the following theorem.
Theorem 2. For  = 0, one has the following results: (i) The alcohol-free equilibrium  0 is globally asymptotically stable if  0 < 1.
(ii) The alcohol-free equilibrium  0 is unstable and the alcohol-present equilibrium  1 exists if  0 > 1.
(ii) If  0 > 1, then the alcohol-free equilibrium  0 is a saddle point.So the equilibrium  0 of system (2a), (2b), (2c), and (2d) is unstable.In this case, the alcohol-present equilibrium  1 exists.Further, the matrix  at  1 can be written as ) ) ) . ( The characteristic equation of system (2a), (2b), (2c), and (2d) at  1 is det( −  1 ) = 0, where  is the identity matrix of size 4 and  is the eigenvalue of matrix  1 .By simple computation, the characteristic equation can be expressed: where Notice that  0 > 1; it can be calculated that the coefficients   ( = 1, 2, 3, 4.) all are positive according to   ( = 1, 2, 3, 4.) and  12 ,  31 all are negative and  13 ,  21 ,  23 ,  24 , and  41 all are positive.Furthermore, by using the Routh-Hurwitz criterion [41], it is easy to see that the eigenvalues of the above characteristic equation have negative real parts if the conditions hold.This shows that the alcohol-present equilibrium  1 is locally asymptotically stable, and the proof is complete.
Moreover, we know that  0 is a threshold value which determines whether alcohol problems will die out or become endemic in the population.The alcohol-free equilibrium  0 undergoes a transcritical bifurcation at  0 = 1.We are now interested in knowing which model parameter has the greatest effect on  0 and hence has the greatest effect in controlling the alcohol consumption.
Using the approach in Chitnis et al. [42], we calculate the normalized forward sensitivity indices of  0 .Let denote the sensitivity index of  0 with respect to the parameter .We get Considering that the natural death rate  is a relatively stable constant, we find that Υ  0  and Υ  0 Λ are the largest sensitivity indexes.We focus on reducing the value of  or Λ whose reduction by 1% would reduce  0 by 1%.In contrast, there is a need to increase  in order to control alcohol problems.In particular, increasing  results in promoting the dissemination rate  of awareness programs; hence, the value of  0 decreases.Therefore, media coverage for the negative effects and the number of mortalities induced by heavy drinking is very important.At the same time, this sensitivity analysis tells us that the efforts through awareness programs to increase prevention, which cause the number of aware populations () to increase, are effective in controlling the spread of habitual alcohol consumption.To clarify this, the numerical simulation results will be also performed in Section 5 (see Figures 2 and 3).

Stability Analysis with 𝜏 > 0.
In this subsection, we will be concerned with the effect of time delay on the stability of the alcohol-present equilibrium  1 .Let () = ((), (), (), ())  ; by linearizing system (2a), (2b), (2c), and (2d) around the alcohol-present equilibrium  1 , we have where By simple computation, the characteristic equation of the linearized system ( 22) is rewritten as follows: where Now, we will derive the conditions for the stability of  1 and the occurrence of Hopf bifurcation.Let  =  ( > 0) be a root of (24).By substituting it into (24) and separating the real and imaginary parts, we can get the following equations: Using the fact that sin 2 () + cos 2 () = 1 and denoting  =  2 , we obtain where If the coefficients   ( = 1, 2, 3, 4.) of ( 27) satisfy the Routh-Hurwitz criterion [41], then we do not get any positive value of .In this case, (24) does not have any purely imaginary roots.In summary, we have the following result.Theorem 3. If  0 > 1 and the coefficients   ( = 1, 2, 3, 4.) of ( 27) satisfy the Routh-Hurwitz criterion, then the alcoholpresent equilibrium  1 is locally asymptotically stable for all  > 0.
Assume that (H2) holds; the following result is true.
Lemma 4. Assume that (H2) holds; then the transversality condition is satisfied.
Proof.Differentiating the two sides of ( 24) with respect to , we get Combining ( 26) with (H2), we compute that Suppose we let  <  0 and  →  0 , which implies that (24) has a root () = () + () such that () > 0, but this contradicts the definition of  0 .Hence, we obtain This completes the proof of the lemma.
Second, the effect of awareness programs on the alcohol consumption behavior is demonstrated in Figure 3.It is apparent that awareness programs are an effective measure in controlling heavy drinking from this figure.We note that the aware population () increase whereas heavy drinkers () decrease with an increase in the value of dissemination rate  of awareness programs among susceptible population while the other parameters remain unchanged.Hence, alcohol problems should be controlled through the execution of awareness programs.
Moreover, according to the above given parameter values, the critical value of time delay  0 = 1.5767 can be obtained by solving the corresponding expression.From Theorem 5, the corresponding waveforms are shown in Figures 4-7.That is, when  = 1.50 <  0 = 1.5767, the alcohol-present equilibrium  1 of system (2a), (2b), (2c), and (2d) is locally asymptotically stable (see Figures 4 and 5, resp.); and Hopf bifurcation occurs once  = 2.50 >  0 = 1.5767 (see Figures 6  and 7, resp.).These figures indicate that the system is stable for lower values of  but for its higher values periodic solutions may arise and system may lose its stability.

Conclusions and Discussions
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 ) has been used for studying alcohol problems and describing the effects of awareness programs by media on the infectious disease or alcohol consumption behavior.However, most of these models are ODE models and do not incorporate the effects of the time delay.Based on the previous works, we establish an alcohol consumption model with awareness programs and time delay in Section 2. Compared to previous models, a key novelty of our model is that we introduce the delay in our  model to describe the time needed for a susceptible individual to become an infectious alcohol user.To be more realistic, we considered interaction between susceptible population and awareness programs with Holling-II functional response function.In addition, we assume that the growth rate of the cumulative density of awareness programs is proportional to the number of mortalities induced by heavy drinking.We obtain the conditions for the existence and stability of two equilibria of our model in Section 3.
At the same time, the condition of transcritical bifurcation is also given when  = 0.In order to study the effect of time delay  > 0 on system (2a), (2b), (2c), and (2d), we compute the linear form of system (2a), (2b), (2c), and (2d) around the alcohol-present equilibrium  1 and analyze the local stability of  1 .In addition, we obtain the existence of Hopf bifurcation by taking time delay  as bifurcation parameter.Our results suggest that the alcoholpresent equilibrium  1 of system (2a), (2b), (2c), and (2d) is locally asymptotically stable for  ∈ [0,  0 ) and unstable for  >  0 .System (2a), (2b), (2c), and (2d) undergoes a Hopf bifurcation at the equilibrium  1 when  =  0 .Furthermore, we study the impacts of awareness programs on the alcohol consumption behavior.Our findings show that awareness programs are an effective measure in controlling heavy drinking.
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.

Figure 2 :
Figure 2: The effect of increasing  and Λ on  0 .