The stationary distribution of a stochastic rumor spreading model

Abstract: In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.


Introduction
Rumor spreading as a social contagion process is very similar to the epidemic diffusion, so most rumor spreading models have evolved from epidemic models, such as SI, SIR and SIS. A classic rumor model was DK model proposed by Daley and Kendall in 1964 [2], in which the population was divided into three classes: people who did not know the rumor, people who spread the rumor and people who know but will never spread the rumor. Maki and Thomson modified the DK model into the MK model [8]. Since then, a number of scholars have proposed various rumor spreading models by improving the traditional epidemic models [1,3,[14][15][16]. In 2019, Tian and Ding [11] formulated an ordinary differential equation (ODE) compartmental model for rumor, where the population was divided into five disjoint classes, namely the ignorants, the latents, the rumor-spreaders, the debunkers and the stiflers. At time t, the numbers in each of these classes are denoted by I(t), L(t), R(t), D(t) and S (t), respectively. The rumor spreading model considered by Tian and Ding [11] can be described by a system of ODEs where Λ is the constant immigration rate of the population, µ is the rumor-contacting rate, k is the debunker-contacting rate, ρ is the rate at which all existing users exit from the five classes (i.e. emigration rate), α is the rumor-spreading rate, β is the debunking rate, γ is the silent rate, δ is the rumor-stifling, ξ is the reversal rate and θ represents the debunking-stifling rate. All parameters are assumed to be independent of time t and positive. The model used in the above study to describe rumor propagation behavior is deterministic model, whereas the random models used to study rumor propagation are few [3]. But in the real world, rumor models are often affected by environmental noise. Especially in emergency events, when rumors are widely spread, the propagation process is affected by many uncertain factors, which increase the volatility of the propagation process. Therefore, it would be necessary and interesting to reveal how the environmental noise affects the rumor spreading model. Following the idea of Jia et al. [3], in this paper, we assume that the stochastic perturbations are of white noise type which are proportional to the system variable, respectively. Then we obtain a stochastic analogue of the deterministic model (1.1) as follows where B i (t)(i = 1, 2, 3, 4, 5) are independent Brownian motions and σ i > 0(i = 1, 2, 3, 4, 5) are their intensities. All the other parameters in system (1.2) have the same meaning as in system (1.1). Throughout this paper, unless otherwise specified, let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions, that is, it is rightly continuous and increasing while F 0 contains all P−null sets, and let B i (t)(i = 1, 2, 3, 4, 5) be scalar Brownian motions defined on the probability space. For the sake of simplicity, we introduce the following notations: With the help of the Lyapunov function methods and the inequality techniques, the existence and uniqueness of an ergodic stationary distribution of the positive solutions to system (1.2) are presented. The main difficulties lies in the construction of Lyapunov function and the construction of a bounded closed domain. The main contribution of this paper are highlighted as follows: (i) a stochastic rumor spreading model is proposed and investigated; (ii) some sufficient conditions for the existence of an ergodic stationary distribution; and (iii) the stationary distribution implies that the rumor can be persistent in the mean. The subsequent part of this paper is as follows: In Section 2, we prove the existence and uniqueness of a global positive solution to system (1.2) with any positive initial value. In Section 3, by constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to model (1.2). In Section 4, some numerical simulations are provided to illustrate our theoretical results. Finally, some concluding remarks are presented to end this paper.

Existence and uniqueness of the global positive solution
To analyze the dynamical behavior of a rumor spreading model, the first concerning thing is whether the solution is global and positive. In this section, motivated by the methods in [9] and we show that there is a unique global positive solution of system (1.2). The key is to construct a Lyapunov function.
Proof. Since the coefficients of system (1.2) satisfy the local Lipschitz condition, we know that, for any initial value (I(0), L(0), R(0), D(0), S (0)) ∈ R 5 + , there is a unique local solution ( where τ e is the explosion time [10]. Now we prove the solution is global, i.e. to prove τ e = ∞ a.s. To this end, let m 0 > 0 be sufficiently large such that each component of (I Throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). Obviously, τ m is an increasing function as m → ∞. Set τ ∞ = lim m→∞ τ m . Then τ ∞ ≤ τ e a.s. If τ ∞ = ∞ a.s. is true, then τ e = ∞ a.s. and (I(t), L(t), R(t), D(t), S (t)) ∈ R 5 + a.s. for all t ≥ 0. In other words, in order to show this assertion, we only need to prove τ ∞ = ∞ a.s. If the assertion is false, then there is a pair of constants where a is a positive constant to be determined later. The nonnegativity of this function can be seen from u − 1 − ln u ≥ 0, ∀u > 0. According to the general Itô formula (see, for example, Theorem 4.
where LV : Choose a = min{ ρ µ , ρ k }, then we obtain and K is a positive constant. Thus, For any m ≥ m 0 , integrating (2.1) on both sides from 0 to τ m ∧ T and then taking expectation yield Let Consequently, where 1 Ω m denotes the indicator function of Ω m . Letting m → ∞ leads to the contradiction This completes the proof.

