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

Abstract

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 \(R_{0}\) is got and analyzed. The basic reproductive number \(R_{0}\) is related to anti-gambling policy and the network topology. Then, gambling-free equilibrium \(E_{0}\) and gambling-prevailing equilibrium \(E_{ +} \) are obtained. The global stability of \(E_{0}\) is analyzed. The global attractivity of \(E_{ +} \) 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 [24].

How to control the phenomenon of online gambling is very important. Some scholars have studied the phenomenon of online gambling from different aspects [59]. 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 [1318]. 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–hesitator–gambler–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.

Figure 1
figure1

The flow diagram of the SHGD model

The transitions of these states are as follows:

  1. (1)

    When a susceptible individual connects with a hesitator or a gambler, he or she can be influenced and become a hesitator with probability \(\beta _{1}\) or \(\beta _{2}\), respectively.

  2. (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 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 \((1 - \chi )\eta \).

  3. (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. (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. (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:

$$ \textstyle\begin{cases} \frac{dS_{k}(t)}{dt} = \delta + \lambda D_{k}(t) + \gamma G_{k}(t) + \chi \varepsilon H_{k}(t) - k\beta _{1}\theta _{1}(t)S_{k}(t) - k\beta _{2}\theta _{2}(t)S_{k}(t) - \delta S_{k}(t), \\ \frac{dH_{k}(t)}{dt} = k\beta _{1}\theta _{1}(t)S_{k}(t) + k\beta _{2}\theta _{2}(t)S_{k}(t) + \psi \varphi G_{k}(t) \\ \hphantom{\frac{dH_{k}(t)}{dt} =}{}- (1 - \chi )\eta H_{k}(t) - \chi \varepsilon H_{k}(t) - \delta H_{k}(t), \\ \frac{dG_{k}(t)}{dt} = (1 - \chi )\eta H_{k}(t) - \psi \mu G_{k}(t) - \gamma G_{k}(t) - \psi \varphi G_{k}(t) - \delta G_{k}(t), \\ \frac{dD_{k}(t)}{dt} = \psi \mu G_{k}(t) - \lambda D_{k}(t) - \delta D_{k}(t), \end{cases} $$
(1)

where \(\theta _{1}(t)\) is the probability of linking to a hesitator at time t and satisfies

$$ \theta _{1}(t) = \frac{\sum_{k} k Q(k)H_{k}(t)}{\sum_{k} sQ(s)} = \frac{1}{ \langle k \rangle } \sum _{k} k Q(k)H_{k}(t), $$
(2)

where \(\theta _{2}(t)\) is the probability of linking to a gambler at time t and satisfies

$$ \theta _{2}(t) = \frac{\sum_{k} k Q(k)G_{k}(t)}{\sum_{k} sQ(s)} = \frac{1}{ \langle k \rangle } \sum _{k} k Q(k)G_{k}(t). $$
(3)

Here, \(\langle k \rangle \) represents the average degree values in the network, and \(Q(k)\) represents the degree distribution. \(H(t) = \sum_{k} Q(k)H_{k} (t)\) is the density of the hesitator, and \(G(t) = \sum_{k} Q(k)G_{k} (t)\) is the density of the gambler. We make \(\rho (t) = \beta _{1}\theta _{1} + \beta _{2}\theta _{2}\). And according to system (1), we can get

$$ \textstyle\begin{cases} \frac{dS_{k}(t)}{dt} = \delta + \lambda D_{k}(t) + \gamma G_{k}(t) + \chi \varepsilon H_{k}(t) - (k\rho (t) + \delta )S_{k}(t), \\ \frac{dH_{k}(t)}{dt} = k\rho (t)S_{k}(t) + \psi \varphi G_{k}(t) - ((1 - \chi )\eta + \chi \varepsilon + \delta )H_{k}(t), \\ \frac{dG_{k}(t)}{dt} = (1 - \chi )\eta H_{k}(t) - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t), \\ \frac{dD_{k}(t)}{dt} = \psi \mu G_{k}(t) - (\lambda + \delta )D_{k}(t). \end{cases} $$
(4)

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:

$$ \textstyle\begin{cases} 0 \le S_{k}(0),H_{k}(0),G_{k}(0),D_{k}(0) \le 1, \\ \rho (0) > 0. \end{cases} $$
(5)

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:

$$ R_{0} = \frac{ \langle k^{2} \rangle }{ \langle k \rangle } \frac{(\psi \mu + \gamma + \psi \varphi + \delta )\beta _{1} + \eta (1 - \chi )\beta _{2}}{(\psi \mu + \gamma + \delta )((1 - \chi )\eta + \chi \varepsilon + \delta ) + \psi \varphi (\chi \varepsilon + \delta )}. $$
(6)

Consider system (4), we can get:

  1. (1)

    There is a gambling-free equilibrium \(E_{0}(1,0,0,0)\) when \(R_{0} < 1\).

  2. (2)

    There is a unique gambling-prevailing equilibrium \(E_{ +} (S_{k}^{*},H_{k}^{*},G_{k}^{*},D_{k}^{*})\) when \(R_{0} > 1\).

Proof

It can be easy to find that system (4) satisfies \(S_{k}(t) = 1 - H_{k}(t) - G_{k}(t) - D_{k}(t)\). According to system (4), we can get

$$ \textstyle\begin{cases} \frac{dH_{k}(t)}{dt} = k\rho (t)(1 - H_{k}(t) - G_{k}(t) - D_{k}(t)) + \psi \varphi G_{k}(t) - ((1 - \chi )\eta + \chi \varepsilon + \delta )H_{k}(t), \\ \frac{dG_{k}(t)}{dt} = (1 - \chi )\eta H_{k}(t) - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t), \\ \frac{dD_{k}(t)}{dt} = \psi \mu G_{k}(t) - (\lambda + \delta )D_{k}(t). \end{cases} $$
(7)

