Dynamics analysis of an online gambling spreading model on scale-free networks

Nowadays, online gambling has a great negative impact on the society. In order to study the effect of people’s psychological factors, anti-gambling policy, and social network topology on online gambling dynamics, a new SHGD (susceptible–hesitator–gambler–disclaimer) online gambling spreading model is proposed on scale-free networks. The spreading dynamics of online gambling is studied. The basic reproductive number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}$\end{document} is got and analyzed. The basic reproductive number R0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{0}$\end{document} is related to anti-gambling policy and the network topology. Then, gambling-free equilibrium E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{0}$\end{document} and gambling-prevailing equilibrium E+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{ +} $\end{document} are obtained. The global stability of E0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{0}$\end{document} is analyzed. The global attractivity of E+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E_{ +} $\end{document} and the persistence of online gambling phenomenon are studied. Finally, the theoretical results are verified by some simulations.


Introduction
Online gambling has emerged with the wide use of network technology. Compared with traditional gambling, the online gambling is stronger interaction, higher concealment, and more difficult to control [1]. Obviously, online gambling spreads more easily and widely than traditional gambling. The widespread spread of online gambling phenomenon has a huge negative impact on society [2][3][4].
How to control the phenomenon of online gambling is very important. Some scholars have studied the phenomenon of online gambling from different aspects [5][6][7][8][9]. King and Barak [10] studied the characteristics of online gambling such as attraction, convenience, and reasons why people participate in gambling. Dickson-Gillespie et al. [11] found that effective educational programs, media campaigns, and public policy would be good for quitting gambling. In addition, we should note the network spread characteristic of online gambling [12]. So, it is important for us to study the spreading dynamics of online gambling. Through the study of online gambling dynamics, we can comprehensively and systematically learn about the spreading mechanism and influence factors, which is more helpful to control the spread of online gambling.
Research on spreading dynamics of online gambling is relatively rare at present. There are some results in information spreading dynamics and disease spreading dynamics [13][14][15][16][17][18]. Liu et al. [19] studied the spread dynamics of word-of-mouth. Wang et al. [20] proposed a network epidemic model for waterborne diseases spread and considered both indirect environment-to-human and direct human-to-human transmission routes. King et al. [21] established a two-way model, studied the influence of some background factors to gambling spread. However, in the research works mentioned above, the persistence of online gambling phenomenon and the global attractivity of online gambling equilibrium are not studied. Meanwhile, some researchers found that the scale-free property is an important property of social networks [22,23]. Obviously, the spread networks of online gambling are based on social networks. So, based on scale-free networks, we study the dynamics of online gambling in the paper. Taking into account people's psychological factors, anti-gambling policy, we present a new comprehensively SHGD (susceptible-hesitatorgambler-disclaimer) online gambling spreading model.
The rest of the paper is as follows: The SHGD online gambling spreading model is presented and described in Sect. 2. The basic reproductive number R 0 , gambling-free equilibrium E 0 , and gambling-prevailing equilibrium E + are derived in Sect. 3. Then, the stability of E 0 , the global attraction of E + , and the persistence of online gambling phenomenon are studied. Some simulations are shown in Sect. 4. We conclude the paper in Sect. 5.

Model formulation
We present a new SHGD (susceptible-hesitator-gambler-disclaimer) online gambling spreading model. The model has the spread sketch in Fig. 1. In the model, nodes are used to stand for individuals, and edges are used to stand for the relationships between individuals. The whole crowd is divided into four different classes, namely susceptible (S), hesitator (H), gambler (G), and disclaimer (D). S nodes represent individuals who are not involved in gambling currently and can be influenced by the online gambling behavior; H nodes represent individuals who know the phenomenon of online gambling and hesitate whether to participate in online gambling and can spread online gambling behavior; G nodes represent individuals who take part in online gambling and can spread the online gambling behavior; D nodes represent the individuals who have given up gambling.
The transitions of these states are as follows: (1) When a susceptible individual connects with a hesitator or a gambler, he or she can be influenced and become a hesitator with probability β 1 or β 2 , respectively. (2) The parameter ε represents the probability that a hesitator becomes a susceptible individual. The parameter η represents the probability that a hesitator becomes a gambler. The parameter χ indicates the influence degree of the anti-gambling policy Figure 1 The flow diagram of the SHGD model to the hesitator. Considering the influence degree of the anti-gambling policy, the hesitator will become a susceptible individual with the probability χε, in contrast, a gambler with the probability (1χ)η. (3) The parameter ϕ represents the probability that a gambler becomes a hesitator. The parameter μ represents the probability that a gambler becomes a disclaimer. The parameter ψ represents the influence degree of the anti-gambling policy to the gambler. Considering the influence degree of the anti-gambling policy, a gambler will become a hesitator or a disclaimer with the probability ψϕ or ψμ, respectively. A gambler will become a susceptible individual with the probability γ when he or she loses interest in online gambling. (4) Because of the psychological factors of the disclaimer, such as forgetting and so on, the disclaimer will become a susceptible individual with the probability λ. (5) The probability δ is the register rate and logout rate. Assume newcomers are susceptible individuals. We define S k (t), H k (t), G k (t), D k (t) as the relative densities of susceptible, hesitator, gambler, and disclaimer nodes at time t, respectively, where k is the node degree. According to the above description and assumption, we can get the SHGD model as follows: where θ 1 (t) is the probability of linking to a hesitator at time t and satisfies where θ 2 (t) is the probability of linking to a gambler at time t and satisfies Here, k represents the average degree values in the network, and Q(k) represents the degree distribution.
is the density of the hesitator, and G(t) = k Q(k)G k (t) is the density of the gambler. We make ρ(t) = β 1 θ 1 + β 2 θ 2 . And according to system (1), we can get According to the normalization conditions, we can know S k (t) + H k (t) + G k (t) + D k (t) = 1. The initial conditions for the system are as follows:

