# Dynamics Analysis of a Betel Nut Addiction Spreading Model on Scale-Free Networks

Authors: He Wang; Tao Li; Xinming Cheng; Yu Kong; Yangmei Lei
Journal: Complexity
 (2020)

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## Abstract

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 numberR0 and equilibria are derived. Theoretical results indicate that the basic reproductive number R0 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 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 addiction-prevailing equilibrium is globally attractive. Finally, numerical simulations confirm the theoretical analysis results.


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## Body

## 1. Introduction

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 [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. They 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 [16–19], 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 [26]. Barabási and Albert studied the impact of neighboring infection on the computer virus spread [27]. 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 Section2, 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 Figure1. 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.Figure 1
The flow diagram of the PMSR model.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 probabilityρ1. 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 ρ2.(2)
The parameterα 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 probabilityφ or get rid of the addiction with probability w1.(4)
The severe addict may get rid of the addiction with probabilityw2.(5)
Considering the relapse of addiction, a recovered individual may convert into a mild addict with probabilityf.(6)
Here, we assume that the immigration rate equals the emigration rate, and the rate constant isμ.We definePkt, Mkt, Skt, and Rkt 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:(1)dPktdt=μ−kαρ1θ1+ρ2θ2Pkt−μPkt,dMktdt=kαρ1θ1+ρ2θ2Pkt+fRkt−φ+μ+w1Mkt,dSktdt=φMkt−w2+μSkt,dRktdt=ω2Skt+w1Mkt−f+μRkt,where θ1t and θ2t denote the probability of a contact pointing to a mild addict and a severe addict, respectively, and(2)θ1t=1k∑k=1∞kHkMkt,(3)θ2t=1k∑k=1∞kHkSkt.Here,k=∑kkHk represents the average degree values of the network and Hk represents the degree distribution. Mt=∑kHkMkt is the density of all the mild addicts and St=∑kHkSkt is the density of all the severe addicts.According to the normalization conditions, we have(4)Pkt+Mkt+Skt+Rkt=1.Obviously, the initial values for system (1) are as follows:(5)Pk0>0,Mk0≥0,Sk0≥0,Ck0≥0,k=1,2,...,n.Then, we obtain that(6)Pkt>0,Mkt≥0,Skt≥0,Ckt≥0,k=1,2,...,nandt≥0.


