Nonlinear Dynamics of Complex SystemsView this Special Issue
Research Article | Open Access
He Wang, Tao Li, Xinming Cheng, Yu Kong, Yangmei Lei, "Dynamics Analysis of a Betel Nut Addiction Spreading Model on Scale-Free Networks", Complexity, vol. 2020, Article ID 3457068, 13 pages, 2020. https://doi.org/10.1155/2020/3457068
Dynamics Analysis of a Betel Nut Addiction Spreading Model on Scale-Free Networks
Medical research has shown that overeating betel nut can be addictive and can damage health. More serious cases may cause mouth cancer and other diseases. Even worse, people’s behavior habit of chewing betel nut may influence each other through social interaction with direct or indirect ways, such as face-to-face communication, Facebook, Twitter, microblog, and WeChat, which leads to the spreading phenomenon of betel nut addiction. In order to investigate the dynamic spreading characteristics of betel nut addiction, a PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model is presented on scale-free networks. The basic reproductive number and equilibria are derived. Theoretical results indicate that the basic reproductive number is significantly dependent on the topology of the underlying networks, and some influence parameters. The existence of equilibria is determined by the basic reproductive number . Furthermore, we prove that if the addiction-elimination equilibrium is globally asymptotically stable. If , the betel nut addiction spreading is permanent, and the addiction-prevailing equilibrium is globally attractive. Finally, numerical simulations confirm the theoretical analysis results.
The habit of chewing betel nut is common in many places around the world [1–8]. It is worth noting that overeating betel nut can be addictive and is harmful to health [8, 9]. It is found that betel nut affects the health of the nervous, gastrointestinal, metabolic, respiratory, and reproductive systems. More serious cases may cause mouth cancer and other diseases. Betel nut is classified in the first category of carcinogens by the International Agency for Research on Cancer [9–11]. Even worse, people’s behavior habit of chewing betel nut may influence each other through social contact, which leads to the spreading phenomenon of betel nut addiction. The widespread spread of betel nut addiction, in turn, exacerbates the damage to people’s health and even the whole society.
It is very important to control the spreading phenomenon of betel nut addiction. In recent years, the research on betel nut addiction has attracted the attention of many scholars and researchers. Murphy et al. confirmed that the formation of the “chewing betel nut” habit could be due to exposure to and influence by betel nut chewers through statistical analysis . As indicated in , the cultural differences can affect the spreading of betel nut addiction in society. Ghani et al. identified the factors which influence the development of betel nut addiction. They also proposed the health policies to prevent addiction . Moss et al.  showed that the critical factors of addiction were contact with addicts and the self-prevention consciousness. Many researchers studied the addictiveness and harmfulness of the “chewing betel nut” habit [16–19], but few focused on the spreading dynamics of betel nut addiction. There are some studies in social information and disease spreading dynamics [20–25]. Liu et al. studied the spreading dynamics of cyber violence . Barabási and Albert studied the impact of neighboring infection on the computer virus spread . These studies can give some help for the analysis of the spreading dynamics of betel nut addiction. Through the study of addiction spreading dynamics, we can comprehensively and systematically learn about the addiction spreading mechanism and influence factors, which is helpful to control the spread of betel nut addiction. Meanwhile, some researchers found that the scale-free property is an important property of social networks [28–32]. Obviously, the spread of betel nut addiction is based on social networks. So, based on scale-free networks, we study the spreading dynamics of betel nut addiction in the paper. Apparently, the marketing strategy of betel nut has a great influence on sales, and at the same time, it affects the spreading of betel nut addiction. Taking into account the influence of betel nut advertising campaigns and the heterogeneity of underlying spreading networks, we present a new comprehensively PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model.
The rest of the paper is as follows. In Section 2, the PMSR betel nut addiction spreading model is proposed and described. In Section 3, the basic reproductive number and equilibriums are derived at first. Then, we analyze the globally asymptotic stability of addiction-elimination equilibrium, the permanence of the addiction spreading, and the global attractivity of addiction-prevailing equilibrium. Section 4 presents the results of our numerical simulation. Finally, we conclude the paper in Section 5.
2. Model Formulation
We present a new comprehensively PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model. The model has the spread sketch in Figure 1. In the model, nodes are used to stand for individuals, and edges are used to stand for the relationships between individuals. The whole population is divided into four distinct classes, namely, potential individuals (P), mild addicts (M), severe addicts (S), and recovered individuals (R). P nodes refer to the people who do not have addictive behavior of chewing betel nut; M nodes refer to the people who have mild addiction of the “chewing betel nut” habit; S nodes refer to the people who have severe addiction of the “chewing betel nut” habit; and R nodes refer to the people who get rid of the addiction.
In the spreading process of betel nut addiction, these states are subjected to the following rules:(1)If a potential individual is influenced by a mild addict with direct or indirect ways, such as face-to-face communication, Facebook, Twitter, microblog, and WeChat, then he or she will convert into a mild addict with probability . Similarly, if a potential individual is influenced by a severe addict with direct or indirect ways, then he or she will convert into a mild addict with probability .(2)The parameter is defined as an influence parameter of betel nut advertisements. In real life, the marketing strategy of betel nut will influence people to try to purchase betel nut.(3)A mild addict may convert into a severe addict with probability or get rid of the addiction with probability .(4)The severe addict may get rid of the addiction with probability .(5)Considering the relapse of addiction, a recovered individual may convert into a mild addict with probability .(6)Here, we assume that the immigration rate equals the emigration rate, and the rate constant is .
