Stability Analysis of a Reaction-DiffusionHeroin EpidemicModel

A reaction-diffusion (R-D) heroin epidemic model with relapse and permanent immunization is formulated. We use the basic reproduction number R0 to determine the global dynamics of the models. For both the ordinary differential equation (ODE) model and the R-D model, it is shown that the drug-free equilibrium is globally asymptotically stable if R0 ≤ 1, and if R0 > 1, the drug-addiction equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.


Introduction
Heroin comes from opioids, commonly known as "opium," derived from the poppy plant. Pure heroin is a white powdery substance or a white crystalline powder. It is well known that long-term consumption and injection of heroin can cause personality disintegration, psychological metamorphosis, and lifespan to be reduced. ere are more and more registered drug users, and this number is growing continuously [1,2]. Heroin abuse and independence has been bringing tremendous pressures on the social and public health systems due to its prevalence all over the world [3,4].
Since the spread of heroin is as contagious as infectious diseases, it is a new trend to study heroin transmission from the perspective of infectious disease dynamics [5][6][7][8][9][10][11][12][13][14][15][16]. In 2007, White and Comiskey established an ODE model for heroin infectious diseases [14]. ey studied the dynamics using a threshold R 0 and showed that prevention is better than treatment. In 2009, this model was reconsidered by Mulone and Straughan [11], and the stability of the positive equilibrium point of the model is obtained by the authors by using the eigenvalue equation and Poincare-Bendixson theory. In 2011, Wang et al. [15] used the bilinear law incidence function instead of standard incidence, and they also analyzed the dynamic behavior of the heroin model. In order to better study the dynamics of heroin infectious diseases, many other different epidemic models have been formulated and studied in various ways [5-10, 12, 13, 16].
As known to all that many people who are detoxifying are not really willing to go, and lots of people still keep in touch with friends who use drugs after they have a successful drug treatment, thus they have a high probability of reusing drugs. Hence, some scholars considered relapse in the drug model and analyzed its stability [12,17]. However, there are some drug users who experienced the harmful effect of drugs and realized the happiness of being an ordinary person after they have a successful drug treatment, such persons will be far away from drugs, so we are optimistic that they will continue this good habit for a lifetime. In addition, there are some people who have been well educated since childhood and have a healthy living environment and strong willpower. So, they do not take drugs from the start to the finish. We call these two types of people permanently immunized against drugs.
In recent years, it has been well recognized that spatial diffusion and environmental heterogeneity have important effects on the persistence and extinction of contagious diseases . In the case of the heroin epidemic model, the spatial distribution of the susceptible or infected person is uneven, and the density may change at any time and place, thus it is more reasonable to use the R-D equations to describe the spread of drug abusers. Moreover, to the best of our knowledge, the heroin infectious disease models in which population density depends on both time and space variables are rarely studied.
In this work, we first formulate an R-D heroin model with relapse and permanent immunization, and then study its global dynamics. We organize this paper as follows: in Section 2, the model is derived and the positive property of the solutions for the model is proved. In Section 3, the model without diffusion is analyzed, and we obtain the basic reproduction number and show the stability of all the equilibria. In Section 4, the stability of the R-D model is obtained. In Section 5, we illustrate our results by some numerical simulations. Finally, we finish this paper with a concluding discussion.

Model Formulation.
We divide the total population into five compartments: S, U 1 , U 2 , Q, and R. Here, S represents the number of susceptible individuals who have never used heroin; U 1 represents the number of heroin users; U 2 represents the number of heroin users undergoing treatment; Q represents the number of people who have used drugs and are not taking drugs at this stage, but may be taking drugs in the future; and R represents the number of people who never use drugs or these successful detoxification people do not take drugs anymore. We assume that drug users are not able to heal themselves through self-control, and if they want to abstain from drugs, they have to go through treatment. We also assume that not all people can be cured completely. If the drug users are successfully cured, individuals in compartment U 2 will enter into the compartment of Q. If the treatment is terminated or failure, the people who failed the treatment will still take drugs. Some of these successful detoxification people will redrug because they cannot resist the temptation of drugs, and some will never take drugs because they know the harm of drug abuse.
us, the total population is given by We give the transfer diagram of the model in Figure 1. According to Figure 1, we obtain the following R-D heroin epidemic model with relapse and permanent immunization: 2 Complexity We assume that the bounded domain Ω ⊂ R 5 has a smooth boundary zΩ. As shown in (2), the homogeneous Neumann boundary conditions indicate that the population movements will not cross the border. Δ is the usual Laplacian operator on R 5 . d i > 0 (i � 1, 2, 3, 4) are the diffusion coefficients. As mentioned above, R is a compartment of complete rehabilitation. e individuals in R are permanently immunized, and we can see that the R equation is uncoupled with the other equations of (2). Hence, the diffusion of R is not considered. We assume that all parameters in the model are positive constants, and the meaning of the parameters is described in Table 1.
To investigate the global dynamic behavior of system (2), we first study its ODE counterpart version as follows: Since the proof of the above lemma is direct, we omit it.