## 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:(7)R0=αf+μρ1μ+w2+ρ2φw2+μw1+f+μ+φw2+f+μμ.
There always exists an addiction-elimination equilibriumE01,0,0,0, and if R0>1, system (1) has a unique addiction-prevailing equilibrium E+Pk∗,Mk∗,Sk∗,Rk∗.Proof.
It can be easy to find that system (1) satisfiesRkt=1−Pkt−Mkt−Skt. According to system (1), we can get(8)dPktdt=μ−kαρ1θ1+ρ2θ2Pkt−μPkt,dMktdt=kαρ1θ1+ρ2θ2Pkt+f1−Pkt−Mkt−Skt−φ+μ+w1Mkt,dSktdt=φMkt−d+μSkt.
Obviously, there is an addiction-elimination equilibriumE01,0,0,0 in system (1). It is easy to verify that system (8) satisfies the conditions A1−A5 in [33]. By using the next generation matrix method [33], system (8) can get(9)dxdt=Zx−Vx,where(10)x=Pk,Mk,SkT,Zx=−kαρ1θ1t+ρ2θ2tPktkαρ1θ1t+ρ2θ2tPkt0,Vx=μPktw1+μ+φMkt−fCktw2+μSkt−φMkt.
At the addiction-elimination equilibriumE0, the Jacobian matrices of Zx and Vx can get(11)Z=JZE0=0Z12Z130Z22Z23000,V=JVE0=V1100V21V22V230V32V33,where(12)Z12=−αρ1k12⋮n1P12P2⋯nPn,Z13=−αρ2k12⋮n1P12P2⋯nPn,Z22=αρ1k12⋮n1P12P2⋯nPn,Z23=αρ2k12⋮n1P12P2⋯nPn.
Here,(13)V11=μI,V21=fI,V22=w1+μ+φ+fI,V23=fI,V32=−φI,V33=w2+μI,where I is the identity matrix. So, we can get the basic reproductive number:(14)R0=ρZV−1=k2kαf+μρ1μ+w2+ρ2φw2+μw1+f+μ+φw2+f+μμ,where k2=∑kk2Pk.
Next, it is clear that system (1) has an equilibrium E0=1,0,0,0k. To get the addiction-prevailing equilibrium E∗Pk∗,Mk∗,Sk∗,Rk∗, system (1) satisfies(15)dPktdt=0,dMktdt=0,dSktdt=0,dRktdt=0.
So, we can know(16)μ−kαρ1θ1+ρ2θ2Pkt−μPkt=0,kαρ1θ1+ρ2θ2Pkt+fRkt−φ+μ+w1Mkt=0,φMkt−w2+μSkt=0,w2Skt+w1Mkt−f+μRkt=0.
According to the above equations, we get(17)Pk=μw1w2+μ+f+μw2+μ+φw2+f+μkαρ1θ1+ρ2θ2w2+μf+μMk,Sk=φw2+μMk,Rk=w2+μw1+φw2w2+μf+μMk.
By using the normalization conditionPk∗t+Mk∗t+Sk∗t+Rk∗t=1, we get(18)Pk∗=μw1w2+μ+f+μw2+μ+φw2+f+μkaρ1θ1∗+ρ2θ2∗w2+μf+μMk∗,Mk∗=f+μw1w2+μ+μf+μw2+μ+φw2+f+μw2+μf+μw1+μ+φ,Sk∗=φw2+μMk∗,Rk∗=w2+μw1+φw2w2+μf+μMk∗.And, for θ1∗=1/k∑k=1∞kPkMk∗t, θ2∗=1/k∑k=1∞kPkSk∗t, we get(19)θ1∗=w2+μφθ2∗,θ2∗=φw2+μθ1∗.
By substituting the second equation of system (16) into equation (2), we get(20)θ1∗=1k∑k=1∞kPkMk∗t=1k∑k=1∞kPkkαρ1θ1∗+ρ2θ2∗w2+μf+μkαρ1θ1∗+ρ2θ2∗+μw2+μf+μ+φf+μ=1k∑k=1∞k2Pkkαf+μw2+μρ1w2+μ+ρ2φθ1∗kαw2+μf+μρ1w2+μ+ρ2φθ1∗+Ak=fθ1∗,where(21)Ak=μw2+μf+μw2+μ+φ+kαφ+kαφf+μw2+μρ1+φρ2θ1∗.
Obviously,θ1∗=0 satisfies system (20). System (20) has a unique nontrivial solution provided that(22)dfθ1∗dθ1∗Θ1∗=0>1,f1≤1.
So, we get(23)R0=k2kαf+μρ1μ+w2+ρ2φw2+μw1+f+μ+φw2+f+μμ>1.
Thus, a unique nontrivial solution exists if and only ifR0>1 According to equation (17), we know(24)0<Pk∞<1,0<Mk∞<1,0<Sk∞<1,0<Rk∞<1.
Therefore, the addiction-prevailing equilibriumE∗Pk∗,Mk∗,Sk∗,Rk∗ is well defined. Then, when R0>1, there is a unique positive equilibrium E∗Pk∗,Mk∗,Sk∗,Rk∗. The proof is completed.Remark 1.
The basic reproductive numberR0 is obtained by equation (23), which depends on the fluctuations of the degree distribution k2/k and some model parameters. When the heterogeneity of the degree distribution k2/k is larger, the basic reproductive number R0 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 R0 increases.Theorem 2.
IfR0<1, the addiction-elimination equilibrium E0 is globally asymptotically stable. If R0>1, the addiction spreading phenomenon is persistent, i.e., there exists a constant η>0, such that(25)limt⟶∞infMt+St=limt⟶∞inf∑kHkMkt+Skt≥η.Proof.
