Global Stability for a Binge Drinking Model with Two Stages

A more realistic two-stage model for binge drinking problem is introduced, where the youths with alcohol problems are divided into those who admit the problem and those who do not admit it. We also consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers. Mathematical analyses establish that the global dynamics of the model are determined by the basic reproduction number, R0. The alcohol-free equilibrium is globally asymptotically stable, and the alcohol problems are eliminated from the population if R0 < 1. A unique alcohol-present equilibrium is globally asymptotically stable if R0 > 1. Numerical simulations are also conducted in the analytic results.


Introduction
Young people's binge drinking problem is a major concern to public health.Recently, US surveys indicate that approximately 90% of college students have consumed alcohol at least once 1 , and more than 40% of college students have engaged in binge drinking 2, 3 .The binge drinking refers to youths in 17-30 age group who drink a large amount of alcohol and become so drunk; they are likely to exhibit antisocial behavior 4 .Although there have been many attempts to reduce the problem, alcohol abuse by college students has persisted and in some cases increased over the past several decades 5 .Prior studies have indicated that heavy alcohol drinkers are likely to engage in risky sexual behaviours and more likely to get sexually transmitted infection than social drinkers 6, 7 .There is a strong medical evidence that treatment of individuals with alcohol problems is a major issue 8-10 .
Thus, it is very important to use a mathematical method to study the binge drinking problems in youths.A simple model for alcohol treatment is presented by Sanchez et al. 11 .Since then, there have been numerous studies investigating campus drinking and the associated consequences 12-15 .Manthey et al. 12 focus on a college campus, divide the student population into three classes: nondrinkers, social drinkers and problem drinkers, and show that campus alcohol abuse may be reduced by minimizing the ability of problem drinkers to directly recruit nondrinkers.Cintron-Arias et al. 13 focus on situations where relapse rates are high and conclude that the systematic removal of individuals from highrisk environments, or the development of programs that limit access or reduce the residence times in such environments or both approaches combined , may reduce the level of alcohol abuse.Mubayi et al. 14 show that if the relative residence times of moderate drinkers are distributed randomly between low-and high-risk environments, then the proportion of heavy drinkers is likely to be higher than expected.Mulone and Straughan 15 investigate a model for binge drinking taking into account admitting and nonadmitting drinkers.But the global stability of binge drinking model is not discussed in the literature.
Motivated by the binge drinking model in 15 , we develop a more realistic model with two stages.Drinking is often encouraged by peer pressure.A susceptible individual acquires alcohol problems through the direct contact with the admitting drinker or the nonadmitting drinker.The new drinker can become either the admitting drinker or the nonadmitting drinker.So we consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers; furthermore, we study the global dynamics of the model.The reason to introduce this new direct transfer is that about one-third of the American population admit to drinking problems, 17.8% of the population admit to the binge drinking problem 16 .Therefore, this fact cannot be neglected in the binge drinking model.
The organization of this paper is as follows.In the next section, the binge drinking model with two stages and some basic properties are derived.In Section 3, the existence and the global stability of equilibria are investigated.Some numerical simulations are given in Section 4. Some discussions are given in Section 5.

System Description
The total population is divided into four compartments, namely, the susceptible compartment of those who do not drink or drink only moderately, denoted by S t , those who drink heavily at least some of the time but do not admit having a problem, denoted by A 1 t , those who drink heavily and admit having a problem, denoted by A 2 t , and those people in treatment, denoted by R t .The total number of population at time t is given by The model structure is shown in Figure 1.The transfer diagram leads to the following system of ordinary differential equations: where μN is the number of individuals entering into the system in a given time interval say each year , so μ represents the rate of entry.Since we are dealing with youths, we assume that the death rate is negligible, and so the leaving rate is also μ.A susceptible individual can be turned to drink through direct contact with an admitting drinker or a nonadmitting drinker.β is the transmission coefficient for the individuals who do not admit to have alcohol problems, γ is the transmission coefficient for the individuals who admit to have these problems.We assume that β > γ due to the fact that they are more likely to be unaware of their condition.α is the rate at which represents those with the alcohol problems admitting to have these problems and then transferring from class A 1 to A 2 , π is that fraction of A 2 who go into treatment, η is that fraction of R who relapse into admitting drinkers we adopt a linear relapse term rather than peer pressure since we argue that the relapse is primarily due to the person , p is the probability of a new drinker to admit having the problem.We may show that N S A 1 A 2 R is constant, and then we introduce the fractions of S, A 1 , A 2 , and R: with s a 1 a 2 r 1.Then the system 2.2 becomes ṙ πa 2 − η μ r.

