Stability Analysis of a Reaction-Diffusion Heroin Epidemic Model

Zhang, Liang; Xing, Yifan

doi:https://doi.org/10.1155/2020/3781425

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Research Article | Open Access

Volume 2020 | Article ID 3781425 | https://doi.org/10.1155/2020/3781425

Stability Analysis of a Reaction-Diffusion Heroin Epidemic Model

Liang Zhang¹and Yifan Xing¹

Academic Editor: Lucia Valentina Gambuzza

Received22 Jan 2020

Revised15 Mar 2020

Accepted04 May 2020

Published08 Jun 2020

Abstract

A reaction-diffusion (R-D) heroin epidemic model with relapse and permanent immunization is formulated. We use the basic reproduction number 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 , and if , the drug-addiction equilibrium is globally asymptotically stable. Some numerical simulations are also carried out to illustrate our analytical results.

1. 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. There 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 [5–16]. In 2007, White and Comiskey established an ODE model for heroin infectious diseases [14]. They studied the dynamics using a threshold 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 [18–38]. 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.

2. The Model

2.1. Model Formulation

We divide the total population into five compartments: . Here, represents the number of susceptible individuals who have never used heroin; represents the number of heroin users; represents the number of heroin users undergoing treatment; 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 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 will enter into the compartment of . 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. Thus, 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:

We assume that the bounded domain has a smooth boundary . As shown in (2), the homogeneous Neumann boundary conditions indicate that the population movements will not cross the border. is the usual Laplacian operator on . are the diffusion coefficients. As mentioned above, is a compartment of complete rehabilitation. The individuals in are permanently immunized, and we can see that the equation is uncoupled with the other equations of (2). Hence, the diffusion of 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:

2.2. The Basic Properties of Model (3)

Lemma 1. If the initial values are positive, then model (3) has positive solutions , , , , and for all .

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 is a solution of model (3) with nonnegative initial conditions, adding the five equations yieldswherewhich indicates thatwhere is the initial value. Thus, , as . This completes the proof.

3. Global Dynamics of the ODE Model (3)

3.1. The 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 of model (3).

Let , then system (3) can be written aswhere

The Jacobian matrices of and at the drug-free equilibrium are as follows:where

Then, the basic reproduction number is

The following result on the local stability of can be obtained directly by Theorem 2 in [41], and we thus omit its proof.

Theorem 1. The drug-free equilibrium is locally asymptotically stable for and unstable for .

Theorem 2. If , the drug-free equilibrium of system (3) is globally asymptotically stable.

Proof. Inspired by [28], we introduce the following Lyapunov function:It follows that the derivative of isHere, we sued the equalities . Since , if , we have . Clearly, if and only if . Substituting into system (3), we get as . According to LaSalle’s invariance principle [42], we obtain that the drug-free equilibrium of system (3) is globally asymptotically stable in if . This completes the proof.

3.2. Existence and Stability of the Drug-Addiction Equilibrium

Let the right side of model (3) be equal to zero. Then we get

Then, the unique drug-addiction equilibrium exists if , where

Theorem 3. If , then the unique drug-addiction equilibrium of system (3) is global asymptotically stable.

Proof. If , then there exists a unique drug-addiction equilibrium . We now introduce a Lyapunov function as follows:where , , and are the positive constants to be determined later. It follows that the derivative of isLet , , , and . Then, we haveThe variables with nonnegative coefficients in (20) are , , , and . If all the coefficients are positive, then is positive. If all the coefficients of , , , and are equal to zero, then we getBy (21), we obtainHence, we haveIt is clear that if and if only if . By the relationships between the arithmetic mean and the geometric mean, we get if and if and only if ; for and if and only if ; for and if and only if . Therefore, if and if and only if . Substituting into the first equation of system (3), we get . It follows from the first equation of (16) that . Therefore, the maximum invariant set of system (2) on set is the singleton . This implies that the largest invariant set where is the singleton . Thus, by LaSalle’s invariance principle in [42], the drug-addiction equilibrium of model (3) is globally asymptotically stable when . This completes the proof.

4. Global Dynamics of the R-D Model (2)

4.1. Positivity and Boundedness of the Solutions

Theorem 4. Let be a solution of system (2) and , where is the maximal existing time. If , then , for all .

Proof. By the first equation of model (2), we getthat is,Since , by Lemma 2.4.1 in [33], we get .
LetThen, the second to the fourth equations of system (2) can be rewritten asDefine operator () on as follows:We now consider the following parabolic system:whereApplying Theorem 4.2.4 in [33] to model (29), we get . So, by the second equation of model (2), we getThat is,Since , according to Lemma 2.4.1 in [33], . Analogously,That is,Similarly, we obtain . If does not hold, then there exist such that . By the fifth equation of model (2), we getSince , . This is a contradiction. Therefore, for all . This completes the proof.