For the addiction-elimination equilibrium, system (8) has the Jacobian matrix as follows:(26)J=A11⋯A1n⋮⋱⋮An1⋯Ann,where(27)A11=αρ1P1−w1+μ+φαρ2P1fφ−w2+μ0w1w2−f+μ,A1n=αρ1Pnαρ2Pn0000000,An1=nαρ1Pnnαρ2Pn0000000,Ann=nαρ1Pn−w1+μ+φnαρ2Pnfφ−w2+μ0w1w2−f+μ.
So, the characteristic polynomial of the addiction-elimination equilibrium is(28)λ+w1+μ+φn−1λ+w2+μn−1λ2+pλ+qλ+f+μn−1=0,where(29)p=w2+μw1+f+μ+φw2+f+μ+f+μw1+μ+μw2+μ+φ−ρ1f+μ+w2+μ+ρ2φα∑i=1niPi−fw1,q=w1+μ+φw2+μf+μ+fφw2−αf+μρ1w2+μ+ρ2φ∑i=1niPi−fw1w2+μ.
WhenR0<1, we can get(30)R0=k2kαf+μρ1μ+w2+ρ2φw2+μw1+f+μ+φw2+f+μμ<1.
It also means(31)w2+μw1+f+μ+φw2+f+μμ>αf+μρ1μ+w2+ρ2φ∑i=1niPi.
In other words, we getp>0 and q>0. According to the above proof, the real eigenvalues λ of matrix J are all negative when R0<1. Furthermore, if and only if R0>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 R0>1. Through the theorem of Lajmanovich and York [34], we can get the results. The proof is completed.Lemma 1 (see [35]).
Ifa>0,b>0 and dxt/dt≥b−ax, when t≥0 and x0≥0, we have limt⟶∞infxt≥b/a; if a>0,b>0 and dxt/dt≤b−ax, when t≥0 and x0≥0, we have limt⟶∞supxt≤b/a.
Next, the global attractivity of the addiction-prevailing equilibrium is discussed. The main result is given in the following theorem.Theorem 3.
Suppose thatPkt,Mkt,Skt is a solution of system (8) satisfying initial conditions Mkt>0 or Skt>0. If R0>1, then(32)limt⟶∞Pkt,Mkt,Skt=Pk∗t,Mk∗t,Sk∗t,where Pk∗t,Mk∗t,Sk∗t is the addiction-prevailing equilibrium of system (8) for k=1,2,…,n.Proof.
In the course of the proving, we followed the methods of reference [35]. In the following, we suppose that k is an integer between 1 and n. According to Theorem 2, there exists a positive constant 0<τ<1/3 and a large enough constant T>0 such that Mkt≥τ and Skt≥τ for t>T. Therefore, θ1t>τ,θ2t>τ for t>T. Substituting this into the first equation of system (8), it is easy to get(33)dPktdt=μ−kαρ1θ1+ρ2θ2Pkt−μPkt≤μ−kαρ1+ρ2τ+μPkt,t>T.
From Lemma1, according to the standard comparison theorem of differential equation theory, for any given positive constant 0<τ1<kρ1+ρ2τ/μ+kaρ1+ρ2τ, there exists t1>T, such that Pkt≤Xk1−τ1 for t>t1, where(34)Xk1=μμ+kαρ1+ρ2τ+2τ1<1.
From system (8), it is easy to obtain(35)dMkdt=kαρ1θ1+ρ2θ2+f1−Mkt−w1+μ+φMkt,t>t1.
So, the constant0<τ2<min1/2,w1+μ+φ/2kαρ1+ρ2τ+w1+μ+φ+f, there exists t2>t1, so Mkt≤Yk1−τ2 for t>t2, where(36)Yk1=2kρ1+ρ2τ+f2kρ1+ρ2τ+f+w1+μ+φ+2τ2<1.
From system (8), it is easy to obtain(37)dSktdt⋅≤φ1−Skt−w2+μSkt,t>t2.
Thus, for constant0<τ3<min1/3,τ2,w2+μ/2φ+w2+μ, there exists t3>t2, so Skt≤Zk1−τ3 for t>t3, where(38)Zk1=φφ+w2+μ+2τ3<1.
Then, replacingMkt≤Yk1−τ2 and Skt≤Zk1−τ3 into the first equation of system (8), we get(39)dPkt⋅dt=μ−kαρ1θ1t+ρ2θ2tPkt−μPkt≥μ−kαρ1+ρ2τ+μPkt,t>T.
Thus, for constant0<τ4<min1/4,μ/2kαρ1+ρ2τ+μ, there exists t4>t3, such that Pkt≥xk1+τ4 for t>t4, where(40)xk1=μαkρ1+ρ2τ+μ−2τ4>0.
Then, substitutingθ1t>τ,θ2t>τ into system (8), it follows that(41)dMktdt⋅≥kαρ1+ρ2τxk1−w1+μ+φMkt+fτ,t>t4.
Hence, for constant0<τ5<min1/5,kαρ1+ρ2τxk1+fτ/2w1+μ+φ, there exists t5>t4, such that Mkt≥yk1+τ5 for t>t5, where(42)yk1=kαρ1+ρ2τxk1+fτw1+μ+φ−2τ5>0.
Therefore,(43)dSk⋅tdt≥φyk1−w2+μSkt,t>t5.