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

$$ \frac{dx}{dt} = j(x) - l(x), $$

where

$$\begin{aligned}& x = (H_{k},G_{k},D_{k})^{T}, \\& j(x) = \begin{pmatrix} k\rho (t)((1 - H_{k} - G_{k} - D_{k}) \\ 0 \\ 0 \end{pmatrix}, \end{aligned}$$
(8)
$$\begin{aligned}& l(x) = \begin{pmatrix} ((1 - \chi )\eta + \chi \varepsilon + \delta )H_{k} - \psi \varphi G_{k} \\ (\psi \mu + \gamma + \psi \varphi + \delta )G_{k} - (1 - \chi )\eta H_{k} \\ (\lambda + \delta )D_{k} - \psi \mu G_{k} \end{pmatrix}. \end{aligned}$$
(9)

At \(E_{0}\), the Jacobian matrices of \(j(x)\) and \(l(x)\) are got

$$\begin{aligned}& J = Dj(E_{0}) = \begin{pmatrix} J_{11} & J_{12} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \end{aligned}$$
(10)
$$\begin{aligned}& L = Dl(E_{0}) = \begin{pmatrix} L_{11} & L_{12} & 0 \\ L_{21} & L_{22} & 0 \\ 0 & L_{32} & L_{33} \end{pmatrix}, \end{aligned}$$
(11)

where

$$\begin{aligned}& J_{11} = \frac{\beta _{1}}{ \langle k \rangle } \begin{pmatrix} Q(1) & 2Q(2) & \cdots & nQ(n) \\ 2Q(1) & 2^{2}Q(2) & \cdots & 2nQ(n) \\ \vdots & \vdots & \ddots & \vdots \\ nQ(1) & 2nQ(2) & \cdots & n^{2}Q(n) \end{pmatrix}, \end{aligned}$$
(12)
$$\begin{aligned}& J_{12} = \frac{\beta _{2}}{ \langle k \rangle } \begin{pmatrix} Q(1) & 2Q(2) & \cdots & nQ(n) \\ 2Q(1) & 2^{2}Q(2) & \cdots & 2nQ(n) \\ \vdots & \vdots & \ddots & \vdots \\ nQ(1) & 2nQ(2) & \cdots & n^{2}Q(n) \end{pmatrix}. \end{aligned}$$
(13)

Here,

$$ \begin{gathered} L_{11} = \bigl((1 - \chi )\eta + \psi \varepsilon + \delta \bigr)I, \qquad L_{12} = - \psi \varphi I,\qquad L_{21} = - \bigl((1 - \chi )\eta \bigr)I,\\ L_{22} = (\psi \mu + \gamma + \psi \varphi + \delta )I,\qquad L_{32} = - \psi \mu I,\qquad L_{33} = ( \lambda + \delta )I, \end{gathered} $$

where I is an identity matrix. So, we can calculate the basic reproductive number denoted by

$$ R_{0} = \rho \bigl(JL^{ - 1}\bigr) = \frac{ \langle k^{2} \rangle }{ \langle k \rangle } \frac{(\psi \mu + \gamma + \psi \varphi + \delta )\beta _{1} + \eta (1 - \chi )\beta _{2}}{(\psi \mu + \gamma + \delta )((1 - \chi )\eta + \chi \varepsilon + \delta ) + \psi \varphi (\chi \varepsilon + \delta )}, $$

where \(\langle k^{2} \rangle = \sum_{k} k^{2}Q(k)\).

Next, it is clear that system (4) has a gambling-free equilibrium \(E_{0}(1,0,0,0)\). To get the gambling-prevailing equilibrium \(E_{ +} (S_{k}^{*},H_{k}^{*},G_{k}^{*},D_{k}^{*})\), system (4) satisfies

$$ \frac{dS_{k}(t)}{dt} = 0,\qquad \frac{dH_{k}(t)}{dt} = 0,\qquad \frac{dG_{k}(t)}{dt} = 0,\qquad \frac{dD_{k}(t)}{dt} = 0. $$

So, we can know

$$ \textstyle\begin{cases} \delta + \lambda D_{k}(t) + \gamma G_{k}(t) + \chi \varepsilon H_{k}(t) - (k\rho (t) + \delta )S_{k}(t) = 0, \\ k\rho (t)S_{k}(t) + \psi \varphi G_{k}(t) - ((1 - \chi )\eta + \chi \varepsilon + \delta )H_{k}(t) = 0, \\ (1 - \chi )\eta H_{k}(t) - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t) = 0, \\ \psi \mu G_{k}(t) - (\lambda + \delta )D_{k}(t) = 0. \end{cases} $$
(14)

According to the above equation, we get

$$ \textstyle\begin{cases} S_{k}(t) = \frac{((1 - \chi )\eta + \psi \varepsilon + \delta )(\psi \mu + \gamma + \psi \varphi + \delta ) - \psi \varphi \eta (1 - \chi )}{k\rho (t)(1 - \chi )\eta } G_{k}(t), \\ H_{k}(t) = \frac{\psi \mu + \gamma + \psi \varphi + \delta }{(1 - \chi )\eta } G_{k}(t), \\ D_{k}(t) = \frac{\psi \mu }{\lambda + \delta } G_{k}(t). \end{cases} $$
(15)

By using to the normalization condition \(S_{k}^{*}(t) + H_{k}^{*}(t) + G_{k}^{*}(t) + D_{k}^{*}(t) = 1\), it gets

$$ \textstyle\begin{cases} S_{k}^{*}(t) = \frac{((1 - \chi )\eta + \chi \varepsilon + \delta )(\psi \mu + \gamma + \delta )(\lambda + \delta ) + \psi \varphi (\lambda + \delta )(\chi \varepsilon + \delta )}{B_{k}}, \\ H_{k}^{*}(t) = \frac{k\rho (t)(\lambda + \delta )(\psi \mu + \gamma + \psi \varphi + \delta )}{B_{k}}, \\ G_{k}^{*}(t) = \frac{k\rho (t)\eta (1 - \chi )(\lambda + \delta )}{B_{k}}, \\ D_{k}^{*}(t) = \frac{k\rho (t)\psi \mu \eta (1 - \chi )}{B_{k}}. \end{cases} $$
(16)