Theorem 5. Let be a solution of model (2) and , where is the maximum time of existence. If , then the solution is bounded on .

Proof. By model (2), we getIt then follows thatBy Green’s formulas and Newman boundary , we haveHence,Let . Then, (39) becomesWhich indicates that , hereThis shows that is bounded. Let , thenBy Theorem 3.1 in [43], there is a positive constant depending on so thatHence, we obtain that are uniformly bounded on .
For the last equation of model (2), let be bounded by . Then,Hence, on . This completes the proof.

Let be a Banach space with a supremum norm:

Define , where , and let be a linear operator on defined by

Then, by [23], we obtain that are the infinitesimal generators of a strongly continuous semigroup in . For , there are

They are also the infinitesimal generator of a strongly continuous semigroup in , wheresince is a Banach space with norm

Let be a nonlinear operator on defined bywhere

Then, model (2) can be rewritten into a more abstract form in as follows:

By Proposition 4.16 in [44], it can be obtained that the unique continuously differentiable solution of the above equations has a maximum interval of existence so thatand either or . By Theorem 5, we obtain holds. Therefore, is a global solution.

4.2. Global Stability of the R-D Model

Since the equation of model (2) is uncoupled with the other equations, model (2) can be reduced by ignoring . It is clear that the R-D heroin epidemic model (2) has a drug-free equilibrium and a unique positive drug-addiction equilibrium if .

Theorem 6. If , the drug-free equilibrium is globally asymptotically stable.

Proof. We give a Lyapunov function as follows:where is given by (14) and . Direct computations show thatIt follows by Green’s identity thatTaking (56) into (55) and by Theorem 2, we obtain that if , . Furthermore, if . By LaSalle’s invariance principle in [42], the drug-free equilibrium is globally asymptotically stable. This completes the proof.

Theorem 7. If , then the drug-addiction equilibrium is globally asymptotically stable.

Proof. We give the following Lyapunov function:where andDirect calculation yieldsBy Theorem 3, we know that if , then . Therefore, if , then . By LaSalle’s invariance principle in [42], the drug-addiction equilibrium is globally asymptotically stable. This completes the proof.

5. Numerical Simulations

In this section, we shall carry some numerical simulations to illustrate our analytic results by using the parameter values in Table 1. We fix .

If we choose , then and . Give different initial values , respectively, and we can see that the drug-free equilibrium is globally asymptotically stable (Figure 2). When , we get , this is verified by the figure. We can see all solutions of the system converge to the drug-free equilibrium . If we keep unchanged and let , then . According to the above discussion, is asymptotically stable, and when , . Figure 3 not only illustrates the stability of , but also shows the number of people who are permanently immunized against drugs, () is greater than the number of susceptible people () in the equilibrium when . By Figures 2 and 3, we can not only clearly see that declined sharply and got to zero finally, but also see that all solutions of the system infinitely close to the drug-free equilibrium . It verifies the existence of .

(a)

(b)

(a)

(b)

If we choose parameters , and initial values and , then . Hence, the endemic equilibrium is global asymptotically stable (Figure 4). This also verifies the existence of .

(a)

(b)

6. 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 the next-generation matrix method, we obtained the basic reproduction number . We obtained the global dynamics of the model by constructing some suitable Lyapunov functions. It is shown when , the drug-free equilibrium is globally asymptotically stable; that is, the drug abuse will be eradicated; when , the endemic equilibrium is globally asymptotically stable, which means that drug abuse will be permanent.

For the ODE system (3), , and represent the detoxification success rate, detoxification failure rate, relapse rate from to abusers, permanent withdrawal rates from to , and permanent withdrawal rates from to , respectively. Figure 5 shows the relationship between and , , , and , with other parameter values as given in Table 1. As shown in Figures 5(a)–5(c), grows with and but decreases with , and it means that if we want to control heroin addiction, we should reduce and and increase , 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), decreases with , 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.

(a)

(b)

(c)

(d)

The effect of and on is shown in Figure 6. It indicates that the values of and have a significant effect on the number of drug-addiction equilibrium. Let us observe Figure 6(a) first, the larger the 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. The 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 on is less than the effect of on , it can still obtain that the larger the is, the fewer the people who use drugs in equilibrium is; 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.

(a)

(b)

The 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 the endemic equilibrium is global stability, and if , the drug-free equilibrium is global stability, which indicates that the R-D model (2) may contain traveling waves connecting the steady states and . 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

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This work was partly supported by the Natural Science Foundation of Shaanxi Province (no. 2020JM-175) and the China Scholarship Council (201806305025).

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Copyright © 2020 Liang Zhang and Yifan Xing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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