Hence, for constant0<τ6<min1/6,τ5,φyk1/2w2+μ, there exists t6>t5, such that Skt≥zk1+τ6 for t>t6, where(44)zk1=φyk1w2+μ−2τ6>0.
Becauseτ is a small positive constant, we can get(45)0<xk1<Pk<Xk1<1,0<yk1<Mk<Yk1<1,0<zk1<Sk<Zk1<1.
Let(46)qj=∑i=1nPiyij+zij,Qj=∑i=1nPiYij+Zij,j=1,2,…,n.
From the above discussion, we have(47)0<q1≤ρ1θ1+ρ2θ2≤Q1<ρ1+ρ2,t>t6.And, we substitute Mkt≥yk1+τ5, Skt≥zk1+τ6 into system (8), we can get(48)dPk⋅tdt≤μ−kαq1+μPkt,t>t6.
Consequently, for constant0<τ7<min1/7,τ6, there exists t7>t6, such that(49)Pkt≤Xk2≜minXk1−τ1,μμ+kαq1+τ7,t>t7.
Thus,(50)dMk⋅tdt≤kαQ1Xk2−w1+φ+μMkt+f1−Mkt−xk1−zk1≤kαQ1Xk2−w1+φ+μMkt+f1−xk1−zk1,t>t7.
Consequently, for constant0<τ8<min1/8,τ7, there exists t8>t7, such that(51)Mkt≤Yk2≜minYk1−τ2,kQ1Xk2+f1−xk1−zk1w1+φ+μ+f+τ8,t>t8.
As a result, it follows that(52)dSk⋅tdt≤φMk2−w2+μSkt,t>t9.
Therefore, for constant0<τ9<min1/9,τ8, we can get(53)Skt≤Zk2≜minZk1−τ3,φYk2w2+μ+τ9,t>t9.
SubstitutingMkt≤Yk2,Skt≤Zk2 and ρ1θ1+ρ2θ2≤Q2 into system (8), where(54)dPk⋅tdt≥μ−kαQ2+μPkt,t>t9.
Hence, for constant0<τ10<min1/10,τ9,μ/2kαQ2+μ, there exists t10>t9, such that Pkt≥xk2+τ10 for t>t10, where(55)xk2=maxxk1+τ4,μμ+kαQ2−2τ10.
Thus,(56)dMk⋅tdt≥kαq1xk2−φ+w1+μMkt+f1−Mkt−xk1−zk1,t>t10.
So, for constant0<τ11<min1/11,τ10,kαq1xk2+f1−xk1−zk1/2w1+φ+μ+f, there exists t11>t10, and Mkt≥yk2+τ11 for t>t11, where(57)yk2=maxyk1+τ5,kαq1xk2+f1−xk1−zk1w+f+φ+μ−2τ11.
Similarly,(58)dSk⋅tdt≥φyk2−w2+μSkt,t>t11.
Therefore, for constant0<τ12<min1/12,τ11,φyk2/2w2+μ, there exists t12>t11, and Skt≥zk2+τ12 for t>t12, where(59)zk2=maxzk1+τ6,φyk2w2+μ−2τ12.
According to the above discussion and analyses, we can obtain six sequences:Xkl, Ykl, Zkl, xkl, ykl, and zkl. 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 H such that, for l≥H,(60)Xkl=μμ+kαql−1+τ6l−5,Ykl=kαQl−1Xkl+f1−xkl−1−zkl−1w1+φ+f+μ+τ6l−4,Zkl=φYklw2+μ+τ6l−3,xkl=μμ+kαQl−2τ6l−2,ykl=kαql−1xkl+f1−Xkh−Zkhw1+φ+f+μ−2τ6l−1,zkl=φyklw2+μ−2τ6l−1.
It is easy to find that(61)xkl≤Pkt≤Xkl,ykl≤Mkt≤Ykl,zkl≤Skt≤Zkl,t>t6l.
Since the sequential limits of system (60) exist, let liml⟶∞Ωkl=Ωk, where Ωkl∈Xkl,Ykl,Zkl,xkl,ykl,zkl,Qkl,qkl and Ωk∈Xk,Yk,Zk,xk,yk,zk,Qk,qk.
Since0<τl<1/l, it has τl⟶0 as l⟶∞. Supposing l⟶∞, it follows from (61) that(62)Xk=μkαq+μ,Yk=kαQXk+f1−xk−zkw1+f+μ+φ,Zk=φw2+μYk,xk=μkαQ+μ,yk=kαqxk+f1−Xk−Zkw1+f+μ+φ,zk=φw2+μyk,where(63)q=∑i=1nPiyi+zi,Q=∑i=1nPiYi+Zi.
Furthermore,(64)Yk=kQαw2+μw2+μkαQμ+fkαq+μ+φfkαq+μ−kαq+μfw2+μw2+μ+φkαq+fφkαq+μkαQ+μw1+f+μ+φw2+μw2+μ,yk=kqαw2+μw2+μkαqμ+fkαQ+μ+φfkαQ+μ−kαQ+μfw2+μw2+μ+φkαQ+fφkαq+μkαQ+μw1+f+μ+φw2+μw2+μ.
Substituting (62) and (64) into Q and q, respectively, we obtain(65)1=φ+w2+μw2+μ∑i=1niPikαw2+μw2+μkαQμ+fkαq+μ+φfkαq+μ−kαq+μfw2+μw2+μ+φkαq+fφkαq+μkαQ+μw1+f+μ+φw2+μw2+μ,1=φ+w2+μw2+μ∑i=1niPikαw2+μw2+μkαqμ+fkαQ+μ+φfkαQ+μ−kαQ+μfw2+μw2+μ+φkαQ+fφkαq+μkαQ+μw1+f+μ+φw2+μw2+μ.
From the above two equations, we can get(66)Q−qφ+w2+μw2+μ∑i=1niPikaq+μ−kaQ+μfw2+μw2+μ+φkaQ+fφkaq+μkaQ+μw1+f+μ+φw2+μw2+μ≡0.
It is clear thatQ=q. So, ∑i=1nPiYi−yi+Zi−zi=0, which is equivalent to Yi=yi and Zi=zi for 1≤i≤n. Then, from systems (61) and (62), it can be concluded that(67)limt⟶∞Pkt=Xk=xk,limt⟶∞Mkt=Yk=yk,limt⟶∞Rkt=Zk=zk.
Finally,Q=q is substituted into system (64). For system (62), we can get Xk=Pk∗, Yk=Mk∗, and Zk=Rk∗. This proof is completed.


