#### Abstract

Smoking subject is an interesting area to study. The aim of this paper is to derive and analyze a model taking into account light smokers compartment, recovery compartment, and two relapses in the giving up smoking model. Stability of the model is obtained. Some numerical simulations are also provided to illustrate our analytical results and to show the effect of controlling the rate of relapse on the giving up smoking model.

#### 1. Introduction

As early as 1889, people have established a model for the spread of infectious diseases. Then the spread rule and trend of the model were studied by analying the stability of the solutions ([1–4] and the references cited therein). De la Sen and Alonso-Quesada [3] present several simple linear vaccination-based control strategies for a SEIR propagation disease model and study the stability of this model. De La Sen et al. [4] discuss a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules, and in this paper the authors were using the good methods to study the dynamic behavior of the model especially the positivity of the model. There are many methods to discuss the stability, one of the most powerful techniques for qualitative analysis of a dynamical system is Direct Lyapunov Method [5]. This method employs an appropriate auxiliary function, called a Lyapunov function. For example, [6–8] use this method to discuss the stability of the model. In addition, there are a number of articles which use Routh-Hurwitz theory to explore the stability, see for example [9, 10].

In recent years, many types of epidemic models are discussed, such as virus dynamics models [11, 12], tuberculosis models [13, 14], and HIV models [15, 16].

Due to the increasing in the number of smokers, tobacco use is also as a disease to be treated. In order to explore the spread rule of smoking, quit smoking model is developed. Castillo-Garsow et al. [17] proposed a simple mathematical model for giving up smoking in the first time. In this model, a total constant population was divided into three classes: potential smokers, that is, people who do not smoke yet but might become smokers in the future (), smokers (), and quit smokers (). Zaman [18] extended the work of Castillo-Garsow et al. [17] by adding the population of occasional smokers in the model, and presented qualitative behavior of the model. Zaman [19] presented the optimal campaigns in the smoking dynamics. They consider two possible control variables in the form of education and treatment campaigns oriented to decrease the attitude towards smoking and first showed the existence of an optimal control for the control problem.

However, in real life, the usual quit smokers are only temporary quit smokers. Some of them may relapse since they contact with smokers again, and the others may become permanent quit smokers. Statistics also show that 15% quit smokers may relapse when they contact with smokers. Enlightening by the previously mentioned cases, we present a model, which extend the models in [17–19] by taking into account the temporary quit smoker compartment () and two kinds of relapses, that is, once a smoker temporary quits smoking he/she may become a light or occasion smoker or a persistent smoker again. First, we derive the basic reproductive number, and discuss the positivity of the solution for the giving up smoking model. Then, we analyze the stability of equilibria by Lyapunov Method and Routh-Hurwitz theory. Finally, by estimating of parameter, we present the numerical simulation. Moreover, the numerical simulation shows that we can greatly improve the effect of quit smoking by using some methods, such as treatment and education, controlling the rate of relapse.

The organization of this paper is as follow: the model is given under some assumption in Section 2. The basic reproductive number, existence, and the stability of equilibria are investigated in Section 3. Some numerical simulations are given in Section 4. The paper ends with a discussion in Section 5.

#### 2. The Model Formulation

##### 2.1. System Description

In this paper, we establish the giving up smoking model as Figure 1. Table 1 presents the parameters description of the model.

From Figure 1, the total population is divided into five compartments, namely, the potential smokers compartment (), light or occasion smokers compartment (), persistent smokers compartment (), temporary quit smokers () that is the people who did some efforts to stop smoking, and quit smokers forever group (). As we know that if you smoke more, the harm of nicotine on the body will be greater. So the death rate is also higher. Hence, we can further assume that . The total population size is , where

The transfer diagram leads to the following system of ordinary differential equations:

##### 2.2. Positivity and Boundedness of Solutions

For system (2), to ensure that the solutions of the system with positive initial conditions remain positive for all , it is necessary to prove that all the state variables are nonnegative. Similar to the proof of [3, 4, 13], we have the following lemma.

Lemma 1. * If , the solutions , , , , of system (2) are positive for all .*

*Proof. * If the conclusion does not hold, then at least one of , , , , is not positive. Thus, we have one of the following five cases.There exists a first time such that
There exists a first time such that
There exists a first time such that
There exists a first time such that
There exists a first time such that

In case (1), we have
which is a contradiction to .

In case (2), we have
which is a contradiction to .

In case (3), we have
which is a contradiction to .

