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Dynamics analysis of an online gambling spreading model on scalefree networks
Advances in Difference Equations volume 2021, Article number: 11 (2021)
Abstract
Nowadays, online gambling has a great negative impact on the society. In order to study the effect of people’s psychological factors, antigambling policy, and social network topology on online gambling dynamics, a new SHGD (susceptible–hesitator–gambler–disclaimer) online gambling spreading model is proposed on scalefree 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 antigambling policy and the network topology. Then, gamblingfree equilibrium \(E_{0}\) and gamblingprevailing 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 [2–4].
How to control the phenomenon of online gambling is very important. Some scholars have studied the phenomenon of online gambling from different aspects [5–9]. King and Barak [10] studied the characteristics of online gambling such as attraction, convenience, and reasons why people participate in gambling. DicksonGillespie 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 [13–18]. Liu et al. [19] studied the spread dynamics of wordofmouth. Wang et al. [20] proposed a network epidemic model for waterborne diseases spread and considered both indirect environmenttohuman and direct humantohuman transmission routes. King et al. [21] established a twoway 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 scalefree 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 scalefree networks, we study the dynamics of online gambling in the paper. Taking into account people’s psychological factors, antigambling 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 \(R_{0}\), gamblingfree equilibrium \(E_{0}\), and gamblingprevailing 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 \(\beta _{1}\) or \(\beta _{2}\), 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 antigambling policy to the hesitator. Considering the influence degree of the antigambling policy, the hesitator will become a susceptible individual with the probability χε, in contrast, a gambler with the probability \((1  \chi )\eta \).

(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 antigambling policy to the gambler. Considering the influence degree of the antigambling 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 \(\theta _{1}(t)\) is the probability of linking to a hesitator at time t and satisfies
where \(\theta _{2}(t)\) is the probability of linking to a gambler at time t and satisfies
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
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:

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

(2)
There is a unique gamblingprevailing 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
Obviously, there is a gamblingfree equilibrium \(E_{0} = \{ (0,0,0)\}_{k}\) in system (7). By using the next generation matrix method [24], system (7) can be written
where
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
where \(\langle k^{2} \rangle = \sum_{k} k^{2}Q(k)\).
Next, it is clear that system (4) has a gamblingfree equilibrium \(E_{0}(1,0,0,0)\). To get the gamblingprevailing equilibrium \(E_{ +} (S_{k}^{*},H_{k}^{*},G_{k}^{*},D_{k}^{*})\), system (4) satisfies
So, we can know
According to the above equation, we get
By using to the normalization condition \(S_{k}^{*}(t) + H_{k}^{*}(t) + G_{k}^{*}(t) + D_{k}^{*}(t) = 1\), it gets
And
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
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:
So, we get
According to Eq. (16), we know \(0 < S_{k}^{*}\), \(H_{k}^{*},G_{k}^{*},D_{k}^{*} < 1\). System (4) has the gamblingprevailing 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 gamblingfree 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 gamblingfree equilibrium, system (7) has the Jacobian matrix of \(3n \times 3n\) as follows:
where
So, the characteristic polynomial of the gamblingfree equilibrium \(E_{0}\) is
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
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. □
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 gamblingprevailing 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
for \(t > T\).
According to the standard comparison theorem in the theory of differential equations, for any given positive constant
there exists \(t_{1} > T\), so \(H_{k}(t) \le M_{k}^{(1)}  \alpha _{1}\) for \(t > t_{1}\), where
From system (7), it is easy to obtain
for \(t > t_{1}\).
So, the constant
there exists \(t_{2} > t_{1}\), so \(G_{k}(t) \le A_{k}^{(1)}  \alpha _{2}\) for \(t > t_{2}\), where
From system (7), it is easy to obtain
for \(t > t_{2}\).
Consequently, for constant
there exists \(t_{3} > t_{2}\) such that \(D_{k}(t) \le V_{k}^{(1)}  \alpha _{3}\) for \(t > t_{3}\), where
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
for \(t > T\).
Therefore, for constant
there exists \(t_{4} > t_{3}\) such that \(H_{k}(t) \ge m_{k}^{(1)} + \alpha _{4}\) for \(t > t_{4}\), where
Therefore
for \(t > t_{4}\).
Hence, for constant
there exists \(t_{5} > t_{4}\) such that \(G_{k}(t) \ge a_{k}^{(1)} + \alpha _{5}\) for \(t > t_{5}\), where
Similarly,
for \(t > t_{5}\).
Consequently, for constant
there exists \(t_{6} > t_{5}\) such that \(D_{k}(t) \ge v_{k}^{(1)} + \alpha _{6}\) for \(t > t_{6}\), where
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
From the above discussion, we have
and \(t > t_{6}\).
And, according to system (7), we can get
for \(t > t_{6}\).
Consequently, for constant \(0 < \alpha _{7} < \min \{ 1 / 7,\alpha _{6}\}\), there exists \(t_{7} > t_{6}\) such that
for \(t > t_{7}\).
Thus,
for \(t > t_{7}\).
Consequently, for constant \(0 < \alpha _{8} < \min \{ 1 / 8,\alpha _{7}\}\), there exists \(t_{8} > t_{7}\) such that
for \(t > t_{8}\).
As a result, it follows that
for \(t > t_{8}\).
Therefore, for constant \(0 < \alpha _{9} < \min \{ 1 / 9,\alpha _{8}\}\), there exists \(t_{9} > t_{8}\) such that
for \(t > t_{9}\).
According to system (7), we can get
for \(t > t_{9}\).
Hence, for constant
there exists \(t_{10} > t_{9}\), and \(H_{k}(t) \ge m_{k}^{(2)} + \alpha _{10}\), \(t > t_{10}\), where
Thus,
for \(t > t_{10}\).
So, for constant
there exists \(t_{11} > t_{10}\), and \(G_{k}(t) \ge a_{k}^{(2)} + \alpha _{11}\), \(t > t_{11}\), where
Similarly,
for \(t > t_{11}\).
Therefore, for constant
there exists \(t_{12} > t_{11}\), and \(D_{k}(t) \ge v_{k}^{(2)} + \alpha _{12}\), \(t > t_{12}\), where
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 \(r \ge L\):
It is easy to find that
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
where
What is more,
where
From the above equation, we get \(U = u\). So,
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
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 scalefree 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.
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.
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.
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.
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.
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.
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.
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 antigambling efforts, the antigambling policy helps to decrease the spread of online gambling, and the antigambling 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 \(R_{0}\), gamblingfree equilibrium \(E_{0}\), and gamblingprevailing equilibrium \(E_{ +} \). If \(R_{0} < 1\), the gamblingfree 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 antigambling policy on the hesitator or the gambler can restrain online gambling spreading, and the antigambling 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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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.
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Kong, Y., Li, T., Wang, Y. et al. Dynamics analysis of an online gambling spreading model on scalefree networks. Adv Differ Equ 2021, 11 (2021). https://doi.org/10.1186/s1366202003165z
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Keywords
 SHGD model
 Heterogeneity
 Psychological factors
 Antigambling policy
 Stability
 Persistence