Lemma 2. All feasible solutions of the model (3) are bounded
and enter the following region: Proof. If (S, U 1 , U 2 , Q, R) is a solution of model (3) with nonnegative initial conditions, adding the five equations yields where which indicates that where N(0) is the initial value. us, 0 ≤ N(t) ≤ (Λ/μ), as t ⟶ ∞. is completes the proof.

e Basic Reproduction Number and Stability of Drug-Free
Equilibrium. It is easy to get the drug-free equilibrium of system (3): We now use the next-generation matrix method formulated in [41] to derive the basic reproduction number R 0 of model (3).
Let l � (U 1 , U 2 , Q, S, R) T , then system (3) can be written as Complexity e Jacobian matrices of P(l) and W(l) at the drug-free equilibrium E 0 are as follows: where en, the basic reproduction number R 0 is Number of susceptible people at location l and time t U 1 (l, t) Number of heroin users at location l and time t U 2 (l, t) Number of heroin users undergoing treatment at location l and time t Q(l, t) Number of people who have used drugs at location l and time t R(l, t) Number of people who never use drugs at location l and time t Λ Recruitment rate of the population 1 [ [41], and we thus omit its proof.

Theorem 1.
e drug-free equilibrium E 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1.
Proof. Inspired by [28], we introduce the following Lyapunov function: It follows that the derivative of V is Here, we sued the equalities Λ � S 0 (μ + α 1 ). Since According to LaSalle's invariance principle [42], we obtain that the drug-free equilibrium E 0 of system (3) is globally asymptotically stable in Ω if R 0 ≤ 1. is completes the proof.

Existence and Stability of the Drug-Addiction Equilibrium.
Let the right side of model (3) be equal to zero. en we get en, the unique drug-addiction equilibrium Theorem 3. If R 0 > 1, then the unique drug-addiction equilibrium E * of system (3) is global asymptotically stable.
Proof. If R 0 > 1, then there exists a unique drug-addiction equilibrium E * . We now introduce a Lyapunov function V as follows: where A, B, and C are the positive constants to be determined later. It follows that the derivative of V is 6 Complexity e variables with nonnegative coefficients in (20) are xy, y, u, and z. If all the coefficients are positive, then _ V is positive. If all the coefficients of xy, y, u, and z are equal to zero, then we get By (21), we obtain Hence, we have It is clear that −  (3), we get 0 � Λ − β 1 S * U 1 − (μ + α 1 )S * . It follows from the first equation of (16) that U 1 � U * 1 . erefore, the maximum invariant set of system (2) on set (x, y, z, u): _ V � 0 is the singleton (1, 1, 1, 1). is implies that the largest invariant set where _ V � 0 is the singleton (S * , U * 1 , U * 2 , Q * , R * ) . us, by LaSalle's invariance principle in [42], the drug-addiction equilibrium E * of model (3) is globally asymptotically stable when R 0 > 1. is completes the proof.  R(l, t)) be a solution of system (2) and S(l, t),

Proof. By the first equation of model (2), we get
that is, Since S(l, 0) > 0 ≡ 0, by Lemma 2.4.1 in [33], we get S(l, t) > 0. Let en, the second to the fourth equations of system (2) can be rewritten as
For the last equation of model (2), let S(l, t) and Q(l, t) be bounded by S and Q. en, Hence, R(t) ≤ ((α 1 S + α 2 Q)/μ) + ‖R(l, 0)‖ ∞ on Ω. is completes the proof. (46) en, by [23], we obtain that B i are the infinitesimal generators of a strongly continuous semigroup e tB i t ≥ 0 in L.
(47) ey are also the infinitesimal generator of a strongly continuous semigroup e tB t ≥ 0 in Y ≔ L 5 , where e tB � e tB 1 , e tB 2 , e tB 3 , e tB 4 , 1 T , since Y is a Banach space with norm Let G be a nonlinear operator on Y defined by where Complexity (51) en, model (2) can be rewritten into a more abstract form in Y as follows: By Proposition 4.16 in [44], it can be obtained that the unique continuously differentiable solution φ: (0, T max ] ⟶ Y of the above equations has a maximum interval of existence (0, T max ] so that and either T max ⟶ ∞ or limsup t⟶T max − 0 ‖φ(t)‖ Y ⟶ +∞. By eorem 5, we obtain T max � +∞ holds. erefore, φ(t) is a global solution.