The basic reproductive number and equilibriums
In the section, we analyze the properties of the SHGD online gambling spreading model. Theorem 1 According to system (4), the basic reproductive number is defined as follows: Consider system (4), we can get: Proof It can be easy to find that system (4) According to system (4), we can get Obviously, there is a gambling-free equilibrium E 0 = {(0, 0, 0)} k in system (7). By using the next generation matrix method [24], system (7) can be written At E 0 , the Jacobian matrices of j(x) and l(x) are got where Here, where I is an identity matrix. So, we can calculate the basic reproductive number denoted by Next, it is clear that system (4) has a gambling-free equilibrium E 0 (1, 0, 0, 0). To get the gambling-prevailing equilibrium So, we can know According to the above equation, we get By using to the normalization condition S * And where ρ(t) = k kQ(k)(β 1 θ 1 + β 2 θ 2 )/ k . By substituting the second equation of system (16) into Eq. (2), we get to have a nontrivial solution, the following conditions should be satisfied: So, we get According to Eq. (16), we know 0 < S * k , H * k , G * k , D * k < 1. System (4) has the gambling- . Then, when the basic regeneration number R 0 > 1, there is a unique positive equilibrium The proof is completed.
Theorem 2 When R 0 < 1, the gambling-free equilibrium E 0 is global asymptotically stable. When R 0 > 1, online gambling phenomenon is persistent, which means there is a constant Proof For simplicity, let Q i = iQ(i)/ k . For the gambling-free equilibrium, system (7) has the Jacobian matrix of 3n × 3n as follows: where So, the characteristic polynomial of the gambling-free equilibrium E 0 is Obviously, when R 0 < 1, q > 0. It also means and In other words, we get s > 0, q > 0, and p > 0. According to the above proof, the real eigenvalues λ of matrix B are all negative when R 0 < 1. Furthermore, there is a unique positive eigenvalue λ of matrix B if R 0 > 1. By using the Perron-Frobenius theorem, the maximal real part of all eigenvalues of λ is positive only if R 0 > 1. Through the theorem of Lajmanovich and York [25], we can get the results. The proof is completed. (7), which satisfies Eq.
Proof In the proof, let us assume that k is integer between 1 and n. According to Theorem 2, a positive constant 0 < α < 1/3 and a sufficiently large constant T > 0 exist to satisfy H k (t) ≥ α and G k (t) ≥ α for t > T. Thus, ρ(t) > α(β 1 + β 2 ) for t > T. Submitting this into the first equation of system (7), it is easy to get for t > T. According to the standard comparison theorem in the theory of differential equations, for any given positive constant From system (7), it is easy to obtain for t > t 1 . So, the constant 0 < α 2 < min 1/2, α 1 , From system (7), it is easy to obtain for t > t 2 .
And, according to system (7), we can get for t > t 6 .
From the above equation, we get U = u. So, which is equivalent to M i = m i and A i = a i for 1 ≤ i ≤ n. Then, from systems (63) and (64), it can be concluded that Finally, U = u is substituted into system (65). For system (64), we can get The proof is completed.

Simulation results and analyses
In this section, the analysis results are illustrated through numerical simulations. Based on a scale-free network, we have Q(k) = ωk -3 in system (1), and the parameter ω satisfies  Figure 2 shows that when R 0 < 1, H 150 and G 150 will equal to zero eventually, which means that the spread of online gambling phenomenon will eventually disappear.
In  The orbits of four states over time when R 0 = 0.9544 < 1 Figure 3 The orbits of four states over time when R 0 = 6.1795 > 1 Figure 4 The orbits of the hesitator or the gambler with different degree when R 0 < 1 Figure 5 The orbits of the hesitator or the gambler with different degree when R 0 > 1 online gambling phenomenon will be persistent. Moreover, more people are involved in gambling with the increasing of degree.
In Fig. 9(a) and (b), we choose δ = 0.1, β 1 = 0.2, β 2 = 0.5, ε = 0.1, η = 0.3, ϕ = 0.2, μ = 0.6, γ = 0.1, λ = 0.1. Apparently, larger χ or ψ can lead to smaller R 0 , χ has a greater impact on R 0 . In other words, within a certain range of anti-gambling efforts, the antigambling policy helps to decrease the spread of online gambling, and the anti-gambling policy for the hesitator is more effective in reducing the spread of online gambling. It is more effective to decrease the spread of online gambling if they work together.

Conclusion
In this paper, we proposed a new SHGD online gambling spreading model and analyzed the spreading dynamics of online gambling. We obtained the basic reproductive number R 0 , gambling-free equilibrium E 0 , and gambling-prevailing equilibrium E + . If R 0 < 1, the  The relationship between the basic reproductive number R 0 and the parameters ψ and χ on scale-free networks gambling-free equilibrium is globally asymptotically stable, i.e., online gambling spreading phenomenon will eventually disappear. If R 0 > 1, the spread of online gambling phenomenon is persistent and globally asymptotically stable, i.e., online gambling is a universal phenomenon. Smaller β 1 and β 2 can lead to the lower number of the disseminator, and β 1 has a greater impact than β 2 . Furthermore, larger χ and ψ can speed up the disappearance of online gambling phenomenon, especially χ . That is, increasing the intensity of the anti-gambling policy on the hesitator or the gambler can restrain online gambling spreading, and the anti-gambling policy on the hesitator is more effective. This research results have important guiding significance in controlling the spreading of online gambling.