Global Dynamics of a Discretized Heroin Epidemic Model with Time Delay

and Applied Analysis 3 Lemma 1. Let (S n , U n , V n ) be any solution of system (2) with initial condition (3); then (S n , U n , V n ) is positive for any n ∈ N and V 0 < P(1 + μ 3 + ξ 2 )/β 3 . Proof. Let (S n , U n , V n ) be any solution of system (2) with initial condition (3). It is evident that system (2) is equivalent to the following iteration system: U n+1 = β 1 (U n ) S n+1 ∑ h k=0 U n−k η k + U n 1 − β 3 V n+1 + μ 2 + P + ξ 1 , V n+1 = V n + PU n+1 1 + β 3 U n+1 + μ 3 + ξ 2 , S n+1 = λ + S n + ξ 1 U n+1 + ξ 2 V n+1 1 + β 1 (U n )∑ h k=0 U n−k η k + μ 1 . (7) In the following, we will use the induction to prove the positivity of solution. When n = 0, we have U 1 = β 1 (U 0 ) S 1 ∑ h k=0 U −k η k + U 0 1 − β 3 V 1 + μ 2 + P + ξ 2 , (8)


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][2][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][6][7][8][9][10][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][14][15][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  1 and  2 by  and , respectively.Thus, we obtain the following continuous heroin epidemic model with a distributed time delay: Abstract and Applied Analysis 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 () : [0, ℎ] → [0, ∞) 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  1 ,  2 , and  3 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  1 ≤ min{ 2 ,  3 }.The constant  > 0 denotes the recruitment rate of susceptible population from the general population.Constant  > 0 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  2 ≥ 0. Constant  3 represents the transmission rate from heroin users in treatment to untreated heroin users. 1 (  ) is the probability per unit time and the transmission is used with the form  1 (  ) +1 ∑ ℎ =0  −   , which includes various delays.By a natural biological meaning, we assume that  1 () is a positive function and that there exists a constant   > 0 such that  1 () is nondecreasing on the interval [0,   ].The integer ℎ ≥ 0 is the time delay.The sequence   : −∞ <   < +∞ ( = 0, 1, . . ., ℎ) is nondecreasing and has bounded.
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.

Basic Properties
For system (2), the heroin-using free equilibrium is given by Define a positive constant  ≡ ∑ ℎ =0   .The stability of  0 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  0 , is then given by  0 = ( −1 ), where  is the spectral radius of  −1 .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.
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  = 0, we have From ( 8)-( 10), we see that, as long as  1 is obtained,  1 and  1 will be obtained too.
In the following, we will examine the existence of endemic equilibrium for a special case of system (2).
admits a heroin-using equilibrium  * = ( * ,  * ,  * ) when  < / 3 , where  * satisfies following equality: 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 Since  ̸ = 0, 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  0 =  1 / 1 ( 2 +  +  1 ) > 1, then it is easy to see that  < 0 and  > 0. According to Descartes' rule of signs, if  ≥ 0, then () = 0 has a positive solution; if  < 0, then () = 0 has two positive solutions.From the expression of  and , we note that  < / 3 .Since This means that () = 0 has a unique positive solution  * ∈ (0, / 3 ).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  1 () =  1 ,  1 is a positive constant.The heroin-using free equilibrium  0 = ( 0 , 0, 0) of system (2) is locally asymptotically stable if  0 < 1 and unstable if  0 > 1.

Global Asymptotic Stability of the Heroin-Using Free Equilibrium
In this section, we still assume that  1 () =  1 > 0, and obtain a sufficient condition for global asymptotic stability of the heroin-using free equilibrium  0 of system (2).Using a Lyapunov function similar to that in [11], we can easily prove the global asymptotic stability of the heroinusing free equilibrium  0 .Theorem 5.If  0 < 1, the drug-using free equilibrium  0 of system (2) is globally asymptotically stable.

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  1 () is related to .Theorem 6.If  0 > 1, then system (2) is permanent for any initial condition (3).
Proof.Firstly, from system (2) and Lemmas 1 and 2, for any  0 > 0, there exists sufficiently large  0 > 0 such that   ≤ / 1 +  0 as  ≥  0 + ℎ.Then, we have . Thus, we have Notice that  0 can be arbitrarily small.Then, we have lim inf Next, let us consider the positive sequences   and   of (2).
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 0 <   ≤ , is a constant which will be given later.Clearly, we only need to consider case (ii).Let positive integers  1 and  2 be sufficiently large that   1 ≥ ,   2 ≥ , and   < , for we have . ( From Lemma 2 and the discussion above, we have The proof is completed.

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  0 ≤ 1 and  1 () 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  1 () =  1 > 0, 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  0 =  1 (0)/ 1 ( 2 +  +  1 ), we see that a decrease in  1 (transmission coefficient from susceptible population) will cause a decrease of the same proportion in  0 .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  0 .The spread of the heroin users can also be controlled by educators, epidemiologists, and treatment Consider the following equation:  −  1  −  1  +  1  +  2  = 0,  1  +  3  − ( 2 +  +  1 )  = 0,  −  3  − ( 3 +  2 )  = 0. (25) Figure 1