## 4. Numerical Simulations

This section illustrates the analytical results through numerical simulations. Based on scale-free network, we haveHk=ck−3 in system (1), and the parameter c satisfies ∑k=1nck−3=1, n=1000.In Figure2, the parameters are chosen as μ=0.3, α=0.2, φ=0.3, w2=0.5, w1=0.3, and f=0.6. We can see that the larger the ρ1 and ρ2, the larger the R0, 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.Figure 2
The influence of parametersρ1 and ρ2 on R0.In Figure3, we choose μ=0.3, α=0.2, φ=0.4, w2=0.4, w1=0.3, f=0.6, ρ1=0.4, and ρ2=0.4, thus the basic reproductive number R0=0.8090<1. We can see that there is almost no spreading of betel nut addiction when R0<1, which means that betel nut addiction will eventually disappear. It also suggests that the addiction-elimination equilibrium E0 is globally asymptotically stable when R0<1. This 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.Figure 3
The time series and orbits of four states withR0<1.In Figure4, we choose μ=0.3, α=0.5, φ=0.4, w2=0.3, w1=0.3, f=0.6, ρ1=0.7, and ρ2=0.7, thus the basic reproductive number R0=3.7163>1. We can see that the number of mild addicts and severe addicts will maintain at a positive constant, i.e., the spreading phenomenon of betel nut addiction is permanent on the scale-free networks when R0>1.Figure 4
The time series and orbits of four states withR0>1.In Figure5, we choose μ=0.4, φ=0.6, w2=0.6, w1=0.4, f=0.2, ρ1=0.2, and ρ2=0.2, thus the basic reproductive number R0<1. We can see that the corresponding M100 and S100 decrease significantly as the parameter α decreases, i.e., a smaller parameter α can weaken the spreading phenomenon of betel nut addiction. This also means that the marketing strategies of betel nut merchants have a great influence on the spreading of betel nut addiction.Figure 5
The prevalence ofM100andS100 versus t to different α with R0<1.
(a)(b)In Figure6, we choose μ=0.3, φ=0.4, w2=0.3, w1=0.3, f=0.6, ρ1=0.7, and ρ2=0.7, thus the basic reproductive number R0>1. Figure 6 describes the change of the mild addicts M100 and severe addicts S100 with different rate α. We can see that when R0>1, M100 and S100 both grow to positive constants with the increasing of parameter α. This 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.Figure 6
The prevalence ofM100 and S100 versus t to different α with R0>1.
(a)(b)


## 5. 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 numberR0 addiction-elimination equilibrium E0, and addiction-prevailing equilibrium E+ If R0<1, the addiction-elimination equilibrium E0 is globally asymptotically stable, i.e., betel nut addiction spreading behavior will eventually disappear. If R0>1, the spreading dynamical behavior of betel nut addiction is persistent and globally asymptotically stable. Moreover, we found that the larger the parameters ρ1 and ρ2, the larger the basic reproductive number R0, 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. Therefore, 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. The research results have important guiding significance in controlling the spreading of betel nut addiction.



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*Source: 3457068-2020-09-27.xml*