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. /e basic reproductive number R0 and equilibria are derived. /eoretical results indicate that the basic reproductive number R0 is significantly dependent on the topology of the underlying networks, and some influence parameters./e existence of equilibria is determined by the basic reproductive number R0. Furthermore, we prove that if R0 < 1 the addiction-elimination equilibrium is globally asymptotically stable. If R0 > 1, the betel nut addiction spreading is permanent, and the addictionprevailing equilibrium is globally attractive. Finally, numerical simulations confirm the theoretical analysis results.


Introduction
e habit of chewing betel nut is common in many places around the world [1][2][3][4][5][6][7][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][10][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. e 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 [12]. As indicated in [13], 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. ey also proposed the health policies to prevent addiction [14]. Moss et al. [15] 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][17][18][19], but few focused on the spreading dynamics of betel nut addiction. ere are some studies in social information and disease spreading dynamics [20][21][22][23][24][25]. Liu et al. studied the spreading dynamics of cyber violence [26]. Barabási and Albert studied the impact of neighboring infection on the computer virus spread [27]. ese studies can give some help for the analysis of the spreading dynamics of betel nut addiction.
rough 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][29][30][31][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. e 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. en, 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.

Model Formulation
We present a new comprehensively PMSR (Potential-Mild-Severe-Recovered) betel nut addiction spreading model. e 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. e 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 ρ 1 . 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 ρ 2 . (2) e parameter α 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 φ or get rid of the addiction with probability w 1 . (4) e severe addict may get rid of the addiction with probability w 2 . (5) Considering the relapse of addiction, a recovered individual may convert into a mild addict with probability f. (6) Here, we assume that the immigration rate equals the emigration rate, and the rate constant is μ.
We define P k (t), M k (t), S k (t), and R k (t) 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 θ 1 (t) and θ 2 (t) denote the probability of a contact pointing to a mild addict and a severe addict, respectively, and Here, 〈k〉 � k kH(k) represents the average degree values of the network and H(k) represents the degree distribution. M(t) � k H(k)M k (t) is the density of all the mild addicts and S(t) � k H(k)S k (t) is the density of all the severe addicts.

Complexity
According to the normalization conditions, we have Obviously, the initial values for system (1) are as follows: en, we obtain that

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:
ere always exists an addiction-elimination equilibrium E 0 (1, 0, 0, 0), and if R 0 > 1, system (1) has a unique addiction- Proof. It can be easy to find that system (1) satisfies Obviously, there is an addiction-elimination equilibrium E 0 (1, 0, 0, 0) in system (1). It is easy to verify that system (8) satisfies the conditions (A 1 ) − (A 5 ) in [33]. By using the next generation matrix method [33], system (8) can get where At the addiction-elimination equilibrium E 0 , the Jacobian matrices of Z(x) and V(x) can get where

Complexity
Here, where I is the identity matrix. So, we can get the basic reproductive number: where 〈k 2 〉 � k k 2 P(k).
Next, it is clear that system (1) has an equilibrium So, we can know According to the above equations, we get By using the normalization condition And, for By substituting the second equation of system (16) into equation (2), we get Obviously, θ * 1 � 0 satisfies system (20). System (20) has a unique nontrivial solution provided that So, we get us, a unique nontrivial solution exists if and only if R 0 > 1 According to equation (17), we know erefore, the addiction-prevailing equilibrium . e proof is completed.

Remark 1.
e basic reproductive number R 0 is obtained by equation (23), which depends on the fluctuations of the degree distribution 〈k 2 〉/〈k〉 and some model parameters. When the heterogeneity of the degree distribution 〈k 2 〉/〈k〉 is larger, the basic reproductive number R 0 is greater, i.e., the larger heterogeneity of the degree distribution can promote the spreading of betel nut addiction. Obviously, as the influence parameter α of betel nut advertisements increases, the basic reproductive number R 0 increases.

Theorem 2.
If R 0 < 1, the addiction-elimination equilibrium E 0 is globally asymptotically stable. If R 0 > 1, the addiction spreading phenomenon is persistent, i.e., there exists a constant η > 0, such that Proof. For the addiction-elimination equilibrium, system (8) has the Jacobian matrix as follows: where So, the characteristic polynomial of the addictionelimination equilibrium is where When R 0 < 1, we can get It also means In other words, we get p > 0 and q > 0. According to the above proof, the real eigenvalues λ of matrix J are all negative when R 0 < 1. Furthermore, if and only if R 0 > 1, there is a unique positive eigen value λ of matrix J. By using the Perron-Frobenius theorem, the maximal real part of all eigenvalues of λ is positive only if R 0 > 1.
rough the theorem of Lajmanovich and York [34], we can get the results. e proof is completed.
Next, the global attractivity of the addiction-prevailing equilibrium is discussed. e main result is given in the following theorem. where is the addiction-prevailing equilibrium of system (8) for k � 1, 2, . . . , n.

Complexity 9
Finally, Q � q is substituted into system (64). For system (62), we can get X k � P * k , Y k � M * k , and Z k � R * k . is proof is completed.

Numerical Simulations
is section illustrates the analytical results through numerical simulations. Based on scale-free network, we have H(k) � ck − 3 in system (1), and the parameter c satisfies n k�1 ck − 3 � 1, n � 1000.
In Figure 2, the parameters are chosen as μ � 0.3, α � 0.2, φ � 0.3, w 2 � 0.5, w 1 � 0.3, and f � 0.6. We can see that the larger the ρ 1 and ρ 2 , the larger the R 0 , i.e., long time contact with mild addicts and severe addicts both can increase the spreading speed of betel nut addiction. is also means that if people's health knowledge is improved, the probability of people chewing betel nut is reduced. e spreading of betel nut addiction will be weakened.
is also means that restricting the marketing strategy of betel nut and controlling the recurrence rate of betel nut addiction will weaken the spreading of betel nut addiction.
is also means that the marketing strategies of betel nut merchants have a great influence on the spreading of betel nut addiction.
In Figure 6, we choose μ � 0.3, φ � 0.4, w 2 � 0.3, w 1 � 0.3, f � 0.6, ρ 1 � 0.7, and ρ 2 � 0.7, thus the basic reproductive number R 0 > 1. Figure 6 describes the change of the mild addicts M 100 and severe addicts S 100 with different rate α. We can see that when R 0 > 1, M 100 and S 100 both grow to positive constants with the increasing of parameter α. is figure shows that a larger α can lead to the larger value of betel nut addiction spreading level. It means that restricting the marketing strategies of betel nut plays a vital role in controlling the spread of betel nut addiction.

Conclusions
In this paper, a new comprehensively PMSR betel nut addiction spreading model has been proposed. By the mean-field theory, we have obtained the basic reproduction number R 0 addiction-elimination equilibrium E 0 , and addiction-prevailing equilibrium E + If R 0 < 1, the addiction-elimination equilibrium E 0 is globally asymptotically stable, i.e., betel nut addiction spreading behavior will eventually disappear. If R 0 > 1, the spreading dynamical behavior of betel nut addiction is persistent and globally asymptotically stable. Moreover, we found that the larger the parameters ρ 1 and ρ 2 , the larger the basic reproductive number R 0 , i.e., long-term and frequent contact with mild addicts and severe addicts can increase the spreading phenomenon of betel nut addiction. Furthermore, a larger α can increase the spread speed and range of betel nut addiction.
erefore, controlling the spread of betel nut addiction requires restricting the marketing strategies adopted by betel nut merchants and widely publicizing the harmfulness of chewing betel nut. e research results have important guiding significance in controlling the spreading of betel nut addiction.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

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