An uncertain SIR rumor spreading model

In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model.


Introduction
Rumor is a statement that has no basis in fact, but is made up out of thin air or according to one's will and spread by some means. In other words, a rumor is a piece of information that is created out of nothing but spreads widely. With the development of the society and the internet, compared with the traditional interpersonal communication, rumor propagation has the characteristics of faster information transmission, wider influence, stronger force, and more concealment. Rumors have a great negative effect on our daily life. Rumors not only convey some inaccurate information to the society, but also cause bad social habits, crimes, and economic losses, as well as pose a great threat to the national and public security [1,2]. At the same time, changes in rumor spreading are sudden and uncertain. How to remove rumors in a timely manner and how to effectively suppress rumors have become a crucial issue to the government and the society. Therefore, it is significant to study uncertain SIR rumor spreading model for rumor control and social stability.
Due to the similarity between rumor spreading process and propagating process of epidemic diseases, an epidemic model has been favored by most domestic and overseas scholars in the research on rumor spreading. Kermack and McKendrick [3] first proposed the classic SIR epidemic model in 1927. Subsequently, based on this research, many scholars used mathematical models to establish SI [4], SIS [4], SIRS [5], SEIRS [6], and other models, all of which are playing an increasingly important role in predicting and controlling infectious diseases. These models provide more ideas and insights for people to study the rumor propagation process. In 1964, based on theoretical analysis and practical obser-vation of infectious disease models, Daley and Kendall found that rumor and infectious disease spreads have many similarities on the surface, but their transmission systems have essential differences. Therefore, Daley and Kendall [7,8] established the classical rumor spreading model for the first time and named it DK model. In the DK model, the general population is divided into three categories: those who don't know rumors (ignorant), those who know and spread rumors (spreader), and those who know but do not spread rumors (stiflers). Later, Maki [9] modified the DK model and established MK model in 1973. This model more truly reflected the rumor propagation process, because it indicated that direct contact between the disseminator and others was the way to spread rumors. Although DK and MK models have rigorous logic in establishing the model process, they ignore many factors that affect the rumor propagation process. Based on the DK, MK, and infectious disease models, more extensive studies have been conducted on rumor propagation from different perspectives, such as small-world networks [10,11], complex networks [12,13], forgetting mechanism [14], memory mechanism [15], retweeting mechanism [16], individual behavior [17], education [18], internal and external influences [19]. In 2016, Zhang et al. [20] used existing epidemic models to study the dynamics of information diffusion and applied them to complex networks, emphasizing the importance of information dissemination. In 2018, based on a nonlinear model of SIS in a complex network environment, Zhan et al. [21] studied the coupling dynamics of epidemic transmission and information diffusion.