And

$$ \begin{aligned} B_{k} &= k\rho (t) \bigl((\lambda + \delta ) (\psi \mu + \gamma + \psi \varphi + \delta ) + \psi \mu \eta (1 - \chi ) + \eta (1 - \chi ) (\lambda + \delta )\bigr) \\ &\quad {}+ \psi \varphi (\lambda + \delta ) (\chi \varepsilon + \delta )+ \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr) (\psi \mu + \gamma + \delta ) (\lambda + \delta ), \end{aligned} $$

where \(\rho (t) = \sum_{k} kQ(k)(\beta _{1}\theta _{1} + \beta _{2}\theta _{2}) / \langle k \rangle \). By substituting the second equation of system (16) into Eq. (2), we get

$$ \theta _{1}^{*}(t) = \frac{1}{ \langle k \rangle } \sum _{k} k^{2} Q(k) \cdot \frac{\rho (t)(\lambda + \delta )(\psi \mu + \gamma + \psi \varphi + \delta )}{B_{k}}. $$

According to \(\theta _{1}(t) = \frac{\sum_{k} k Q(k)H_{k}(t)}{\sum_{k} sQ(s)} = \frac{1}{ \langle k \rangle } \sum_{k} k Q(k)H_{k}(t)\) and \(\theta _{2}(t) = \frac{\sum_{k} k Q(k)G_{k}(t)}{\sum_{k} sQ(s)} = [4] \frac{1}{ \langle k \rangle } \sum_{k} k Q(k)G_{k}(t)\), we can get \(\theta _{2}^{*} = \frac{\eta (1 - \chi )}{\psi \mu + \gamma + \psi \varphi + \chi } \theta _{1}^{*}\). Then, let \(\theta _{1}^{*} \stackrel{{{{{{{{\Delta }}}}}}}}{=} f(\theta _{1}^{*})\), obviously, \(\theta _{1}^{*} = 0\) is a solution. In order for \(\theta _{1}^{*} \stackrel{{{{{{{{\Delta }}}}}}}}{=} f(\theta _{1}^{*})\) to have a nontrivial solution, the following conditions should be satisfied:

$$ \frac{df(\theta _{1}^{*})}{d\theta _{1}^{*}} \bigg|_{\theta _{1}^{*} = 0} > 1\quad \text{and}\quad f(1) \le 1. $$
(17)

So, we get

$$ R_{0} = \frac{ \langle k^{2} \rangle }{ \langle k \rangle } \frac{(\psi \mu + \gamma + \psi \varphi + \delta )\beta _{1} + \eta (1 - \chi )\beta _{2}}{(\psi \mu + \gamma + \delta )((1 - \chi )\eta + \chi \varepsilon + \delta ) + \psi \varphi (\chi \varepsilon + \delta )} > 1. $$

According to Eq. (16), we know \(0 < S_{k}^{*}\), \(H_{k}^{*},G_{k}^{*},D_{k}^{*} < 1\). System (4) has the gambling-prevailing equilibrium \(E_{ +} (S_{k}^{*},H_{k}^{*},G_{k}^{*},D_{k}^{*})\). Then, when the basic regeneration number \(R_{0} > 1\), there is a unique positive equilibrium \(E_{ +} (S_{k}^{*},H_{k}^{*},G_{k}^{*},D_{k}^{*})\). 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 \(\phi > 0\), \(\lim \inf_{t \to \infty } \sum_{k} (H(t) + G(t)) \ge \phi \).

Proof

For simplicity, let \(Q_{i} = iQ(i) / \langle k \rangle \). For the gambling-free equilibrium, system (7) has the Jacobian matrix of \(3n \times 3n\) as follows:

$$ G = \begin{pmatrix} B_{11} & \cdots & B_{1n} \\ \vdots & \ddots & \vdots \\ B_{n1} & \cdots & B_{nn} \end{pmatrix}, $$
(18)

where

$$\begin{aligned}& B_{11} = \begin{pmatrix} - ((1 - \chi )\eta + \chi \varepsilon + \delta ) + \beta _{1}Q_{1} & \psi \varphi + \beta _{2}Q_{1} & 0 \\ (1 - \chi )\eta & - (\psi \mu + \gamma + \psi \varphi + \delta ) & 0 \\ 0 & \psi \mu & - (\lambda + \delta ) \end{pmatrix}, \end{aligned}$$
(19)
$$\begin{aligned}& B_{1n} = \begin{pmatrix} \beta _{1}Q_{n} & \beta _{2}Q_{n} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \end{aligned}$$
(20)
$$\begin{aligned}& B_{n1} = \begin{pmatrix} n\beta _{1}Q_{1} & n\beta _{2}Q_{1} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \end{aligned}$$
(21)
$$\begin{aligned}& B_{nn} = \begin{pmatrix} - ((1 - \chi )\eta + \chi \varepsilon + \delta ) + n\beta _{1}Q_{n} & \psi \varphi + n\beta _{2}Q_{n} & 0 \\ (1 - \chi )\eta & - (\psi \mu + \gamma + \psi \varphi + \delta ) & 0 \\ 0 & \psi \mu & - (\lambda + \delta ) \end{pmatrix}. \end{aligned}$$
(22)

So, the characteristic polynomial of the gambling-free equilibrium \(E_{0}\) is

$$ (z + \lambda + \delta )^{n - 1}(z + \psi \mu + \gamma + \psi \varphi + \delta )^{n - 1}\bigl(z + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr)^{n - 1}\bigl(z^{3} + sz^{2} + pz + q\bigr) = 0, $$
(23)

where \(s = ((1 - \chi )\eta + \chi \varepsilon + \delta ) + (\psi \mu + \gamma + \psi \varphi + \delta ) + (\lambda + \delta ) - \beta _{1}\sum_{i = 1}^{n} iQ_{i}\), and

$$\begin{aligned}& \begin{aligned} p &= \bigl((1 - \chi )\eta + \chi \varepsilon + \lambda + 2 \delta \bigr) (\psi \mu + \gamma + \psi \varphi + \delta ) + \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr) (\lambda + \delta ) \\ &\quad {}- \psi \varphi \eta (1 - \chi ) - \bigl((\psi \mu + \gamma + \psi \varphi + \lambda + 2\delta )\beta _{1} + \eta (1 - \chi )\beta _{2}\bigr)\sum_{i = 1}^{n} iQ_{i}, \end{aligned} \end{aligned}$$
(24)
$$\begin{aligned}& \begin{aligned} q &= \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr) ( \psi \mu + \gamma + \psi \varphi + \delta ) (\lambda + \delta ) - \psi \varphi \eta (1 - \chi ) (\lambda + \delta ) \\ &\quad {}- \bigl((\psi \mu + \gamma + \psi \varphi + \delta )\beta _{1} + \eta (1 - \chi )\beta _{2}\bigr) (\lambda + \delta ) \sum_{i = 1}^{n} iQ_{i}. \end{aligned} \end{aligned}$$
(25)