In case (4), we have
which is a contradiction to .

In case (5), we have
which is a contradiction to .

Thus, the solutions , , , , of system (2) remain positive for all .

Lemma 2. * All feasible solution of the system (2) are bounded and enter the region
*

*Proof. *Let be any solution with nonnegative initial condition: adding the first four equations of (2), we have
It follows that
where represents initial values of the total population. Thus , as . Therefore all feasible solutions of system (2) enter the region
Hence, is positively invariant, and it is sufficient to consider solutions of system (2) in . Existence, uniqueness, and continuation results of system (2) hold in this region. It can be shown that is bounded and all the solutions starting in approach enter or stay in .

#### 3. Analysis of the Model

In this section, we will analyze the existence of equilibria of system (2).

##### 3.1. The Existence of Equilibria and the Basic Reproduction Number

The model (2) has a smoking-free equilibrium given by (see Theorem 3 (1))

In the following, the basic reproduction number of system (2) will be obtained by the next generation matrix method formulated in [21].

Let ; then system (2) can be written as where The Jacobian matrices of and at the smoking-free equilibrium are, respectively, where In order to simplify the calculation, letting , we obtain The basic reproduction number, denoted by , is thus given by Throughout this paper, we denote

Theorem 3. *For the giving up smoking model (2), there exist the following three types of equilibrium.*(1)* For all parameter values, system (2) exists the smoking-free equilibrium .*(2)* If , there exists the occasion smoking equilibrium , and there exists no occasion smoking equilibrium if .*(3)* If , and , then system (2) has positive smoking-present equilibrium where .**Moveover, satisfies the following equality:
**
where is a positive solution of , where
*

*Proof. *It follows from system (2) that
(1) Letting in (27), we can obtain the smoking-free equilibrium .(2) If , letting in (27), we can obtain the occasion smoking equilibrium .(3)From third equation of the system (27), we obtain
By adding the third equation and the fourth equation, we get
From this we have
From the second equation of the system (27) we get
Substituting into (31), we get
It follows from the fifth equation that
Let
From (28) we can see that if , then . From (30) we can see that if , then ; these show that . From (32), we can see that if , then . Therefore, if , , , , . In the following, we prove that the existence of positive solutions of . From (34), we know
If , then
where
Clearly, is the minimum point of . For , we have
Hence,
where
Then
as we know that and . If , then for any , we have , so is a strictly monotone increasing function on . As we know that and , so
Hence, for any , so ; that is, is a strictly monotone increasing function on . Therefore, has an unique positive solutions on . The proof is completed.

##### 3.2. Qualitative Analysis

In this part we will discuss the qualitative behavior of the giving up smoking model (2).

###### 3.2.1. Stability of the Smoking-Free Equilibrium

Theorem 4. *If , the smoking-free equilibrium is globally asymptotically stable.*

* Proof. *We introduce the following Lyapunov function:
The derivative of is given by
If , then , so we get .

As we know, , so we obtain with equality only if , and . By LaSalle invariance principle [20, 22], is globally asymptotically stable. Thus, for system (2), the smoking-free equilibrium is globally asymptotically stable if .

###### 3.2.2. Stability of the Occasion Smoking Equilibrium

In this part, we will consider an occasional smoking equilibrium ; that is, only potential smokers and occasionally smokers are not zero, and the other compartments are zero.

Theorem 5. * If , the occasion smoking equilibrium is locally asymptotically stable.*

*Proof. *The Jacobian matrix of the giving up smoking model (2) around is given by
The characteristic polynomial is , where and . Therefore by Routh-Hurwitz criteria we deduce that the roots of the polynomial have negative real part when , which shows that the system is locally asymptotically stable if .

Theorem 6. *If and , the occasion smoking equilibrium is globally asymptotically stable.*

* Proof. *We introduce the following Lyapunov function:
The derivative of is given by
If , we can obtain . As we know , if , so we obtain with equality only if and . By LaSalle invariance principle [20, 22], is globally asymptotically stable. This completes the proof.

###### 3.2.3. Stability of the Smoking-Present Equilibrium

Theorem 7. * Under the condition (3) of the Theorem 3, if satisfy one of four relations as follows:
**
Then smoking-present equilibrium is globally asymptotically stable.*

*Proof. *At the equilibrium point, the expressions on the right-hand side of system (2) give us following relations:
We now consider a candidate Lyapunov function such that
Then and the derivative of are given by
Using the relations in (49) we have
We consider the following variable substitutions by letting
The derivative of reduces to
as we know that , when satisfy one of four relations as follows:
This implies that with equality only if and , that is, , . By LaSalle invariance principle [20, 22], is globally asymptotically stable. This completes the proof.