The deterministic rumor spreading model has made great achievements in all aspects, but some scholars have found that in the process of rumor propagation, rumor spread form is largely influenced by the environment noise. Human contact is not fixed and can be disturbed by some random factors. Therefore, the use of deterministic rumor propagation model to describe the rumor propagation process will reduce the accuracy of rumor propagation results. There are a lot of studies on stochastic infectious disease model, but there are few studies on stochastic rumor propagation. Tornatore et al. [22] built a stochastic SIR model with or without distributed time delay and studied the stability of disease-free equilibrium of the model. Miao et al. [23] proposed a dynamic behavior of stochastic SIR infectious disease with horizontal and vertical characteristics by using stochastic differential equation theory and inequality technique. Ball et al. [24] considered the propagation of a Markovian SIR model over the configuration model network. In 2016, Chen [25] used Brownian movement to describe environmental noise to establish a random rumor propagation model with parametric perturbation. In 2020, Huo and Dong [26] analyzed the propagation mechanism of the rumor spreading model in media reports with white noise interference based on the classical rumor propagation model. However, it is unreasonable to use stochastic differential equations with Wiener process to study the uncertain factors in rumor propagation process in a short time. In this paper, we use another effective mathematical tool, uncertainty theory, to study the uncertainty factors in SIR rumor spreading model.
The uncertainty theory was established by Liu [27,28] in 2007 and revised in 2009. Uncertainty theory is a new branch of mathematics based on normality, duality, subadditivity, and product axioms. It is used to describe and deal with the phenomenon of uncertainty [29]. In this theory, uncertain phenomena are characterized by uncertain variables, and uncertain variables are described by uncertain distributions. In 2008, Liu [30] proposed the uncertain process. An uncertain process is actually a series of uncertain variables with time as the index, which describes the dynamic behavior of the uncertain phenomenon. Then, in 2009, Liu [28] proposed a special class of uncertain processes called Liu process, which is a stationary, independent increment process whose increments are normal uncertain variables [31]. Liu process is the uncertain counterpart of Wiener process. Uncertain differential equation is a kind of differential equation with Liu process, which was first proposed by Liu [30] in 2008. In 2010, Chen and Liu [32] proved the existence and uniqueness theorem of solutions for uncertain differential equations under Lipschitz and linear growth conditions. Moreover, some scholars presented some methods to solve special uncertain differential equations [32][33][34]. Based on the Lipschitz continuity of Liu process, Yao et al. [35] gives a sufficient condition for the stability of uncertain differential equations. In many cases, uncertain differential equations are difficult to solve to obtain analytic solutions. Yao and Chen [36] first put forward the concept of an α-path and Yao-Chen fomula, which can transform uncertain differential equations into ordinary differential equations, and one can use Euler method to solve the uncertain differential equation. Based on the uncertain process in the uncertainty theory and the knowledge of an uncertain differential equation, we propose an uncertain SIR rumor spreading model.
In this paper, the uncertain differential equation and Liu process in the uncertainty theory is used to study the rumor spreading for the first time, and establish the uncertain SIR rumor spreading model. The rest of the paper is arranged as follows. In Sect. 2, the basic definitions and theorems of uncertain variable, uncertain process and uncertain differential equation in uncertainty theory are introduced. Section 3 establishes the uncertain SIR rumor spreading model with one Liu process, and gives the derivation process of the model. Section 4 proves the existence and uniqueness of the solutions of uncertain SIR rumor spreading differential equations. Section 5 proves that the uncertain SIR rumor spreading model is stable in measure under certain conditions. In Sect. 6, we give a numerical solution of the model. Section 7 introduces a paradox of stochastic rumor spreading model that shows it is unreasonable to use Wiener process to describe the uncertain disturbance in rumor propagation. In Sect. 8, a brief discussion is given.

