Abstract
We derive a discretized heroin epidemic model with delay by applying a nonstandard finite difference scheme. We obtain positivity of the solution and existence of the unique endemic equilibrium. We show that heroin-using free equilibrium is globally asymptotically stable when the basic reproduction number , and the heroin-using is permanent when the basic reproduction number .
1. Introduction
As we all know, the use of heroin and other drugs in Europe and more specifically in Ireland and the resulting prevalence are well documented [1–3]. It shows that the use of heroin is very popular and causes many preventable deaths. Heroin is so soluble in the fat cells that it crosses the blood-brain barrier within 15–20 seconds, rapidly achieving a high level syndrome in the brain and central nervous system which causes both the “rush” experience by users and the toxicity. Heroin-related deaths are associated with the use of alcohol or other drugs [4]. Treatment of heroin users is a huge burden on the health system of any country.
We often study infectious diseases with mathematical and statistical techniques; see, for example, [5–11]; however, little has been done to apply this method to the heroin epidemics. In 1979, Mackintosh and Stewart [9] considered an exponential model which is simplified from infectious disease model of Kermack and McKendrick to illustrate how the heroin-using spreads in epidemic fashion. They arranged a numerical simulation to show how the dynamics of spread are influenced by parameters in the model. White and Comiskey [5] attempted to extend dynamic disease modeling to the drug-using career and formulated an ordinary differential equation. They divided the whole population into three classes, namely, susceptible, heroin users, and heroin users undergoing treatment. Their model allows a steady state (constant) solution which represents an equilibrium between the number of susceptible, heroin users, and heroin users in treatment. Furthermore, this ODE model was revisited by Mulone and Straughan [12]; the authors proved that this equilibrium solution is stable both linearly and nonlinearly under the realistic condition in which relapse rate of those in treatment returning to untreated drug use is greater than the prevalence rate of susceptible becoming drug users. Recently, the study of the global properties and permanence of continuous heroin epidemic models attracted the researchers and have some very good results; see [13–16]. Specially, Samanta [15] considered a model with time-dependent coefficients and with different removal rates for three different classes, introduced some new threshold values and , and obtained the permanence of heroin-using career.
Motivated by Samanta [15] and Zhang and Teng [8], we alter a nonautonomous heroin epidemic model with time delay to an autonomous heroin epidemic model. For convenience, we replace and by and , respectively. Thus, we obtain the following continuous heroin epidemic model with a distributed time delay: where , , and represent the number of susceptible, heroin users not in treatment, and heroin users in treatment, respectively. We assume that the time taken to become heroin user is . The function is nondecreasing and has bounded variation such that .
For understanding more realistic phenomenon of heroin, a little complicated epidemic model is helpful. By applying Micken’s nonstandard discretization method [17] to continuous heroin epidemics model with time delay (1), we derive the following discretized heroin epidemic model with a distributed time delay: where is the susceptible class, is the class of heroin users not in treatment, and is the class of heroin users in treatment at th step. Since the sufficient condition can be obtained, independently of the choice of a time step-size, we let the time step-size be one for the sake of simplicity. The nonnegative constants , , and denote the death rate of the susceptible, heroin users not in treatment, and heroin users in treatment class, respectively. Throughout the paper, it is biologically natural to assume that . The constant denotes the recruitment rate of susceptible population from the general population. Constant is the proportion of heroin users who enter the treatment class. The individuals in treatment who stop using heroin are susceptible at a constant rate . Constant represents the transmission rate from heroin users in treatment to untreated heroin users. is the probability per unit time and the transmission is used with the form , which includes various delays. By a natural biological meaning, we assume that is a positive function and that there exists a constant such that is nondecreasing on the interval . The integer is the time delay. The sequence is nondecreasing and has bounded.
The initial conditions of the system (2) are given by where . Again, by biological meaning, we further assume that for all .
The paper is organized as follows. In Section 2, we prove the positivity and boundedness of the solution of system (2). In Section 3, we deal with the global asymptotic stability of the heroin-using free equilibrium. In Section 4, we consider the permanence of the discrete epidemic model applying Wang’s technique. In the discretized epidemic model, sufficient condition for global asymptotic stability and permanence are the same as for the original continuous epidemic model. We give some numerical examples and conclusion in Sections 5 and 6.
2. Basic Properties
For system (2), the heroin-using free equilibrium is given by Define a positive constant . The stability of is studied by using the next generation method in [7]. The associated matrix (of the new heroin-using terms) and the M-matrix (of the remaining transfer terms) are given as follows, respectively: Clearly, is nonnegative, is a nonsingular M-matrix, and has sign pattern. The associated basic reproduction number, denoted by , is then given by , where is the spectral radius of . It follows that
Now, we will consider the positivity and boundedness of solution to system (2). For most continuous epidemic models, positivity of the solution is clear, but, for system (2), the positivity of the sequences , and holds in some condition.
Lemma 1. Let be any solution of system (2) with initial condition (3); then is positive for any and .
Proof. Let be any solution of system (2) with initial condition (3). It is evident that system (2) is equivalent to the following iteration system:
In the following, we will use the induction to prove the positivity of solution. When , we have
From (8)–(10), we see that, as long as is obtained, and will be obtained too.
If , from (9), we directly obtain and, from (10), we further obtain that . Furthermore, we also have .
Let ; then, from (8)–(10), we see that satisfies the following equation:
where
Substituting in gets
Since , we have
and because of , thus
Substituting and in yields
Here, constants , , , , and are as follows:
We take the limit on both sides of the above equation:
this means that has at least one positive solution . So, we have . Therefore, the positivity of , , and is finally obtained. When , we have
a similar argument as in the above for , , and ; we also can obtain that , , and . Lastly, by using the induction, we can finally obtain that , , and , for all .
