The analysis of a drug transmission model with family education and public health education.

In this paper, we formulate a six dimensional drug transmission model to study the effect of family education and public health education. The dynamical behaviors of the model are discussed in terms of the basic reproduction number R 0 . By constructing Lyapunov functions, we obtain the drug-free equilibrium is globally asymptotically stable if R 0 ≤ 1 and the drug addiction equilibrium is globally asymptotically stable if R 0 > 1 . Sensitivity analyses are performed to seek for effective control measures for drug spread. The analysis show that both the family and public health education can influence the spread of drug transmission. However, the combination of family and public health education is more effective to reduce the prevalence of drugs. Some numerical simulations are given to confirm the obtained theoretical results.

Introduction

Drug abuse is one of the most serious health and social problems around the world and has attracted governments' attentions. More and more people are infected by various drugs. Curiosity, thrill, seeking negative mentality and environmental impact are the main key factors to tempt susceptible individuals to contact with drugs. The U.S. Centers for Disease Control and Prevention in 2015 reported that from 2002 to 2013, the number of Americans over the age of 12 who snort heroin increased 63 and the deaths increased three times in America (News xinhuanet). According to “China's drug situations report in 2015”, by the end of 2015, the registered drug addicts are 2.345 million which increased by 14.6 compared to 2014 (Legal people). In addition, the crimes caused by taking drugs are growing rapidly, such as theft, robbery, self-injury, violence and traffic accidents and so on, which severely damage public security and public safety. It was reported by South African police statistics that the drug-related crimes increased from 621 to more than 3000 from 2002 to 2006 in Cape Town (Kapp, 2008). In addition, the drug abuse also aggravates the spread of infectious diseases such as hepatitis and AIDS (Mushanyu, Nyabadza, & Stewart, 2015; Nyabadza, Njagarah, & Smith, 2013). The AIDS patients increased from 13.1 in 2003 to in 2005 in the Western Cape Province of South Africa reported by antenatal survey in 2005 (Parry Charles et al., 2009). How to analyze the transmission mechanism of drugs better and present effective prevention measures is a problem that should be solved urgently.

Mathematical models play very important role in dealing with epidemics and biology, see (Cen, Feng, & Zhao, 2014; Driessche, Watmough, & van den Driessche, 2002; Feng, Cen, Zhao, & Velasco-Hernandez, 2015; Funk, Salathe, & Jansen, 2010; Huo, Chen, & Wang, 2016; Li, Graef, & Wang, 1999; Li, Zhao, & Li, 2013; Li, Zhao, & Zhu, 2015) and references therein. The process of epidemics could be displayed by mathematical models vividly. As a kind of new epidemics, drug abuse and drinking have the similar characteristics with traditional epidemics, and several different mathematical models for drugs had been formulated and studied, see (Liu & Zhang, 2011; Muroya, Li, & Kuniya, 2014; White & Comiskey, 2007). White and Comiskey (2007) first presented a heroin model, in which they divided the total population into susceptible individuals, drug users not in treatment and drug users in treatment. They obtained the basic reproduction number and gave the sensitivity analysis of . Then they got the conclusion that prevention was better than cure. Based on the work of (White and Comiskey, 2007), Liu and Zhang (2011) took distributed delays into a heroin model. It was shown that the stabilities of equilibria and uniform persistence were related to the basic reproduction number . In view of “light drug” and “heavy drug”, Muroya et al. (2014) introduced an SIRS model and discussed the global stability by applying Lyapunov function techniques. The other drug models are referred to (Fang, Li, Martcheva, & Cai, 2014; Huang & Liu, 2013; Kalula & Nyabadza, 2011; Mulone & Straughan, 2009; Samanta, 2011; Wang, Yang, & Li, 2011).

It is well known that the social behaviour plays a vital role when infectious diseases appear and outbreak. The common ways are isolation, vaccination and publicity to the health safety and so on. Some scholars took into account of public health educational campaigns in their models. Bhunu, Mushayabasa, Kojouharov, and Tchuenche (2010) analyzed an HIV/AIDS model with the impact of educational programs. Mukandavire and his partners in (Mukandavire, Garira, & Tchuenche, 2009) studied the effects of public health educational campaigns on the spread of HIV/AIDS. They proved the campaigns could slow down the epidemic. For drinking, Xiang, Song, and Huo (2015) dealt with the global properties of a drinking model with public health education. They concluded that public health education is one of the effective measures to control the drinking problems.

Motivated by above works, we develop a six dimensional drug model with family education and public health education in the paper. According to analyze the model, we attempt to provide an effective suggestion for controlling drug transmission. The organization of this paper is as follows. In the next section, the model is built and some basic properties are derived. In Sec.3, the basic reproduction number is obtained and the stability of the drug-free equilibrium of system is discussed. In Sec.4, the existence and the stability of the drug-addiction equilibrium are investigated. In Sec.5, some numerical simulations are carried out to illustrate the theoretical results and the sensitivity analysis is performed with respect to model parameters. Some discussions and summaries are given in last section.