Obviously, when \(R_{0} < 1\), \(q > 0\). It also means

$$ \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr) + (\psi \mu + \gamma + \psi \varphi + \delta ) + (\lambda + \delta ) > \beta _{1}\sum _{i = 1}^{n} iQ_{i} $$
(26)

and

$$ \begin{aligned}& \bigl((1 - \chi )\eta + \chi \varepsilon + \lambda + 2\delta \bigr) (\psi \mu + \gamma + \psi \varphi + \delta ) + \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr) (\lambda + \delta ) \\ &\quad > \psi \varphi \eta (1 - \chi ) + \bigl((\psi \mu + \gamma + \psi \varphi + \lambda + 2\delta )\beta _{1} + \eta (1 - \chi )\beta _{2} \bigr)\sum_{i = 1}^{n} iQ_{i}. \end{aligned} $$
(27)

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. □

Theorem 3

([26])

Suppose that \((H_{k}(t),G_{k}(t),D_{k}(t))\) is the solution of system (7), which satisfies Eq. (5) with \(H_{k}(0) > 0\) or \(G_{k}(0) > 0\). If \(R_{0} > 1\), then \(\lim_{t \to \infty } (H_{k}(t),G_{k}(t),D_{k}(t)) = (H_{k}^{*},G_{k}^{*},D_{k}^{*})\), where \((H_{k}^{*},G_{k}^{*},D_{k}^{*})\) is the gambling-prevailing equilibrium of system (7) for \(k = 1,2, \ldots ,n\).

Proof

In the proof, let us assume that k is integer between 1 and n. According to Theorem 2, a positive constant \(0 < \alpha < 1 / 3\) and a sufficiently large constant \(T > 0\) exist to satisfy \(H_{k}(t) \ge \alpha \) and \(G_{k}(t) \ge \alpha \) for \(t > T\). Thus, \(\rho (t) > \alpha (\beta _{1} + \beta _{2})\) for \(t > T\). Submitting this into the first equation of system (7), it is easy to get

$$ \frac{dH_{k}(t)}{dt} \le k(\beta _{1} + \beta _{2}) \bigl(1 - H_{k}(t)\bigr) - \bigl((1 - \chi )\eta + \chi \varepsilon + \delta \bigr)H_{k}(t) $$
(28)

for \(t > T\).

According to the standard comparison theorem in the theory of differential equations, for any given positive constant

$$ 0 < \alpha _{1} < \frac{(1 - \chi )\eta + \chi \varepsilon + \delta }{2[k(\beta _{1} + \beta _{2}) + ((1 - \chi )\eta + \chi \varepsilon + \delta )]}, $$
(29)

there exists \(t_{1} > T\), so \(H_{k}(t) \le M_{k}^{(1)} - \alpha _{1}\) for \(t > t_{1}\), where

$$ M_{k}^{(1)} = \frac{k(\beta _{1} + \beta _{2})}{k(\beta _{1} + \beta _{2}) + ((1 - \chi )\eta + \chi \varepsilon + \delta )} + 2\alpha _{1} < 1. $$
(30)

From system (7), it is easy to obtain

$$ \frac{dG_{k}(t)}{dt} \le (1 - \chi )\eta \bigl(1 - G_{k}(t)\bigr) - ( \psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t) $$
(31)

for \(t > t_{1}\).

So, the constant

$$ 0 < \alpha _{2} < \min \biggl\{ 1 / 2,\alpha _{1}, \frac{\psi \mu + \gamma + \psi \varphi + \delta }{2((1 - \chi )\eta + (\psi \mu + \gamma + \psi \varphi + \delta ))}\biggr\} , $$
(32)

there exists \(t_{2} > t_{1}\), so \(G_{k}(t) \le A_{k}^{(1)} - \alpha _{2}\) for \(t > t_{2}\), where

$$ A_{k}^{(1)} = \frac{\eta (1 - \chi )}{\eta (1 - \chi ) + (\psi \mu + \gamma + \psi \varphi + \delta )} + 2\alpha _{2} < 1. $$
(33)

From system (7), it is easy to obtain

$$ \frac{dD_{k}(t)}{dt} \le \psi \mu \bigl(1 - D_{k}(t)\bigr) - (\lambda + \delta )D_{k}(t) $$
(34)

for \(t > t_{2}\).

Consequently, for constant

$$ 0 < \alpha _{3} < \min \biggl\{ 1 / 3,\alpha _{2}, \frac{\lambda + \delta }{2(\psi \mu + (\lambda + \delta ))}\biggr\} , $$
(35)

there exists \(t_{3} > t_{2}\) such that \(D_{k}(t) \le V_{k}^{(1)} - \alpha _{3}\) for \(t > t_{3}\), where

$$ V_{k}^{(1)} = \frac{\psi \mu }{\psi \mu + (\lambda + \delta )} + 2\alpha _{3} < 1. $$
(36)

Then, replacing \(H_{k}(t) \ge \alpha \), \(G_{k}(t) \ge \alpha \) and \(\rho (t) > \alpha (\beta _{1} + \beta _{2})\) into the first equation of system (7), we get

$$ \begin{aligned} \frac{dH_{k}(t)}{dt} &\ge k\alpha (\beta _{1} + \beta _{2}) \bigl(1 - H_{k}(t) - G_{k}(t) - D_{k}(t)\bigr)\\ &\quad {} + \psi \varphi G_{k}(t) - \bigl((1 - \chi ) \eta + \chi \varepsilon + \delta \bigr)H_{k}(t) \\ & \ge k\alpha (\beta _{1} + \beta _{2}) \bigl(1 - A_{k}^{(1)} - V_{k}^{(1)}\bigr) + \psi \varphi A_{k}^{(1)}\\ &\quad {} - \bigl(k\alpha (\beta _{1} + \beta _{2}) + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr)H_{k}(t) \end{aligned} $$
(37)

for \(t > T\).

