An uncertain SEIR rumor model

Abstract

In this paper, an uncertain SEIR rumor model driven by one uncertain process is formulated to investigate the influence of perturbation in the transmission of rumor. Firstly, the deduced process of the uncertain SEIR rumor model is presented. Then, we proposed the existence and uniqueness theorem for the solution of the model. Moreover, the study of the stability of the uncertain SEIR rumor model was carried out, and then we came to the conclusion that the model stable in mean. In addition, computer algorithm and numerical simulation is used to verify the accuracy of the theoretical results. The simulation results show that the proposed model can explain the trend of rumor propagation correctly and describe the rumor propagation accurately. Finally, we have compared the propagation process of the uncertain rumor model and the deterministic model according to the numerical algorithm, and drew the conclusion that the model with uncertain perturbation fluctuates around the deterministic model.

Keywords

Uncertain differential equation Liu process Uncertain SEIR model Existence and uniqueness Stability

1 Introduction

A large number of rumor spreading examples show that the spread of rumors do great harm to to the country, society and individuals. For example, rumors will cause social panic, economic decline and residents’ emotional instability, which will lead to social unrest and the breeding of violence [1 –3]. Therefore, it is of great practical significance to study the mechanism of rumor propagation in theory for the management and control of rumor propagation. Many scholars study the mechanism of rumor propagation through mathematical modeling and present some classical models [4, 5]. Due to the analogy between contact and transmission, rumor propagation is similar to epidemic propagation, so rumor propagation can be regarded as the process of “ thought infection ”. In the 1960s, Daley and Kendall proposed a mathematical model of rumor propagation, which is called the DK model [6]. Subsequently, Maki and Thompson improved the DK model and formed the MT model in 1973 [7].

After the above two models, The research on rumor model mainly comes from epidemic models. The classic epidemic models are mainly divided into the following three categories: SIS, SIR and SEIR, there has been a lot of research on these three classic models. Zhang et al. [8] proposed an SIS epidemic model by considering the feedback mechanism. Huo et al. [9] proposed a fractional SIR model by considering the birth rate and the mortality rate. Sun et al. [10] incorporated saturation contact rate into the SEIR model and analyzed the dynamic law of the model under this circumstance. The establishment and analysis of deterministic infectious disease models have yielded fruitful results, but with the deepening of the research, experts found that the study of fixed coefficient model is no longer sufficient. The disturbances in the environment affect the accuracy of results but cannot be ignored in modeling, so it is more reasonable to study the rumor model with varying coefficients.

Numerous scholars have explored infectious disease models with random effects. In particular, the SEIR model attracts more and more attention recently and there have been lots of improved models based on it. Yuan et al. [11] proved the stochastically asymptotically stability of the multi-group SEIR and SIR models with random perturbation. Liu et al. [12] established a class of two-group stochastic SEIR model with infinite delays. Witbooi [13] proposed an SEIR model with independent stochastic perturbations. Han et al. [14] studied the distribution of random SEIR infectious disease models with saturated morbidity. Zhang et al. [15] established a stochastic SEIR model with jumps.

Of course, there are also lots of scholars who put forward modified rumor models based on SEIR infectious disease model. Li et al. [16] studied the rumor spreading of a SEIR model in complex social networks with hesitating mechanism. Komi [17] considered the educational level of the population in the SEIR rumor model. Zhu et al. [18] proposed a modified SEIR rumor model and discussed its stability. Xia et al. [19] proposed a modified SEIR model by considering the vagueness and attractiveness of rumors. Sun et al. [20] established a SEIR rumor spreading model with demographics on scale-free networks and proved its global stability. There are very few researches on the SEIR model with variable coefficients, but if this problem can be solved, it will be a great theoretical breakthrough.

In this paper, another effective mathematical tool, uncertainty theory, is used to deal with the model with uncertain parameters. Uncertainty theory was established by Liu [21] in 2007 and perfected by Liu [24] in 2009. It is a theory based on normality, duality, subadditivity and product axioms. The concept of an uncertain process was introduced by Liu [22] in 2008. And Liu process was proposed by Liu [21] to deal with white noise. In 2008, Liu [22] proposed the concept of uncertain differential equations, which are driven by Liu processes. In 2010, Chen and Liu [25] studied the existence and uniqueness of solutions of uncertain differential equations, and proposed and verified the existence and uniqueness theorem. In 2012, based on Chen and Liu’s research, Gao [26] investigated the existence and uniqueness of solutions of uncertain differential equations with the local Lipschitz condition, and proposed and verified the theorem. Liu presented the concept of stability in measure of uncertain differential equation in [23]. In 2015, Yao et al. [27] consider the case that the uncertain differential equation is stable by means and give a sufficient condition. Liu and Chen [25] have made great breakthroughs in the analytical solution of linear uncertain differential equations. Liu [28] and Yao [29] have made great contributions to the analytical solutions of nonlinear uncertain differential equations and put forward their respective opinions. Yao and Chen [31] put forward a feasible idea on the numerical solution of uncertain differential equations. In 2015, Zhu [32] proposed the concept of uncertain fractional differential equation and applied it to the interest rate model. And in the past two years, Jin et al. [33] studied the extreme values for solution to an uncertain fractional differential equation for the Caputo type and proposed an uncertain stock model. The framework of uncertainty theory is becoming wider and wider.

