Stochastic stability and instability of rumor model

1 Introduction

With the rapid development of economy, more and more people choose website to deliver information. For example, during the corona virus disease 2019, there were many rumors about the birthland of the virus. Thus, it is an important issue to study the propagation law of rumor.

Rumor was regarded as collective problem-solving, [1–4]. Recently, due to the development of self-media, the spread of rumors has become increasingly severe. Some companies, individuals, or officials have to step forward to refute rumors. By analyzing the transmission channels of rumors, establishing mathematical models and analyzing their dynamic behavior, it is possible to effectively control the spread of rumors. Let S ( t , x ) and I ( t , x ) denote the densities of the rumor susceptible users and the rumor infected users with a distance of x at time t , respectively. The model describing rumor propagation can be written as

where A , β , μ , η and α are positive constants, see [5] for the meaning of these parameters. Similar to epidemic model, the parameter

is the basic reproduction number of system (1.1) and ℛ 0 will play an important role in analyzing the dynamic behavior of (1.1).

In order to gain a more comprehensive understanding of the rumor propagation model, let us first review the development history of rumor models. The rumor propagation models can be traced back to 1960s. By dividing people into three types: ignorant, spreaders and stiflers, Daley and Kendall [6] established a stochastic rumor propagation model. Based on the fact that rumors are spread through two-way contact between spreaders and other people in the crowd, Maki and Thomson [7] generalized the Daley-Kendall model by using Markov chain. The dynamic behavior of rumor propagation in small world networks was first studied by Zanette [8], where the spreading threshold is given similar to disease models. Borrowing the idea from epidemic model, a new Susceptible-Infected-Hibernator-Removed model was introduced by Zhao et al. [9]. By considering the impact of education rate on the rumor spreading mechanism, Afassinou [10] established a new rumor propagation model, also see [11]. Just recently, Jain et al. [12] considered the rumor model on homogeneous social network incorporating delay in expert intervention and government action, also see [13].

Upon comparing with the epidemic model, like SIS [14–16], model (1.1) needs more attention. El Fatini et al. [17] considered stochastic stability and instability of an epidemic model, also see [18]. In this work, we focus on the stability and instability of equilibrium of (1.1). In order to do so, we find

1. If ℛ 0 < 1 , a unique free-disease equilibrium E 0 = A μ , 0 exists for system (1.1), and is globally asymptotically stable.

2. If ℛ 0 > 1 , except for E 0 , system (1.1) has a unique positive equilibrium point E * = ( S * , I * ) , where I * and S * satisfy

   I * = ( ℛ 0 − 1 ) μ ( μ + η ) β ( μ + η ) + μ α , S * = μ + η β + α I * β = μ + η β + α β ( ℛ 0 − 1 ) μ ( μ + η ) β ( μ + η ) + μ α .

where B ( t ) is one-dimensional Brownian motion, σ is the intensity of the Brownian motion and other parameters are similar to those in (1.1).

The present study is organized as follows. In Section 2, the well-known results on stochastic stability is introduced. Sections 3 and 4 are concerned with the stability and instability of equilibrium E 0 . Numerical solutions are obtained in Section 5.

2 Preliminaries on stochastic stability

In this section, we begin by recalling some definitions and theorems about equilibrium states of a stochastic differential equations. We will borrow the definition of stability from [19]. Consider the following d -dimensional stochastic system:

where f ( t , X ) is a function in R d , defined in [ t 0 , + ∞ ) × R d and g ( t , X ) is a d × m matrix, f and g are locally Lipschitz functions in X and B = { B ( t ) } t ≥ 0 is an m -dimensional Wiener process. We assume that X = 0 is a solution of system (2.1).

Throughout this work, let X n , X be random variables, X n converges exponentially almost surely to X means that there exists a constant ε > 0 such that

Similar to [17], we let K be the family of all continuous non-decreasing functions and α : R + → R + satisfy α ( 0 ) = 0 and α ( r ) > 0 if r > 0 . For l > 0 , set S l = { x ∈ R d : ∣ x ∣ < l } . We call a continuous function V ( t , x ) a positive-definite on [ t 0 , ∞ ) × S l if V ( t , 0 ) ≡ 0 and for some α ∈ K , it holds that

And if − V is positive-definite, then V is said to be negative-definite. Assume that V ( t , x ) is a continuous non-negative function, then V ( t , x ) is said to be decrescent if for some α ∈ κ , it holds that

Let 0 < l ≤ ∞ . Let C 1 , 2 ( R + × S l ; R + ) be the family of all nonnegative functions V ( t , x ) defined on R + × S l such that they are continuously differentiable once in t and twice in x . The differential operator L associated with system (2.1) is written as

For V ∈ C 1 , 2 ( R + × S l ; R + ) , it is easy to see that

For x ( t ) ∈ S l , Itô’s formula implies that

where

3 Stability of equilibrium E 0

In this section, we consider the stability of equilibrium E 0 by using suitable Lyapunov functions. It is noted that the definition of stability of equilibrium E 0 is 0 replaced by E 0 in Definition 2.1 (1). In order to do that, we first need to verify the existence of global positive solution to (1.1). Define the set

The proof of Proposition 3.1 is standard [21] and we omit it here. Actually, by using the Lyapunov analysis method, we can show that model (1.3) has a local positive solution; and then we can prove that this solution is global.