2.4
As system 2.4 is equivalent to system 2.2 , we only need to study system 2.4 .

Invariant Region
Adding all the equations of system 2.4 gives Thus, the total population N is a constant.Since system 2.4 monitors human population, it is plausible to assume that all its state variables and parameters are nonnegative for all t ≥ 0. Further, it can be shown that the region is positively invariant.Thus, each solution of system 2.4 , with initial conditions in Ω, remains there for t ≥ 0. Therefore, the ω-limit sets of the solutions of system 2.4 , in Ω, are contained in Ω.Furthermore, in Ω, the usual existence, uniqueness, and continuation results hold for the system, so that the system 2.4 is well posed mathematically and epidemiologically 17 .So we consider the dynamics of system 2.4 on the set Ω in this paper.

Positivity of Solutions
For system 2.4 , it is necessary to prove that all the state variables are positive, so that the solutions of the system with positive initial conditions remain positive for all t ≥ 0. We thus state the following lemma.
Proof.Under the given initial conditions, it is easy to prove that the solutions of system 2.4 are positive; if not, we assume a contradiction: that there exists a first time t 1 such that there exists a t 2 , there exists a t 3 , and there exists a t 4 , r t 4 0, r t 4 < 0, s t > 0, a 1 t > 0, a 2 t > 0, 0 < t < t 4 .

2.10
In the first case, we have which is a contradiction, meaning that s t > 0, t ≥ 0. In the second case, we have which is a contradiction, meaning that a 1 t > 0, t ≥ 0. Similarly, it can be shown that a 2 t > 0 and r t > 0 for all t ≥ 0.
Thus, the solutions s t , a 1 t , a 2 t , and r t of system 2.4 remain positive for all t > 0.

Analysis of the Model
There are one alcohol-free equilibrium E 0 and one alcohol-present equilibrium E * for system 2.4 .

Alcohol-Free Equilibrium and the Reproduction Number
The model has an alcohol-free equilibrium given by E 0 1, 0, 0, 0 .

3.1
In the following, the basic reproduction number of system 2.4 will be obtained by the nextgeneration matrix method formulated in 18 .Let x a 1 , a 2 , r, s T , then system 2.4 can be written as where

3.3
The Jacobian matrices of F x and V x at the alcohol-free equilibrium E 0 are, respectively, where The model reproduction number, denoted by R 0 , is thus given by Following Theorem 2 of 18 , we have the following result on the local stability of E 0 .
Theorem 3.1.The alcohol-free equilibrium E 0 is locally asymptotically stable for R 0 < 1 and unstable otherwise.

Global Stability of E 0
Theorem 3.2.For system 2.4 , the alcohol-free equilibrium E 0 is globally asymptotically stable if R 0 < 1.
Proof.We use the comparison theorem to prove the global stability of the alcohol-free equilibrium.The rate of change of the variables a 1 , a 2 , r of system 2.4 can be rewritten as where F and V are defined in 3.5 .Since s ≤ 1 for all t ≥ 0 in Ω, then Since the eigenvalues of the matrix F − V all have negative real parts this comes from the local stability results in Lemma 1 in 18 , then system 2.4 is stable whenever R 0 < 1.So, a 1 , a 2 , r → 0, 0, 0 as t → ∞.By the comparison theorem 19 , it follows that a 1 , a 2 , r → 0, 0, 0 and s → 1 as t → ∞.The s, a 1 , a 2 , r → E 0 as t → ∞.So, E 0 is globally asymptotically stable for R 0 < 1.