Therefore, for constant

$$ 0 < \alpha _{4} < \min \biggl\{ 1 / 4,\alpha _{3}, \frac{k\alpha (\beta _{1} + \beta _{2})(1 - A_{k}^{(1)} - V_{k}^{(1)}) + \psi \varphi A_{k}^{(1)}}{2[k\alpha (\beta _{1} + \beta _{2}) + (1 - \chi )\eta + \chi \varepsilon + \delta ]}\biggr\} , $$
(38)

there exists \(t_{4} > t_{3}\) such that \(H_{k}(t) \ge m_{k}^{(1)} + \alpha _{4}\) for \(t > t_{4}\), where

$$ m_{k}^{(1)} = \frac{k\alpha (\beta _{1} + \beta _{2})(1 - N_{k}^{(1)} - V_{k}^{(1)}) + \psi \varphi N_{k}^{(1)}}{k\alpha (\beta _{1} + \beta _{2}) + (1 - \chi )\eta + \chi \varepsilon + \delta } - 2\alpha _{4} > 0. $$
(39)

Therefore

$$ \frac{dG_{k}(t)}{dt} \ge \eta (1 - \chi )m_{k}^{(1)} - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t) $$
(40)

for \(t > t_{4}\).

Hence, for constant

$$ 0 < \alpha _{5} < \min \biggl\{ 1 / 5,\alpha _{4}, \frac{\eta (1 - \chi )m_{k}^{(1)}}{2(\psi \mu + \gamma + \psi \varphi + \delta )}\biggr\} , $$
(41)

there exists \(t_{5} > t_{4}\) such that \(G_{k}(t) \ge a_{k}^{(1)} + \alpha _{5}\) for \(t > t_{5}\), where

$$ a_{k}^{(1)} = \frac{\eta (1 - \chi )x_{k}^{(1)}}{\psi \mu + \gamma + \psi \varphi + \delta } - 2\alpha _{5} > 0. $$
(42)

Similarly,

$$ \frac{dD_{k}(t)}{dt} \ge \psi \mu a_{k}^{(1)} - (\lambda + \delta )D_{k}(t) $$
(43)

for \(t > t_{5}\).

Consequently, for constant

$$ 0 < \alpha _{6} < \min \biggl\{ 1 / 6,\alpha _{5}, \frac{\psi \mu a_{k}^{(1)}}{2(\lambda + \delta )}\biggr\} , $$
(44)

there exists \(t_{6} > t_{5}\) such that \(D_{k}(t) \ge v_{k}^{(1)} + \alpha _{6}\) for \(t > t_{6}\), where

$$ v_{k}^{(1)} = \frac{\psi \mu a_{k}^{(1)}}{\lambda + \delta } - 2\alpha _{6} > 0. $$
(45)

Because α is a small constant, we can get \(0 < m_{k}^{(1)} < M_{k}^{(1)} < 1\), \(0 < a_{k}^{(1)} < A_{k}^{(1)} < 1\), and \(0 < v_{k}^{(1)} < V_{k}^{(1)} < 1\).

Let

$$ u^{(j)} = \sum_{i = 1}^{n} Q_{i}\bigl(\beta _{1}m_{i}^{(j)} + \beta _{2}a_{i}^{(j)}\bigr),\qquad U^{(j)} = \sum _{i = 1}^{n} Q_{i}\bigl(\beta _{1}M_{i}^{(j)} + \beta _{2}A_{i}^{(j)} \bigr),\quad j = 1,2, \ldots . $$
(46)

From the above discussion, we have

$$ 0 < u^{(1)} \le \rho (t) \le U^{(1)} < \beta _{1} + \beta _{2} $$

and \(t > t_{6}\).

And, according to system (7), we can get

$$ \frac{dH_{k}(t)}{dt} \le kU^{(1)}\bigl(1 - a_{k}^{(1)} - v_{k}^{(1)}\bigr) + \psi \varphi a_{k}^{(1)} - \bigl(kU^{(1)} + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr)H_{k}(t) $$
(47)

for \(t > t_{6}\).

Consequently, for constant \(0 < \alpha _{7} < \min \{ 1 / 7,\alpha _{6}\}\), there exists \(t_{7} > t_{6}\) such that

$$ H_{k}(t) \le M_{k}^{(2)} \stackrel{{{{{{{{\Delta }}}}}}}}{=} \min \biggl\{ M_{k}^{(1)} - \alpha _{1},\frac{kU^{(1)}(1 - a_{k}^{(1)} - v_{k}^{(1)}) + \psi \varphi y_{k}^{(1)}}{kU^{(1)} + (1 - \chi )\eta + \chi \varepsilon + \delta } + \alpha _{7}\biggr\} $$
(48)

for \(t > t_{7}\).

Thus,

$$ \frac{dG_{k}(t)}{dt} \le \eta (1 - \chi )M_{k}^{(2)} - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t) $$
(49)

for \(t > t_{7}\).

Consequently, for constant \(0 < \alpha _{8} < \min \{ 1 / 8,\alpha _{7}\}\), there exists \(t_{8} > t_{7}\) such that

$$ G_{k}(t) \le A_{k}^{(2)} \stackrel{{{{{{{{\Delta }}}}}}}}{=} \min \biggl\{ A_{k}^{(1)} - \alpha _{2},\frac{\eta (1 - \chi )M_{k}^{(2)}}{\psi \mu + \gamma + \psi \varphi + \delta } + \alpha _{8}\biggr\} $$
(50)

for \(t > t_{8}\).

As a result, it follows that

$$ \frac{dD_{k}(t)}{dt} \le \psi \mu A_{k}^{(2)} - (\lambda + \delta )D_{k}(t) $$
(51)

for \(t > t_{8}\).

Therefore, for constant \(0 < \alpha _{9} < \min \{ 1 / 9,\alpha _{8}\}\), there exists \(t_{9} > t_{8}\) such that

$$ D_{k}(t) \le V_{k}^{(2)} \stackrel{{{{{{{{\Delta }}}}}}}}{=} \min \biggl\{ V_{k}^{(1)} - \alpha _{3},\frac{\psi \mu A_{k}^{(2)}}{\lambda + \delta } + \alpha _{9}\biggr\} $$
(52)

for \(t > t_{9}\).