Preliminaries
In this section, we review some definitions and useful results about uncertain variable, uncertain process, and uncertain differential equation.
Beside, the product uncertain measure on the product σ -algebra L was defined by Liu [28].
Axiom 4 (Product axiom) Let ( i , L i , M i ) be uncertainty spaces for i = 1, 2, · · · , then the product uncertain measure M is an uncertain measure satisfying where i are arbitrarily chosen events from L i for i = 1, 2, · · · , respectively.
Definition 2 (Liu [27]) An uncertain variable is a measurable function ξ from an uncertainty space ( , L, M) to the real number set , i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ |ξ (γ ) ∈ B} is an event.
Definition 3 (Liu [27]) The uncertainty distribution of an uncertain variable ξ is defined by (x) = M{ξ ≤ x} for any real number x.
Definition 4 (Liu [27]) An uncertain variable ξ is called normal if it has a normal uncertainty distribution where e and σ are real numbers with σ > 0. In particular, if e = 0 and σ = 1, ξ is called standard normal.
If the inverse function -1 exists and is unique for each α ∈ (0, 1), then it is called the inverse uncertainty distribution of ξ . In this case, the uncertainty distribution is said to be regular.

Uncertain process
Definition 6 (Liu [30]) Let T be an index set, and ( , L, M) be an uncertainty space. An uncertain process X t is a measurable function from T × ( , L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {X t ∈ B} = {γ ∈ |X t (γ ) ∈ B} is an event.
Definition 7 (Liu [31]) An uncertain process X t is said to have an uncertainty distribution t (x) if at each time t, the uncertain variable X t has the uncertainty distribution t (x).
Definition 8 (Liu [28]) An uncertain process C t is said to be a Liu process if (i) C 0 = 0 and almost all sample paths are Lipschitz continuous.
(ii) C t has stationary and independent increments. (iii) Every increment C s+t -C s is a normal uncertain variable with an uncertainty distribution

Uncertain differential equation
Definition 9 (Liu [30]) Let X t be an uncertain process. For any partition of closed interval provided that the limit exists almost surely and is finite.
Definition 10 (Liu [28]) Let X t be an uncertain process and C t be a Liu process. For any provided that the limit exists almost surely and is finite.
Definition 11 (Liu [30]) Suppose C t is a Liu process, and f and g are two given functions. Then is called an uncertain differential equation. An uncertain process that satisfies equation (1) identically at each time t is called a solution of the uncertain differential equation.
Definition 12 (Yao and Chen [36]) The α-path (0 < α < 1) of an uncertain differential with respect to t that solves the corresponding ordinary differential equation where -1 (α) is the inverse uncertainty distribution of standard normal uncertain variable, i.e., Theorem 1 (Yao and Chen [36]) Assume that f (t, x) and g(t, x) are continuous functions. Let X α t be an α-path of the uncertain differential equation Then M{X s ≤ X α s } = α, i.e., X s has an inverse uncertainty distribution Definition 13 (Yao [37]) Let X t be an uncertain process. If for each α ∈ (0, 1), there exists a real function X α t such that then X t is called a contour process. X α t is called the α-path of the uncertain process X t .

Model formulation
DK model is a classical rumor spreading model. On the basis of the DK model, the forgetting mechanism is added to establish the deterministic model of SIR rumor spreading. The ordinary differential equation SIR model is given as follows: In a closed environment, the total population is grouped into three categories: ignorants, spreaders, and stiflers, which are represented by I, S, and R, respectively: • Rumor ignorant (I), individuals who know nothing about rumors; • Rumor spreader (S), individuals who know rumors and will spread them; • Rumor stifler (R), individuals who know rumors but don't spread them. The population sizes of I, S, and R individuals at time t are denoted by I t , S t , and R t , respectively. The parameters in the above model are expressed as follows: • β, the spreading rate that an ignorant turns into a spreader after he comes into contact with the spreader; • λ, the rate that the spreader converts into a stifler after contacting with the rumor stifler; • η, the rate that a spreader becomes a stifler because he forgotten or lost interest in the rumor. It is worth to mention that the deterministic SIR rumor spreading model and uncertain SIR rumor spreading model proposed in this manuscript are suitable to the following cases: 1. The total population size is considered as a constant, N . Suppose that there is no entry and exit in the total population. 2. Assume that the mixture of individuals in the population is sufficiently uniform. 3. Suppose that one spreader is in contact with another, and their status does not change. The SIR rumor spreading process is shown in Fig. 1. However, in real life, rumor spreading is affected by environmental noise. This paper, based on the ordinary differential equation (2), introduces the Liu process to describe the uncertain factors in the environment, the rate λ of a spreader turning into a stifler affected by uncertainties. In this part, we change the rate λ into an uncertain variable λ t .
Let the time interval [0, t] be divided uniformly into n subintervals such that 0 = t 0 < t 1 < t 2 < · · · < t n = t, and the interval length is t = t i+1t i , i = 0, 1, 2, . . . , n -1. We assume that where λ is a constant, C t is a Liu process representing the uncertain disturbance, and σ is a constant denoting the intensity of C t . An increase in the number of spreader individuals, who are the original spreaders in the interval (t i , t i+1 ], is equal to Then we have Letting t → 0, the above equation can be written as an uncertain integral equation, The number of stiflers in the interval (t i , t i+1 ] is equal to Then we have Letting t → 0, an uncertain integral equation can be obtained as The number of ignorants in the interval (t i , t i+1 ] is equal to Then we have Letting t → 0, an uncertain integral equation can be obtained as We obtain a system of uncertain integral equations It is equal to a system of uncertain differential equations, called the uncertain SIR rumor spreading model, as follows: where I t , S t , R t are the numbers of ignorant, spreader, and stifler individuals, respectively, C t is a Liu process representing the uncertain disturbers, σ is a constant denoting the intensity of C t , and β, λ, and η are constants.