Now, we define the total population as . Then, from system (2), we know that
Notice the assumption that ; we obtain
If , it is easy to see that , for all large . If , from right hand side of system (2), we obtain
Hence, we have and there exists such that . Then, we may use this as a starting value instead of . This argument leads to the following result.
Lemma 2. For any solution of system (2), the total population satisfies thus is ultimately bounded.
Let ; then is the positive invariant set to the solution of system (2).
In the following, we will examine the existence of endemic equilibrium for a special case of system (2).
Lemma 3. Assume that is a constant. If , system (2) admits a heroin-using equilibrium when , where satisfies following equality:
Proof. Consider the following equation: From the first equation and the second equation of the system (25), we have From the second equation and the third equation of the system (25), we obtain thus, Since , from the second equation of the system, we have Substituting in (28), we obtain Substituting and in (26) yields a quadratic equation of as follows: where the coefficients are given by Since , then it is easy to see that and . According to Descartes’ rule of signs, if , then has a positive solution; if , then has two positive solutions. From the expression of and , we note that . Since This means that has a unique positive solution . Therefore, there exists a unique positive solution of system (2).
For the local stability of the equilibria, we refer to Theorem 2 in [7] and have the following results.
Theorem 4. Assume that , is a positive constant. The heroin-using free equilibrium of system (2) is locally asymptotically stable if and unstable if .
3. Global Asymptotic Stability of the Heroin-Using Free Equilibrium
In this section, we still assume that , and obtain a sufficient condition for global asymptotic stability of the heroin-using free equilibrium of system (2).
Using a Lyapunov function similar to that in [11], we can easily prove the global asymptotic stability of the heroin-using free equilibrium .
Theorem 5. If , the drug-using free equilibrium of system (2) is globally asymptotically stable.
Proof. Let us take the following Lyapunov function: where are the constants to be defined later and . Using system (2), the difference of satisfies From , for all , we have Let us choose such that these constants satisfy the following inequalities: From (37), we have ; since , then the following inequality is true: that is, Since , which implies that , we can choose ; here, is a sufficiently small positive number such that . Since and , we can choose to satisfy (41). We may further choose to satisfy (38). Therefore, is negative definite and is equal to zero if and only if , , and . The proof is complete.
4. Permanence of System (2)
The system (2) is said to be permanent if there are positive constants and such that hold for any sequence of the system (2), and the same inequalities hold for and . For each class , and , and are independent of initial conditions.
Following the method used by Wang in [6], we will prove the permanence of system (2) for the general case; that is, assume that is related to .
Theorem 6. If , then system (2) is permanent for any initial condition (3).
Proof. Firstly, from system (2) and Lemmas 1 and 2, for any , there exists sufficiently large such that as . Then, we have
Let . Thus, we have
Notice that can be arbitrarily small. Then, we have
Next, let us consider the positive sequences and of (2). According to these sequences, we define
Then, for , we obtain
Since , there exist and such that
note that
We claim that it is impossible that holds for all . The function gives the smallest integer not less than . Suppose the contrary, for . Consider
From Lemma 1, satisfies
and we have that, for , we have
Hence, for , we have
Let . Now, we will show that for all . In fact, there is an integer such that
However, for , we have
Which is a contradiction. Thus, for . Therefore, for ,
which implies that as . But, from Lemma 2 and (46), there exists a sufficiently large integer such that, for ,
which is a contradiction. Hence, the claim is proved.
In the rest, we only need to consider the following two cases:(i) for all large .(ii) oscillates about for all large .
We show that for all large , where , is a constant which will be given later. Clearly, we only need to consider case (ii). Let positive integers and be sufficiently large that , , and , for .
If , since
we have
Hence, for .
If , we can easily obtain that for . Assume that there exists an integer such that
However, for ,
This is a contradiction to the proposition that . Therefore, for . Since these positive integers and are chosen in an arbitrary way, we conclude that for all large in case (ii). Hence, .
Note that, from that third equation of system (2), we have
From Lemma 2 and the discussion above, we have
The proof is completed.
5. Numerical Example
In order to confirm the validity of our results, we consider the following heroin epidemic model with a discrete time delay: Now, we present a numerical example. For the sake of simplicity, we choose the parameters as , , , , , , , , and ; we get . Figure 1 shows that the disease free equilibrium of the system (64) is globally asymptotically stable when . Figure 2 shows that the system (64) is permanent when .
6. Conclusions
In this paper, we have modified the Samanta heroin epidemic model into an autonomous heroin epidemic model with distributed time delay. Further, we established a discretized heroin epidemic model with time delay, sufficient conditions have been obtained to ensure the global asymptotic stability of heroin-using free equilibrium when and is replaced by a positive constant. We also carried out some discussion about the heroin-using equilibrium, but our results are only restricted to the existence of this equilibrium for , a special case of system (2). The stability of heroin-using equilibrium is yet to be studied. As a main result of this paper, we obtained the permanence of the system (2). From the expression of , we see that a decrease in (transmission coefficient from susceptible population) will cause a decrease of the same proportion in . If the rate of migration or recruitment is restricted into susceptible community, the spread of the disease can also be kept under control by reducing and thereby decreasing . The spread of the heroin users can also be controlled by educators, epidemiologists, and treatment providers to increase the values of (removal rate of heroin users not in treatment who stop using heroin but are susceptible) and (proportion of heroin users who enter treatment) and thereby to decrease . This analysis tells us that prevention is better than cure; efforts to increase prevention are more effective in controlling the spread of habitual drug use than efforts to increase the numbers of individuals accessing treatment.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the National Natural Science Foundation of China (Grants nos. 11261056, 11261058, and 11271312).