According to system (7), we can get

$$ \frac{dH_{k}(t)}{dt} \ge ku^{(1)}\bigl(1 - A_{k}^{(2)} - V_{k}^{(2)}\bigr) + \psi \varphi A_{k}^{(2)} - \bigl(ku^{(1)} + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr)H_{k}(t) $$
(53)

for \(t > t_{9}\).

Hence, for constant

$$ 0 < \alpha _{10} < \min \biggl\{ 1 / 10,\alpha _{9}, \frac{ku^{(1)}(1 - A_{k}^{(2)} - V_{k}^{(2)}) + \psi \varphi A_{k}^{(2)}}{2(ku^{(1)} + (1 - \chi )\eta + \chi \varepsilon + \delta )}\biggr\} , $$
(54)

there exists \(t_{10} > t_{9}\), and \(H_{k}(t) \ge m_{k}^{(2)} + \alpha _{10}\), \(t > t_{10}\), where

$$ m_{k}^{(2)} = \max \biggl\{ m_{k}^{(1)} + \alpha _{4},\frac{ku^{(1)}(1 - A_{k}^{(2)} - V_{k}^{(2)}) + \psi \varphi A_{k}^{(2)}}{ku^{(1)} + (1 - \chi )\eta + \chi \varepsilon + \delta } - 2\alpha _{10}\biggr\} . $$
(55)

Thus,

$$ \frac{dG_{k}(t)}{dt} \ge (1 - \chi )\eta m_{k}^{(2)} - (\psi \mu + \gamma + \psi \varphi + \delta )G_{k}(t) $$
(56)

for \(t > t_{10}\).

So, for constant

$$ 0 < \alpha _{11} < \min \biggl\{ 1 / 11,\alpha _{10}, \frac{(1 - \chi )\eta m_{k}^{(2)}}{2(\psi \mu + \gamma + \psi \varphi + \delta )}\biggr\} , $$
(57)

there exists \(t_{11} > t_{10}\), and \(G_{k}(t) \ge a_{k}^{(2)} + \alpha _{11}\), \(t > t_{11}\), where

$$ a_{k}^{(2)} = \max \biggl\{ a_{k}^{(1)} + \alpha _{5},\frac{(1 - \chi )\eta m_{k}^{(2)}}{\psi \mu + \gamma + \psi \varphi + \delta } - 2\alpha _{11}\biggr\} . $$
(58)

Similarly,

$$ \frac{dD_{k}(t)}{dt} \ge \psi \mu a_{k}^{(2)} - (\lambda + \delta )D_{k}(t) $$
(59)

for \(t > t_{11}\).

Therefore, for constant

$$ 0 < \alpha _{12} < \min \biggl\{ 1 / 12,\alpha _{11}, \frac{\psi \mu a_{k}^{(2)}}{2(\lambda + \delta )}\biggr\} , $$
(60)

there exists \(t_{12} > t_{11}\), and \(D_{k}(t) \ge v_{k}^{(2)} + \alpha _{12}\), \(t > t_{12}\), where

$$ v_{k}^{(2)} = \max \biggl\{ v_{k}^{(1)} + \alpha _{6},\frac{\psi \mu a_{k}^{(2)}}{\lambda + \delta } - 2\alpha _{12}\biggr\} . $$
(61)

According to the above discussion and analyses, we can obtain six sequences: \(\{ M_{k}^{(r)}\}\), \(\{ A_{k}^{(r)}\}\), \(\{ V_{k}^{(r)}\}\), \(\{ m_{k}^{(r)}\}\), \(\{ a_{k}^{(r)}\}\), and \(\{ v_{k}^{(r)}\}\). We can find that the first three sequences are monotone increasing and the last three sequences are strictly monotone decreasing, and there is a sufficiently large positive integer L such that, for \(r \ge L\):

$$ \textstyle\begin{cases} M_{k}^{(r)} = \frac{kU^{(r - 1)}(1 - a_{k}^{(r - 1)} - v_{k}^{(r - 1)}) + \psi \varphi a_{k}^{(r - 1)}}{kU^{(r - 1)} + (1 - \chi )\eta + \chi \varepsilon + \delta } + \alpha _{6r - 5}, \\ A_{k}^{(r)} = \frac{(1 - \chi )\eta M_{k}^{(r)}}{\psi \mu + \gamma + \psi \varphi + \delta } + \alpha _{6r - 4}, \\ V_{k}^{(r)} = \frac{\psi \mu A_{k}^{(r)}}{\lambda + \delta } + \alpha _{6r - 3}, \\ m_{k}^{(r)} = \frac{ku^{(r - 1)}(1 - A_{k}^{(r)} - V_{k}^{(r)}) + \psi \varphi A_{k}^{(r)}}{ku^{(r - 1)} + (1 - \chi )\eta + \chi \varepsilon + \delta } - 2\alpha _{6r - 2}, \\ a_{k}^{(r)} = \frac{(1 - \chi )\eta m_{k}^{(r)}}{\psi \mu + \gamma + \psi \varphi + \delta } - 2\alpha _{6r - 1}, \\ v_{k}^{(r)} = \frac{\psi \mu a_{k}^{(r)}}{\lambda + \delta } - 2\alpha _{6r}. \end{cases} $$
(62)

It is easy to find that

$$ \textstyle\begin{cases} m_{k}^{(r)} \le H_{k}(t) \le M_{k}^{(r)}, \\ a_{k}^{(r)} \le G_{k}(t) \le A_{k}^{(r)}, \\ v_{k}^{(r)} \le D_{k}(t) \le V_{k}^{(r)}, \end{cases}\displaystyle \quad \text{where } t > t_{6r}. $$
(63)