Existence of ergodic stationary distribution
When considering rumor propagation model, we are also interested to know when the rumor will persist and prevail in a population. In the deterministic models, it can be solved by proving the rumorepidemic equilibrium of the corresponding model is a global attractor or globally asymptotically stable. But there is no rumor-epidemic equilibrium in system (1.2). In this section, based on the theory of Khasminskii( [7]), we prove that there is a stationary distribution which reveals that the rumor will persist in the mean. Here we present some theory about the stationary distribution which is introduced in ( [7]).
x ∈ R l and A ∈ B and B denotes the σ− algebra of Borel sets in R l .
) Let X(t) be a regular time-homogeneous Markov process in R l described by the stochastic differential equation: The diffusion matrix of the process X(t) is defined as follows: A2: there exists a nonnegative C 2 -function V such that LV is negative for any R n \ D, where L denotes the differential operator defined by Then for all x ∈ R n , where f (·) is a function integrable with respect to the measure π.

Moreover, one has
Making use of (3.1)-(3.3), we then derive that For the convenience, we define Now we are in the position to construct a bounded closed domain U such that the condition A2 in Lemma 3.1 holds. To this end, we define a compact set as follows where > 0 is a sufficiently small constant such that
Remark 3.1. Comparing with the threshold parameter R 0 in [11], the parameter R s 0 in stochastic system (1.2) is less than R 0 , which reveals that the extinction of the rumor is much easier than in the corresponding deterministic model (1.1). Moreover, taking attention to the expression of R s 0 , we can control the rumor propagation by environmental white noise.

Numerical simulations
In this section, we give some examples to illustrate the obtained theoretical results. We illustrate our findings by the Milstein's Higher Order Method developed in [5]. According to this method, we can get the following discretization equation of system (1.2): where time increment ∆t > 0, and ξ 1,k , ξ 2,k , ξ 3,k , ξ 4,k , ξ 5,k are independent Gaussian random variables which follows N(0, 1). However, we would like to point out that the values of parameters of system (1.2) and the initial values in the following numerical simulations are chosen for illustration purposes and are not taken from any real life data for any rumors.
By simple calculation, we have R s 0 = 1.213 > 1, which means that the conditions of Theorem 3.1 hold. Therefore, system (1.2) has a unique ergodic stationary distribution. Figures 1 and 2   In what follows, we start comparing the stochastic system and deterministic system. In Figure 3, the rumor-spreaders in stochastic system are shown in red lines, compared with the rumor-spreaders in deterministic system which are shown in blue lines. It reveals that the environmental disturbance may help to curb the outbreak of rumors. Further, if we increase the environmental noise, the simulation results in Figure 4 suggests that the extinction of rumor-spreaders in stochastic system is much more easier than that in the corresponding deterministic system.

Conclusions
In the current paper, we have studied a stochastic rumor spreading model. We have established sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to system (1.2) by using the stochastic Lyapunov function method. The ergodic property can help us better understand cycling phenomena of a rumor spreading model, and so describe the persistence of a rumor spreading model in practice. More precisely, we have obtained the following result: • Assume Then, for any initial value (I(0), L(0), R(0), D(0), S (0)) ∈ R 5 + , system (1.2) has a unique stationary distribution π(·) and the ergodicity holds. The stationary distribution indicates that the rumor can become persistent in vivo. The theoretic work extended the results of the corresponding deterministic system. The results show that the rumors will maintain its persistence if the environmental noise is sufficiently small, while large stochastic noise can suppress the spread of rumors.
Some interesting topics deserve further consideration. As we all know, time-delay occurs frequently in many practical engineering systems, which is usually the source of oscillation, instability and poor performance of the systems. Now time-delay has been considered into many stochastic models (see, for example, [4,6,12,13]). We leave time-delay case for our future work.