Existence of the Alcohol-Present Equilibrium
If R 0 > 1, system 2.4 has a unique alcohol-present equilibrium E * s * , a * 1 , a * 2 , r * , where 3.9

Global Stability of the Alcohol-Present Equilibrium
Theorem 3.3.If R 0 > 1, the alcohol-present equilibrium E * is globally asymptotically stable.
Proof.To study the global stability of the alcohol-present equilibrium, motivated by 20-22 , we use a Lyapunov function V as follows:

3.11
The positive constants x 1 , x 2 , x 3 , and x 4 are chosen such that the coefficients of sa 1 , sa 2 , a 1 , a 2 , and r are equal to zero, that is, x 3 η − η μ x 4 0.

3.13
We regroup terms in V such that V V1 V2 V3 , where

3.15
We can rewrite V2 as

3.16
Similarly, we can rewrite V3 as Discrete Dynamics in Nature and Society −y 3.17 where Let V3 Va Vb Vc Vd Ve , with each term representing the expression enclosed in a pair of big square brackets.Using the arithmetic-geometric mean inequality, we obtain

Numerical Simulation
To illustrate the analytic results obtained above, we give some simulations using the parameter values in Table 1.Numerical results are displayed in the following figures.First, we choose p 0.8, α 0.2; numerical simulation gives R 0 0.4091 < 1, then Theorem 3.2 indicates that youths do not have alcohol problems.Figure 2 confirms this conclusion.Second, we choose p 0.1, α 0.4; numerical simulation gives R 0 1.8031 > 1; then Theorem 3.3 shows that alcohol problems persist.Figure 3 further validates the conclusion.Finally, we choose α 0.1; numerical simulation gives the relation between p and R 0 .Figure 4 confirms that R 0 shows a decline, while the number of admitting drinkers increases.From the figures above, we find that making more people admit having alcohol problems can reduce the alcohol problems.Then, the treatment of admitting drinkers is an effective measure in alcohol problems.

Discussion
We have formulated a binge drinking model with two stages and investigated their dynamical behaviors.Depending on the basic reproduction number R 0 , the steady state is either the alcohol free or the alcohol-present.By using the comparison theorem, we prove that all solutions converge to E 0 when the basic reproduction number is less than one, that is, the alcohol problems disappear eventually.By constructing the Lyapunov function, we prove that the unique alcohol-present equilibrium is globally stable, that is, the alcohol problems will persist in the population, and the number of binge drinking individuals tends to be a positive constant when the basic reproduction number exceeds one.
To better understand the binge drinking model with two stages, we consider the more realistic model and investigate the global stability of equilibria of the model for binge drinking problem.The global stability is not discussed in the literature 15 .
The basic reproductive number in this paper is given by It is similar to the basic reproductive number in 15 .Comparing with the two expressions, we see that R 0 is a function on p.In Figure 4, we find that the basic reproductive number R 0 decreases when the number of admitting drinkers increases.In general, the nonadmitting drinkers are also unlikely to seek help.Making more people admit to have alcohol problems through the awareness programs, such as education and media programs is an effective measure to control the alcohol problems.In deriving our model 2.2 , we consider the direct transfer from the class of susceptible individuals towards the class of admitting drinkers and assume the relapse from the class of treatment individuals towards the class of admitting drinkers.However, if we hope to include another relapse from the class of treatment individuals towards the class of nonadmitting drinkers, we could modify 2.2 to the following model: where Q t represents the population that has stopped drinking permanently at time t, say quit drinkers.The δ term is the rate from the class of treatment individuals towards the class of quit drinkers.We leave these works for the future.

Table 1 :
Description and estimation of parameters.