Since the sequential limits of system (62), thus let \(\lim_{t \to \infty } \Omega _{k}^{(r)} = \Omega _{k}\), where \(\Omega _{k} \in \{ M_{k},A_{k}, V_{k},m_{k},a_{k},v_{k},U_{k},u_{k}\}\) and \(\Omega _{k}^{(r)} \in \{ M_{k}^{(r)},A_{k}^{(r)},V_{k}^{(r)},m_{k}^{(r)},a_{k}^{(r)},v_{k}^{(r)},U_{k}^{(r)},u_{k}^{(r)}\}\). Since \(0 < \alpha _{r} < 1 / r\), it has \(\alpha _{r} \to 0\) as \(r \to \infty \). Supposing \(r \to \infty \), it follows from (62) that

$$ \textstyle\begin{cases} M_{k} = \frac{kU(1 - a_{k} - v_{k}) + \psi \varphi a_{k}}{kU + (1 - \chi )\eta + \chi \varepsilon + \delta },\qquad A_{k} = \frac{(1 - \chi )\eta M_{k}}{\psi \mu + \gamma + \psi \varphi + \delta },\qquad V_{k} = \frac{\psi \mu A_{k}}{\lambda + \delta }, \\ m_{k} = \frac{ku(1 - A_{k} - V_{k}) + \psi \varphi A_{k}}{ku + (1 - \chi )\eta + \chi \varepsilon + \delta },\qquad a_{k} = \frac{(1 - \chi )\eta m_{k}}{\psi \mu + \gamma + \psi \varphi + \delta },\qquad v_{k} = \frac{\psi \mu a_{k}}{\lambda + \delta }, \end{cases} $$
(64)

where

$$ u = \sum_{i = 1}^{n} Q_{i} (\beta _{1}m_{i} + \beta _{2}a_{i}),\qquad U = \sum _{i = 1}^{n} Q_{i} (\beta _{1}M_{i} + \beta _{2}A_{i}). $$

What is more,

$$ \textstyle\begin{cases} M_{k} = \frac{1}{G_{k}}[ kU(\lambda + \delta )^{2}(\psi \mu + \gamma + \psi \varphi + \delta )^{2}(ku + (1 - \chi )\eta + \chi \varepsilon + \delta ) \\ \hphantom{M_{k} =}{}+ ku(1 - \chi )(\lambda + \delta )(\psi \mu + \gamma + \psi \varphi + \delta )(\psi \varphi \eta (\lambda + \delta ) - kU(\eta (\lambda + \delta ) + \psi \mu \eta )) ], \\ A_{k} = \frac{1}{G_{k}}[ kU\eta (1 - \chi )(\psi \mu + \gamma + \psi \varphi + \delta )(\lambda + \delta )^{2}(ku + (1 - \chi )\eta + \chi \varepsilon + \delta ) \\ \hphantom{A_{k} =}{}+ ku\eta (\lambda + \delta )(1 - \chi )^{2}(\psi \varphi \eta (\lambda + \delta ) - kU(\eta (\lambda + \delta ) + \psi \mu \eta )) ], \\ m_{k} = \frac{1}{G_{k}}[ ku(\lambda + \delta )^{2}(\psi \mu + \gamma + \psi \varphi + \delta )^{2}(kU + (1 - \chi )\eta + \chi \varepsilon + \delta ) \\ \hphantom{m_{k} =}{}+ kU(1 - \chi )(\lambda + \delta )(\psi \mu + \gamma + \psi \varphi + \delta )(\psi \varphi \eta (\lambda + \delta ) - ku(\eta (\lambda + \delta ) + \psi \mu \eta )) ], \\ a_{k} = \frac{1}{G_{k}}[ ku\eta (1 - \chi )(\psi \mu + \gamma + \psi \varphi + \delta )(\lambda + \delta )^{2}(kU + (1 - \chi )\eta + \chi \varepsilon + \delta ) \\ \hphantom{a_{k} =}{}+ kU\eta (\lambda + \delta )(1 - \chi )^{2}(\psi \varphi \eta (\lambda + \delta ) - ku(\eta (\lambda + \delta ) + \psi \mu \eta )) ], \end{cases} $$
(65)

where

$$ \begin{aligned} G_{k} &= (\lambda + \delta )^{2}(\psi \mu + \gamma + \psi \varphi + \delta )^{2}\bigl(ku + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr) \bigl(kU + (1 - \chi )\eta + \chi \varepsilon + \delta \bigr) \\ &\quad {}- (1 - \chi )^{2}\bigl(\psi \varphi \eta (\lambda + \delta ) - kU\bigl(\eta (\lambda + \delta ) + \psi \mu \eta \bigr)\bigr)\\ &\quad {}\times \bigl(\psi \varphi \eta (\lambda + \delta ) - ku\bigl(\eta (\lambda + \delta ) + \psi \mu \eta \bigr)\bigr). \end{aligned} $$

From the above equation, we get \(U = u\). So,

$$ \sum_{i = 1}^{n} Q_{i}\bigl[\beta _{1}(M_{i} - m_{i}) + \beta _{2}(A_{i} - a_{i})\bigr] = 0, $$
(66)

which is equivalent to \(M_{i} = m_{i}\) and \(A_{i} = a_{i}\) for \(1 \le i \le n\). Then, from systems (63) and (64), it can be concluded that

$$ \lim_{t \to \infty } H_{k}(t) = M_{k} = m_{k},\qquad \lim_{t \to \infty } G_{k}(t) = A_{k} = a_{k},\qquad \lim_{t \to \infty } D_{k}(t) = V_{k} = v_{k}. $$

Finally, \(U = u\) is substituted into system (65). For system (64), we can get \(M_{k} = H_{k}^{*}\), \(A_{k} = G_{k}^{*}\), and \(V_{k} = D_{k}^{*}\). 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 \(Q(k) = \omega k^{ - 3}\) in system (1), and the parameter ω satisfies \(\sum_{k = 1}^{n} \omega k^{ - 3} = 1\), \(n = 1000\).

In Fig. 2, we choose \(\delta = 0.2\), \(\beta _{1} = 0.1\), \(\beta _{2} = 0.1\), \(\varepsilon = 0.4\), \(\chi = 0.3\), \(\eta = 0.5\), \(\varphi = 0.1\), \(\psi = 0.7\), \(\mu = 0.6\), \(\gamma = 0.3\), \(\lambda = 0.1\) and obtain the basic reproductive number \(R_{0} = 0.9544 < 1\). 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.