*Remark 8. * It is possible for condition (48) to fail, in which case the global stability of the interior equilibrium of system (2) has not been established. Figure 2, however, seems to support the idea that the interior equilibrium of system (2) is still globally asymptotically stable even in this case.

#### 4. Numerical Simulation

In this section, some numerical results of system (2) are presented for supporting the analytic results obtained previously. Our data are taken from [18], we also consider the data from Statistical Yearbook of the World Health [23] and Report of the Global Tobacco Epidemic [24]. Now, we give the data in Table 2.

According to the survey, the world population over the age of 15 is about 5.5 billion; this population is recorded as potential smokers, the smoking rate was 34%. Hence, we will consider 5.5 billion, 2.2 billion, 1.87 billion, 0.058 billion, and 0.01 billion as the initial values of the five compartments.

Using the data in Table 2 we can get images in Figure 2.

The data in Table 2 satisfy the condition (3) of Theorem 3; we can see that the smoking-present equilibrium is globally asymptotically stable.

For appropriate adjustment parameters, we choose , then the smoking-free equilibrium is globally asymptotically stable (Figure 3).

If we choose , numerical simulation gives and ; the occasion smoking equilibrium is globally asymptotically stable (Figure 4).

At last, we choose , numerical simulation gives ; then the smoking-free equilibrium is globally asymptotically stable (Figure 5).

#### 5. Discussion

We have formulated a giving up smoking model with relapse and investigate their dynamical behaviors. By means of the next generation matrix, we obtain their basic reproduction number, , which plays a crucial role. By constructing Lyapunov function, we prove the global stability of their equilibria: when the basic reproduction number is less than or equal to one, all solutions converge to the smoking-free equilibrium; that is, the smoking dies out eventually; when the basic reproduction number exceeds one, the occasion smoking equilibrium is stable; that is, the smoking will persist in the population, and the number of infected individuals tends to a positive constant.

In this paper, we consider two relapses. One is relapsed into light smokers and the other is relapsed into persistent smokers. If we employ some ways, such as medical care or education, to reduce the relapse rate, then, the number of the quit smokers will increase. We choose , we can get images in Figure 6.

Comparison of Figures 2 and 6, we can see the difference between them. In Figure 6, the number of recovery and quit smokers is increasing obviously. , , , and are approaching the stable state earlier than the case of Figure 2.

Through numerical simulation, we clearly recognize that if we control the rate of relapse, then the efficiency of giving up smoking will be greatly improved.

For system (2), reflects recruitment number, reflects the contact rate between potential smokers and occasion smokers, and denotes the deaths rate of potential smokers. denotes the deaths rate of light or occasion smokers. These four parameters will directly affect the values of the basic reproductive number. Furthermore, when and increase, the number smokers will increase, that is, increases. When we reduce the mortality caused by medical treatment, the number of permanent quit smokers will increase. Hence, will decrease. Figure 7 shows the relation between the basic reproduction number and , Figure 8 shows the relation between the basic reproduction number and , Figure 9 shows the relation between the basic reproduction number and , Figure 10 shows the relation between the basic reproduction number and . From Figures 11, 12, and 13, we can also see that if and increase, then will decrease. And if and increase, then will increase. Biologically, this means that to reduce the relapse rate and the deaths rate of nicotine by medical treatment, education and legal constraints are very important.

Compared with [18], in this paper we add two bilinear relapse rates. Hence, our model is more closer to real life. In [18], the author only discussed the local asymptotic stability of occasional smoking equilibrium. In this paper, we give the proof of the global asymptotic stability of occasional smoking equilibrium, adding two bilinear relapse rates based on [18], our model becomes more complex. This brought difficulties to the discussion of the existence and stability of the endemic equilibrium. From (3) of Theorem 3, we know that if , and , then system (2) has positive smoking-present equilibrium where . Hence, the numerical values of and are playing an important role in the existence of . Next, we simulate the relationship between and under the conditions , . If , we plot the relationship between and in Figure 14. If , we plot the relationship between and in Figure 15. From Figures 14 and 15, we can know that the point which locates above the line is the positive smoking-present equilibrium.

#### Acknowledgments

This work was partially supported by the NNSF of China (10961018), the NSF of Gansu Province of China (1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China, and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.