Global Stability of the R-D Model.
Since the R equation of model (2) is uncoupled with the other equations, model (2) can be reduced by ignoring R. It is clear that the R-D heroin epidemic model (2) has a drug-free equilibrium E r � ((Λ/μ + α 1 ), 0, 0, 0) and a unique positive drug-addic-

Theorem 6.
If R 0 ≤ 1, the drug-free equilibrium E r is globally asymptotically stable.
According to the above discussion, E 0 is asymptotically stable, and when α 1 > μ, (Λ/μ + α 1 ) < (α 1 Λ/μ(μ + α 1 )). Figure 3 not only illustrates the stability of E 0 , but also shows the number of people who are permanently immunized against drugs, (R(t)) is greater than the number of susceptible people (S(t)) in the equilibrium E 0 when α 1 > μ. By  Figures 2 and 3, we can not only clearly see that U 1 (t) declined sharply and got to zero finally, but also see that all solutions of the system infinitely close to the drug-free equilibrium E 0 . It verifies the existence of E 0 .

Discussion
In this paper, we formulated a novel R-D heroin epidemic model, which incorporates the relapse compartment and permanent immunization compartment. With the help of Complexity the next-generation matrix method, we obtained the basic reproduction number R 0 . We obtained the global dynamics of the model by constructing some suitable Lyapunov functions. It is shown when R 0 ≤ 1, the drug-free equilibrium is globally asymptotically stable; that is, the drug abuse will be eradicated; when R 0 > 1, the endemic equilibrium is globally asymptotically stable, which means that drug abuse will be permanent. For the ODE system (3), k 1 , β 2 , k 2 , α 1 , and α 2 represent the detoxification success rate, detoxification failure rate, relapse rate from Q to abusers, permanent withdrawal rates from S to R, and permanent withdrawal rates from Q to R,  Population size   Population size Population size respectively. Figure 5 shows the relationship between R 0 and k 1 , k 2 , β 2 , and α 1 , with other parameter values as given in Table 1. As shown in Figures 5(a)-5(c), R 0 grows with k 2 and β 2 but decreases with k 1 , and it means that if we want to control heroin addiction, we should reduce k 2 and β 2 and increase k 1 , that is, increase the detoxification success rate and pay attention to people with a history of drug abuse to reduce their mental dependence on heroin. Moreover, as shown in Figure 5(d), R 0 decreases with α 2 , and it means that if we want to control the heroin addiction, we should increase publicity to let people understand the harmful effect of heroin and take the initiative to stay away from drugs. e effect of α 1 and α 2 on U 1 is shown in Figure 6. It indicates that the values of α 1 and α 2 have a significant effect on the number of drug-addiction equilibrium. Let us observe Figure 6(a) first, the larger the α 1 is, the fewer the people who use drugs in equilibrium is, and this tells us that we should not only pay attention to drug abuse but should pay more attention to those who do not use drugs. e government should strengthen publicity to raise people's awareness of drug prevention. Comparing Figure 6(b) with Figure 6(a), we can get that although the effect of α 2 on U 1 is less than the effect of α 1 on U 1 , it can still obtain that the larger the α 2 is, the fewer the people who use drugs in equilibrium is;   Figure 6: Effect of (a) α 1 and (b) α 2 on U 1 . therefore, we also should pay attention to the people who have already quit drug abuse, so that they can stay away from drugs instead of reusing drugs. e diffusion phenomena make it more difficult for governments to control drugs. Fortunately, the R-D heroin epidemic model (2) still exists as a drug-free equilibrium, and it convinces us that the spread of drugs can be stopped. Additionally, the global dynamic behaviors of the R-D model (2) show that if R 0 > 1 the endemic equilibrium E * r is global stability, and if R 0 ≤ 1, the drug-free equilibrium E r is global stability, which indicates that the R-D model (2) may contain traveling waves connecting the steady states E r and E * r . We shall conduct further research on this issue in the future.

Data Availability
All the data have been included in the paper; therefore, there are no other data available.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this article.