Figure 2
figure2

The orbits of four states over time when \(R_{0} = 0.9544 < 1\)

In Fig. 3, we choose \(\delta = 0.1\), \(\beta _{1} = 0.2\), \(\beta _{2} = 0.5\), \(\varepsilon = 0.1\), \(\chi = 0.1\), \(\eta = 0.3\), \(\varphi = 0.2\), \(\psi = 0.4\), \(\mu = 0.6\), \(\gamma = 0.1\), \(\lambda = 0.1\) and obtain \(R_{0} = 6.1795 > 1\). The figure shows that when \(R_{0} > 1\), \(H_{150}\) and \(G_{150}\) will maintain positive recently, and the online gambling phenomenon will not disappear.

Figure 3
figure3

The orbits of four states over time when \(R_{0} = 6.1795 > 1\)

In Fig. 4(a) and (b), we choose \(\delta = 0.2\), \(\beta _{1} = 0.1\), \(\beta _{2} = 0.1\), \(\varepsilon = 0.4\), \(\chi = 0.3\), \(\eta = 0.5\), \(\varphi = 0.1\), \(\psi = 0.7\), \(\mu = 0.6\), \(\gamma = 0.3\), \(\lambda = 0.1\) and obtain \(R_{0} = 0.9544 < 1\). The figure shows trends of the hesitator \(H(t)\) and the gambler \(G(t)\) over time with different degree. And when \(R_{0} < 1\), online gambling phenomenon will ultimately disappear. In addition, the larger the degree is, the faster the spread of online gambling behavior.

Figure 4
figure4

The orbits of the hesitator or the gambler with different degree when \(R_{0} < 1\)

In Fig. 5(a) and (b), we choose \(\delta = 0.1\), \(\beta _{1} = 0.2\), \(\beta _{2} = 0.5\), \(\varepsilon = 0.1\), \(\chi = 0.1\), \(\eta = 0.3\), \(\varphi = 0.2\), \(\psi = 0.4\), \(\mu = 0.6\), \(\gamma = 0.1\), \(\lambda = 0.1\) and obtain \(R_{0} = 6.1795 > 1\). The figure shows trends of the hesitator \(H(t)\) and the gambler \(G(t)\) over time with different degree. And when \(R_{0} > 1\), online gambling phenomenon will be persistent. Moreover, more people are involved in gambling with the increasing of degree.

Figure 5
figure5

The orbits of the hesitator or the gambler with different degree when \(R_{0} > 1\)

In Fig. 6(a) and (b), we choose \(\delta = 0.1\), \(\beta _{1} = 0.2\), \(\beta _{2} = 0.5\), \(\varepsilon = 0.1\), \(\chi = 0.1\), \(\eta = 0.3\), \(\varphi = 0.2\), \(\mu = 0.6\), \(\gamma = 0.1\), \(\lambda = 0.1\). The figure shows the change of the hesitator \(H(t)\) and the gambler \(G(t)\) with different probability ψ. With the growth of ψ, \(H(t)\) will increase but \(G(t)\) will fall to a constant. Apparently, larger ψ can decrease the number of gamblers.

Figure 6
figure6

Prevalence \(H_{100}(t)\), \(G_{100}(t)\) versus t corresponding to different ψ

In Fig. 7(a) and (b), we choose \(\delta = 0.1\), \(\beta _{1} = 0.2\), \(\beta _{2} = 0.5\), \(\varepsilon = 0.1\), \(\eta = 0.3\), \(\varphi = 0.2\), \(\psi = 0.6\), \(\mu = 0.6\), \(\gamma = 0.1\), \(\lambda = 0.1\). The figure shows the change of the hesitator \(H(t)\) and the gambler \(G(t)\) with different probability χ. With the growth of χ, \(H(t)\) and \(G(t)\) will fall to a constant. Apparently, larger χ can decrease the number of the hesitator and the gambler.

Figure 7
figure7

Prevalence \(H_{100}(t)\), \(G_{100}(t)\) versus t corresponding to different χ

In Fig. 8, the parameters are chosen as \(\delta = 0.2\), \(\varepsilon = 0.4\), \(\chi = 0.3\), \(\eta = 0.5\), \(\varphi = 0.1\), \(\psi = 0.7\), \(\mu = 0.6\), \(\gamma = 0.3\), \(\lambda = 0.1\). We can see that larger \(\beta _{1}\) or \(\beta _{2}\) can lead to larger \(R_{0}\), and \(\beta _{1}\) has a greater impact on \(R_{0}\). That is to say, the larger number of the hesitator \(H(t)\) and the gambler \(G(t)\) can speed up the spread of online gambling.

Figure 8
figure8

The relationship between the basic reproductive number \(R_{0}\) and the parameters \(\beta _{1}\) and \(\beta _{2}\) on scale-free networks

In Fig. 9(a) and (b), we choose \(\delta = 0.1\), \(\beta _{1} = 0.2\), \(\beta _{2} = 0.5\), \(\varepsilon = 0.1\), \(\eta = 0.3\), \(\varphi = 0.2\), \(\mu = 0.6\), \(\gamma = 0.1\), \(\lambda = 0.1\). 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 anti-gambling 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.

Figure 9
figure9

The relationship between the basic reproductive number \(R_{0}\) and the parameters ψ and χ on scale-free networks

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 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 \(\beta _{1}\) and \(\beta _{2}\) can lead to the lower number of the disseminator, and \(\beta _{1}\) has a greater impact than \(\beta _{2}\). 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.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Acknowledgements

We thank the referees and the editor for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper.

Funding

This work is supported in part by the National Natural Science Foundation of China under grants 61672112 and 61873287.

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Contributions

YK performed the analysis and wrote the manuscript; TL designed the study; YW and XC developed the methodology; HW and YL helped perform the analysis with constructive discussions. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Tao Li.

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The authors declare that they have no competing interests.

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Kong, Y., Li, T., Wang, Y. et al. Dynamics analysis of an online gambling spreading model on scale-free networks. Adv Differ Equ 2021, 11 (2021). https://doi.org/10.1186/s13662-020-03165-z

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Keywords

  • SHGD model
  • Heterogeneity
  • Psychological factors
  • Anti-gambling policy
  • Stability
  • Persistence