Next some sufficient conditions for the extinction of the rumor will be given. The almost sure exponential convergence of solutions to the equilibrium state E 0 is also considered.

The proof of Corollary 3.1 is easy. First, it follows from Theorem 3.2 that I ( t ) converges almost surely exponentially to 0. Then, considering the first equation of (1.1), we can obtain the limit of S ( t ) . Corollary 3.1 shows that if R 0 < 1 , the rumor will disappear.

4 Instability of equilibrium E 0

The instability of the equilibrium E 0 implies that the rumor will become popular. A sufficient condition for the instability of the free equilibrium E 0 of (1.1) will be given in the following result.

It follows from Theorem 4.1 that if condition (4.1) holds, then the equilibrium E 0 is unstable in probability. Condition (4.1) shows that if the conversion rate is suitable large, then the rumor will be popular. By the above limiting behavior of the rumor, there is no result. In the next theorem, inspired by [17,22], we consider the oscillation of solutions of (1.3) around the positive equilibrium E * = ( S * , I * ) , of the deterministic system (1.1). We will prove that the solution of (1.3) remains close to E * under the condition that the intensity of noises is relatively small.

5 Numerical simulations

In this section, by using the improved Milstein method [23], we simulate the positive solution to (1.3) with the time step Δ = 0.048 . Denote

Denote X k = ( S k , I k ) T . We obtain the following scheme:

where Y k = X k + Δ g ( X k ) , time increment Δ > 0 and ξ k , k = 1 , 2 , … , are independent Gaussian random variables obeying the distribution N ( 0 , 1 ) .

Let A = 0.5 , μ = 0.3 , η = 0.2 , and α = 0.1 .

In Figure 1, we obtain an example of one sample path of the solution to system (1.3) using the above parameters and the initial values S ( 0 ) = 0.8 and I ( 0 ) = 0.2 . In addition, we assume that β = 0.4 and σ = 0.4 . In this case, we obtain ℛ s = 8 9 < 1 . The simulation verifies the analytic result shown in Theorem 3.1. In other words, the rumor will disappear with probability one.

Figure 1

One sample path of the solution of system (1.3).

We still choose β = 0.2 and initial values in S ( 0 ) = 0.8 and I ( 0 ) = 0.2 . So, ℛ 0 = 2 3 < 1 . Consequently, Theorem 3.2 shows that the rumor-free steady state will attract all positive solutions of system (1.3). In Figures 2 and 3, σ = 0.2 and σ = 2 clearly support the above result, respectively. Meanwhile, it is easy to show that the stronger the white noise, the faster the rumor disappears.

Figure 2

Rumor-free of (1.3) with σ = 0.2 .

Figure 3

Rumor-free of (1.3) with σ = 2 .

To support the result of Theorem 4.1, we choose β = 1 , σ = 0.5 and S ( 0 ) = 0.99 , I ( 0 ) = 0.05 . Clearly, β = 1 > 0.4 + 5 24 = μ + η A μ + α + 1 2 σ 2 A μ . Hence, Theorem 4.1 shows that the rumor-free equilibrium state E 0 is instable in probability, which is well observed in Figure 3. It is noted that the initial data are chosen, respectively, very close to the rumor-free steady state E 0 .

Finally, we will illustrate the effect of the environment fluctuations on the stability of the positive equilibrium state E * of the corresponding deterministic system (1.3). In order to do that, the parameters σ = 0.01 , β = 0.4 are chosen, and consequently, ℛ 0 = 4 ⁄ 1 > 1 . Note that σ is enough small, Theorem 4.2 implies that the difference between the solution of the stochastic system (1.3) and the equilibrium in time average is also small. Figure 4 supports this result, where the solution of system (1.3) oscillates around the positive equilibrium E * for a long time.

Figure 4

Solutions of (1.3) oscillates around the endemic equilibrium E * .

6 Conclusion

In this work, a stochastic rumor model is introduced:

Like epidemic model, the basic reproduction number of system (6.1) is defined as ℛ 0 = β A μ ( μ + η ) . It is easy to prove that if ℛ 0 < 1 , system (6.1) has a unique free-disease equilibrium E 0 = ( A μ , 0 ) , which is globally asymptotically stable; and if ℛ 0 > 1 , in addition to E 0 , system (6.1) has a unique positive equilibrium point E * = ( S * , I * ) , which is defined in Section 1.

Our main results are as follows. The stability of equilibrium E 0 is established in Section 3 by using suitable Lyapunov functions. The instability of equilibrium E 0 is obtained in Section 4, which implies that the rumor will become popular. Some numerical examples are given in Section 5. Our results provide theoretical support for controlling the spread of rumors.