Lemma 1 If, in the uncertain SIR rumor spreading model (4), C t is a Liu process, then we have
where K(γ ) is the Lipschitz constant of the sample path C t (γ ) for each γ ∈ .
Proof According to model (3), we have For every sample γ , it can be seen from the definition of Liu process that C t (γ ) is Lipschitz continuous with respect to t. So there exists a finite number K(γ ), called Lipschitz constant, such that The lemma is proved.
Proof Let (I t (γ ), S t (γ ), R t (γ )) and (I * t (γ ), S * t (γ ), R * t (γ )) be two solutions of the uncertain SIR rumor spreading model (4) with different initial values (I 0 , S 0 , R 0 ) and (I * 0 , S * 0 , R * 0 ), respectively. On the basis of Lipschitz continuous sample path C t (γ ), we have According to the strong Lipschitz condition, we have where K(γ ) is the Lipschitz constant of C t (γ ). By the Grönwall's inequality, we have Then max{|I t (γ ) - as long as max{|I 0 - In other words, and the uncertain differential equation (4) is stable in measure. The theorem is proved.

Numerical algorithms
In this section, we use a numerical algorithm based on Euler method to solve the uncertain SIR rumor spreading model. For the uncertain SIR rumor spreading model (4), the most important thing is to obtain spectra of α-paths of I t , S t , and R t .
where I t is the population size of ignorant individuals, S t is the population size of spreaders, R t is the population size of stifler individuals,β is the rumor spread rate,λ is the probability of not spreading rumors,η is the forgetting rate, W t is a standard Wiener process, σ is a constant that represents the intensity of W t , and time t ≥ 0. It has been widely accepted that Wiener process is used to describe the environmental noise in the rumor spreading process, but is this really justified? By deforming the model (7), one can obtain ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ dI t dt = -I t S t , dS t dt =βI t S t -ηS t -(λ +σ dW t dt )S t R t , dR t dt =ηS t + (1 -β)I t S t + (λ +σ dW t dt )S t R t .
Letλ =λ +σ dW t dt , which is called the effective rumor avoidance rate. It can be seen that the stochastic model (7) uses Wiener process to describe the noise influence ofλ in the process of rumor spreading, next we use W t t instead of dW t dt . From W t ∼ N (0, t), we can get that is a normal random variable, with expected value 0 and variance 1/ t. And from the properties of the normal distribution, we havė When t → 0, we have whereλ ∈ [0, 1],σ is a constant, and (·) is the standard normal distribution function. It means that Pr λ ∈ [0, 1] ≈ 0, Pr λ / ∈ [0, 1] ≈ 1.
That is to say, the probability that the effective rumor avoidance rateλ is less than 0 or greater than 1 is almost 1. However, from the point of view of rumor spreading,λ is the effective avoidance rate of rumor spreader turning into rumor stifler. In the actual rumor spreading process, it should be bounded on [0, 1]. That is, λ = Number of spreaders converted to rumor stiflers per unit time The total number of people who spread rumors .
Thus, it is incorrect to use Brownian motion to describe the uncertain disturbance in rumor propagation. Therefore, it is not reasonable to use stochastic SIR rumor spreading model (7) to describe the process of rumor spreading.

Conclusion
This paper mainly proposed an uncertain SIR rumor spreading model by uncertain differential equation and Liu process. It first deduced the process of establishing uncertain SIR rumor spreading model. Then the existence and uniqueness of solutions of uncertain differential equation SIR rumor spreading model were proved, and the model was proved to be stable in measure when it satisfied certain conditions. Furthermore, we transformed the uncertain differential equation into an ordinary differential equation by using Yao-Chen formula. And through numerical simulation, several examples were given to compare the influence of different parameters on the uncertain SIR rumor spreading model. The results showed that in daily life, we can predict the time when rumor spreading reaches its peak and rumor dissipates through uncertain SIR rumor spreading model. Finally, a paradox of stochastic SIR rumor spreading model was introduced.