We define , , , and as the relative densities of potential individuals, mild addicts, severe addicts, and recovered individuals nodes of degree k at time t, respectively. According to the above description and assumption, we can get the PMSR model as follows:where and denote the probability of a contact pointing to a mild addict and a severe addict, respectively, and
Here, represents the average degree values of the network and represents the degree distribution. is the density of all the mild addicts and is the density of all the severe addicts.
According to the normalization conditions, we have
Obviously, the initial values for system (1) are as follows:
Then, we obtain that
3. Stability Analysis of the Model
In this section, we analyze the dynamic properties of the PMSR betel nut addiction spreading model.
Theorem 1. According to system (1), the basic reproductive number is defined as follows:There always exists an addiction-elimination equilibrium , and if , system (1) has a unique addiction-prevailing equilibrium .
Proof. It can be easy to find that system (1) satisfies . According to system (1), we can getObviously, there is an addiction-elimination equilibrium in system (1). It is easy to verify that system (8) satisfies the conditions in . By using the next generation matrix method , system (8) can getwhereAt the addiction-elimination equilibrium , the Jacobian matrices of and can getwhereHere,where is the identity matrix. So, we can get the basic reproductive number:where .
Next, it is clear that system (1) has an equilibrium . To get the addiction-prevailing equilibrium , system (1) satisfiesSo, we can knowAccording to the above equations, we getBy using the normalization condition , we getAnd, for , , we getBy substituting the second equation of system (16) into equation (2), we getwhereObviously, satisfies system (20). System (20) has a unique nontrivial solution provided thatSo, we getThus, a unique nontrivial solution exists if and only if According to equation (17), we knowTherefore, the addiction-prevailing equilibrium is well defined. Then, when , there is a unique positive equilibrium . The proof is completed.
Remark 1. The basic reproductive number is obtained by equation (23), which depends on the fluctuations of the degree distribution and some model parameters. When the heterogeneity of the degree distribution is larger, the basic reproductive number is greater, i.e., the larger heterogeneity of the degree distribution can promote the spreading of betel nut addiction. Obviously, as the influence parameter of betel nut advertisements increases, the basic reproductive number increases.
Theorem 2. If , the addiction-elimination equilibrium is globally asymptotically stable. If , the addiction spreading phenomenon is persistent, i.e., there exists a constant , such that
Proof. For the addiction-elimination equilibrium, system (8) has the Jacobian matrix as follows:whereSo, the characteristic polynomial of the addiction-elimination equilibrium iswhereWhen , we can getIt also meansIn other words, we get and . According to the above proof, the real eigenvalues of matrix are all negative when . Furthermore, if and only if , there is a unique positive eigen value of matrix . By using the Perron–Frobenius theorem, the maximal real part of all eigenvalues of is positive only if . Through the theorem of Lajmanovich and York , we can get the results. The proof is completed.
Lemma 1 (see ). If and , when and , we have ; if and , when and , we have .
Next, the global attractivity of the addiction-prevailing equilibrium is discussed. The main result is given in the following theorem.
Proof. In the course of the proving, we followed the methods of reference . In the following, we suppose that is an integer between 1 and n. According to Theorem 2, there exists a positive constant and a large enough constant such that and for . Therefore, for . Substituting this into the first equation of system (8), it is easy to getFrom Lemma 1, according to the standard comparison theorem of differential equation theory, for any given positive constant , there exists , such that for , whereFrom system (8), it is easy to obtainSo, the constant , there exists , so for , whereFrom system (8), it is easy to obtainThus, for constant , there exists , so for , whereThen, replacing and into the first equation of system (8), we getThus, for constant , there exists , such that for , whereThen, substituting into system (8), it follows thatHence, for constant , there exists , such that for , whereTherefore,Hence, for constant , there exists , such that for , whereBecause is a small positive constant, we can getLetFrom the above discussion, we haveAnd, we substitute , into system (8), we can getConsequently, for constant , there exists , such thatThus,Consequently, for constant , there exists , such thatAs a result, it follows thatTherefore, for constant , we can getSubstituting and into system (8), whereHence, for constant , there exists , such that for , whereThus,So, for constant , there exists , and for , whereSimilarly,Therefore, for constant , there exists , and for , whereAccording to the above discussion and analyses, we can obtain six sequences: , , , , , and . We can find that the first three sequences are monotonically increasing, and the last three sequences are strictly monotonically decreasing, and there is a sufficiently large positive integer such that, for ,It is easy to find thatSince the sequential limits of system (60) exist, let , where and .
Since , it has as . Supposing , it follows from (61) thatwhereFurthermore,Substituting (62) and (64) into and , respectively, we obtainFrom the above two equations, we can getIt is clear that . So, , which is equivalent to and for . Then, from systems (61) and (62), it can be concluded thatFinally, is substituted into system (64). For system (62), we can get , , and . This proof is completed.
4. Numerical Simulations
This section illustrates the analytical results through numerical simulations. Based on scale-free network, we have in system (1), and the parameter satisfies , .
In Figure 2, the parameters are chosen as , , , , , and . We can see that the larger the and , the larger the , i.e., long time contact with mild addicts and severe addicts both can increase the spreading speed of betel nut addiction. This also means that if people’s health knowledge is improved, the probability of people chewing betel nut is reduced. The spreading of betel nut addiction will be weakened.
In Figure 3, we choose , , , , , , , and