The model

The model description

In order to describe the spread process of drugs, we formulate a six dimensional drug model. The total population is divided into six compartments: susceptible individuals who do not accept the health education(S), susceptible individuals who accept the health education (C), light drug addicts (L), heavy drug addicts (H), drug users in treatment (T) and quit drug-users permanently (R), namely, the total population at time t is given by . The meanings of and R are demonstrated concisely in Table 1. There is evidence that drug abuse phenomenon tends towards younger (Comiskey, 1999), so we focus on the population over the age of 15 old. Generally speaking, the health education includes the family education and the public health education. The individuals before age 15 access to the drug safety education mainly from families, and after the age of 15, the corresponding education comes from society or school, which could be called public health education. We suppose that the new recruits entering the susceptible humans denoted by is a part of the general population. A proportion of these humans do not accept family education and the complementary proportion q accept family health education. Meanwhile, the proportion μ of S will be educated by public health and enter into class C. and are the effective contact rates when susceptible humans (S) contact with light drug addicts (L) and heavy drug addicts (H) respectively. Generally, we assume because a susceptible individual is easier captured by a heavy drug addict. Once someone accepts public health education, he will reject drugs and avoid contacting with addicts consciously, so the effective contacts rates would reduce. Therefore the public health educated susceptible individuals (C) turn to light addicts (L) at rates and where . Based on the aforementioned modeling assumptions, the transfer diagram of the proposed model is shown in Fig. 1.

Table 1

The state variables for model.

Variable	Description
S(t)	The number of susceptible drug individuals who do not accept the public health education at time t.
C(t)	The number of susceptible drug individuals who accept the public health education at time t.
L(t)	The number of light drug addicts at time t.
H(t)	The number of heavy drug addicts at time t.
T(t)	The number of drug users in treatment at time t.
R(t)	The number of permanent quit drug individuals at time t.

Fig. 1

The transfer diagram of drug model with health education.

Corresponding, the proposed model is defined by the following ordinary differential equations:

For convenience, we list all the parameters and their meanings in Table 2.

Table 2

Description of parameters.

Parameter	Description
d	Natural birth/death rate of humans.
Λ	The total size of new recruits.
q	popularizing rate of family education.
μ	The rate that susceptible individuals accept public health education.
β_1	The effective contact rate that susceptible individuals contact with light drug addicts.
β_2	The effective contact rate that susceptible individuals contact with heavy drug addicts.
ξ	The factor which results the effective contact rates decrease.
d_1	The death rate related the light drug addiction.
d_2	The death rate related the heavy drug addiction.
γ	The treatment rate of light drug addicts.
π	The transformation rate from light addicts to heavy addicts.
δ	The permanent quit drug rate because of public health education.
θ	The treatment rate of heavy addicts.
σ	The relapse rate after treatment.
m	The permanent quit drug rate after treatment.

Since the last equation of system (1) does not have any contribution to the other equations of system (1), denote , , , then from now on we only concentrate on the dynamics analysis of following reduced system:with the initial condition

From the point of view of biology, we only care about the dynamics of system (2) in the feasible region as follows:

In the next subsection, it can be showed that the set is positively invariant.

The basic properties

For system (2), it is necessary to prove all of solutions with initial conditions (3) remain positive for all . We first state the following conclusions:

All the solutions of system (2) with initial conditions (3) are nonnegative for all . For system (2), the set is positively invariant.

Suppose is any solution of model (2), we have

With the continuity of the solution of system (2), for all .

Adding all the equations of system (2), it is obtained

Thus from Birkhoff's and Rota's theorems (Birkhoff & Rota, 1998), it follows as , so the region is a positively invariant set.

The drug-free equilibrium

The basic reproduction number is an important index in epidemiology. It is a threshold value which could represent how many secondary infections result from the introduction of one infected individual into a population of susceptible (Diekmann & Heesterbeek, 2000). In the section, we calculate the basic reproduction number and give the kinetic properties of the drug-free equilibrium .

The basic reproduction number

There always exists a drug-free equilibrium of system (2), where . Next, the basic reproduction number of system (2) will be obtained applying the next generation matrix (Van den Driessche & Watmough, 2002) and the local stability of drug-free equilibrium will be established.

Rewrite , then it follows from system (2) that , where

Thus the corresponding linearized matrixes evaluated at drug-free equilibrium areand

Obviously, is non-negative. By directly calculate, the basic reproduction number , where . Here, we can interpret the dimensionless quantity is the drug generation number which could measure the average number of new drug users generated by a drug addict in a population of susceptible individuals.

By Theorem 2 in (Van den Driessche & Watmough, 2002), the following conclusion is obvious:

For system (2), the drug-free equilibrium is locally asymptotically stable if and unstable if .

The global stability of the drug-free equilibrium

In this subsection, the global stability of the drug-free equilibrium will be established by applying the Lyapunov functional technique.then the derivative of V is given bywhereand

For system (2), the drug-free equilibrium is globally asymptotically stable when .

Define Lyapunov function as follows:

It is easy to see when . Furthermore, if and only if . Meanwhile, by arithmetic geometric average inequality, we have

Therefore, for , and holds if and only if and . The largest invariant set in G is the singleton . By LaSalle (1976), the drug-free equilibrium is globally asymptotically stable.

The drug addiction equilibrium of system (2)

Now, we shall investigate the existence of the drug addiction equilibrium of system (2). The global stability of is considered in detail using Lyapunov function.

The existence of the drug addiction equilibrium

Let the right of system (2) is equal to zero, that isthen the drug addiction equilibrium is determined by system (5). Solving all equations of system (5), we get

Summing the first three equations of system (5), we get is the root of following equation:where

It is easy to verify that if , then , there is no positive root of equation (8); if , then , equation (8) exists a unique positive root . The conclusion is described by the following theorem:

When , the system (2) exist a unique positive equilibrium , where are described as (6) and (7), is the unique positive root of equation (8).

The global stability of the drug addiction equilibrium

In the subsection, we show the drug addiction equilibrium is globally asymptotically stable under certain conditions by using Lyapunov function.

When and , the equilibrium of system (2) is globally asymptotically stable.

Denote and . Construct the Lyapunov function to the drug addiction equilibrium of system (2) as follows:

Calculating the derivative of V along solution of system (2), we obtain

Since , thenand since , then

Using arithmetic geometric average inequality, we obtain

The holds if and only if . Therefore the singleton drug-addiction equilibrium is the maximum invariant set of system (2) on the set . By LaSalle (1976), the drug addiction equilibrium is globally asymptotically stable.

Sensitivity and numerical simulation

In the section, some numerical simulations are present in order to illustrate the theoretical results obtained above. We show the stabilities of equilibria of system (2). Some parameters of the model come from reference (Kalula & Nyabadza, 2011; Nyabadza et al., 2013), , and the rest of the parametric values are estimated for numerical requirement.

Fix , then , the drug-free equilibrium is globally asymptotically stable (see Fig. 2(a)).

Fig. 2

The stabilities of equilibria of system (2).

Fix , then , the drug addiction equilibrium is globally asymptotically stable (see Fig. 2(b)).

For infectious diseases, if the basic reproduction number can decrease below unity, the diseases would be controlled. In order to achieve the goal, it is essential to consider the sensitivity of parameters, especially q and μ. First, we calculate the following partial derivatives.

It is showed that decreasing the relapse rate σ could reduce the basic reproduction number, and increasing the quit drug rate permanently also achieve the same result. Furthermore, we pay attention to the effect of family education and public health education on the drug model in the paper, so the partial derivatives of with respect to q and μ are calculated as follows:

From the results we can see that the basic reproduction number can be less than unit if reinforce the public health education. At last, we simulate the relationship among the basic reproduction number , the popularizing rate of family education q and the public health education rate μ.

Fix , then the relationship of , q and μ is showed in Fig. 3. The image shows q and μ have an inversely proportional relationship with , which is consistent with above theoretical calculation.

Fig. 3

The relationship among the basic reproduction number , q and μ.

Discussion and summary

In the paper, we formulated a drug model in order to study the effect of the family and public health education on drugs spread. In the model, the drug addicts were divided into light drug addicts and heavy drug addicts and the permanent quit drug was considered. Due to the influence of public health education, the effective contact rates would decrease when susceptible individuals who accept the public health education contact with drug addicts, therefore, the factor is brought into and . Before accepting public health education, some susceptible individuals undergo the family education, so the proportions q and denote the initial new recruit rate entering susceptible humans. By means of the next generation matrix, we got the basic reproduction number of system (2). The global stability of system (2) has been proved by using the Lyapunov function. When , the drug-free equilibrium is globally asymptotically stable; when , the drug-addiction equilibrium is globally stable which indicates the drug problem will persist.

We also focus on the effect of family education and public health education on the dynamics of drug model. The sensitivity analysis implies both of them influence the spread of drug transmission. From Fig. 3, it is clear to see that increasing q and μ simultaneously could decrease the basic reproduction number more rapidly compared to just simply increasing q or μ, which could help government to take effect measures for controlling the drug spread.

Conclusion

From a six dimensional drug transmission model, we study the effect of family education and public health education. The analysis show that both the family and public health education can influence the spread of drug transmission. However, the combination of family and public health education is more effective to reduce the prevalence of drugs than one of them simply.

Conflicts of interest

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.