Considering the interference of environmental factors in the spreading of rumors, this paper proposes an uncertain SEIR rumor model with only one uncertain disturbance, and presents and verifies the existence and uniqueness of the solution of the model. Furthermore, the stability of the model is verified, and the result shows that the model is stable in mean. In addition, some sample simulations are presented to illustrate the main conclusions. The rest of this paper is organized as follows. In Section 2, some basic definitions and theorems that will be used later are introduced. In Section 3, we propose an uncertain SEIR model based on the uncertainty theory. Then in Section 4, we verify that the solutions of the uncertain SEIR model exist and and have only one set of solutions. In Section 5, the stability of the model is verified. In Section 6, Euler method and computer algorithm are used to verify the above conclusions. At last, we make a brief summary of this paper.

2 Preliminary

Definition 1. [21, 23] Let L be a σ-algebra on a nonempty set Γ . The uncertain measure ℳ is a set function ℳ : L → [0, 1] if it follows the following axioms,

Axiom 1. (Normality Axiom) ℳ {Γ} =1 for the universal set Γ .

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ .

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯ , we have $\begin{matrix} M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} . \end{matrix}$

Axiom 4. (Product Axiom) Let (Γ_k, L_k, ℳ _k) be uncertainty spaces for k = 1, 2, … The product uncertain measure ℳ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from Λ_k for k = 1, 2, …, respectively.

Definition 2. [23] Let C_t be a Liu process. Then Liu integral of the uncertain process X_t with respect to C_t is $\int_{a}^{b} X_{t} {dC}_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite for any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

Theorem 1. [30] Let C_t be a Liu process on an uncertainty space (Γ, L, ℳ) . Then there exists an uncertain variable L such that L (γ) is a Lipschitz constant of the sample path C_t (γ) for each γ $lim_{x \to + \infty} M {L \leq x} = 1$ and $M {γ \in Γ | L (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$

Definition 3. [22] Suppose C_t is a Liu process, and f and g are two given functions. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ is called an uncertain differential equation.

Theorem 2. [25] The equation in Definition 2 has a unique solution if the coefficients f (t, x) and g (t, x) satisfy the Lipschitz condition $| f (t, x) - f (t, y) | + | g (t, x) - g (t, y) | \leq H | x - y |,$ $\forall x, y \in R, t \geq 0$ and the linear growth condition $| f (t, x) | + | g (t, x) | \leq H (1 + | x |), \forall x \in R, t \geq 0$ for some constant H .

Definition 4. [31] Let α be a number s.t. 0 < α < 1 . The equation in Definition 2 with initial value X₀ is said to have an α-path $X_{t}^{α}$ with respect to t if it is a solution to the following equation $d X_{t}^{α} = f (t, X_{t}^{α}) d t + | g (t, X_{t}^{α}) | Φ^{- 1} (α) d t,$ where Φ^-1 (α) is the inverse uncertainty distribution of standard normal uncertain variable, that is, $Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1 .$

Definition 4. [27] The equation in Definition 2 with different initial values X₀ and Y₀ is called stable in mean if $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] = 0,$ where X_t and Y_t are any two solutions.

3 Model formulation

Firstly, we consider the following ordinary differential equations (ODE) of SEIR model,

${\begin{matrix} d S_{t} & = - ρ S_{t} I_{t} d t \\ d E_{t} & = (ρ S_{t} I_{t} - β E_{t}) d t \\ d I_{t} & = (β E_{t} - φ I_{t}) d t \\ d R_{t} & = φ I_{t} d t, \end{matrix}$ (1)

In the above SEIR rumor model, there are four states in the system at time t. The states S (susceptible), E (exposed), I (infectious) and R (removed) refers to the proportion of people in the system who are unaware of rumors, who has received rumors but not decided whether to spread them, who knows the rumor and spreads it and who do not believe or do not spread rumors, respectively.

The propagation process between the four states are defined as follows:

Susceptible (S) contacts the infectious (I) and becomes the exposed (E) with rate ρ.

After a period of incubation, exposed (E) becomes the infectious (I) with rate β.

The rate that infectious (I) stops spreading rumors and becomes removed (R) is φ.

In the following discussion, S_t, E_t, I_t and R_t represent the proportion of the total population at time t, respectively. Supposing that the removed (R) has permanent immunity, and the total population N_t = 1 . S_t, E_t, I_t and R_t are all continuous differentiable functions of t.

In this paper, we introduce only one noise in (ODE) (1) and transform the deterministic problem into a corresponding uncertain problem. Let β be the rate of the exposed (E) changing to the infectious (I), which is affected by the individual’s knowledge level and the ability to tell right from wrong. However, these factors are basically stable in a short term. Therefore, we assume that β is constant in the following discussion. In addition, we already know that φ is the rate of infectious (I) turning to removed (R) and it is related to individual interest or memory. Nevertheless, these factors have nothing to do with environmental factors in a short term. Thus, we assume that φ is constant. In the real situation, ρ always goes up and down around some value because of environmental interference. In this sense, ρ can be viewed as a uncertain variable ρ_t which is made up of ρ and uncertain disturbance.

Let the time interval [0, t] be divided uniformly into n subintervals such that 0 = t₀ < t₁ < t₂ < ⋯ < t_n = t, and the interval length is Δt = t_i+1 - t_i, i = 0, 1, 2, ⋯ , n - 1 . In this paper, the uncertain disturbance is introduced in (ODE) (1). We assume that $ρ_{t} = ρ + σ \frac{Δ C_{t}}{Δ t},$ where ρ is a positive constant, C_t is a Liu process representing the uncertain disturbance, and σ > 0 is a constant denoting the intensity of C_t .

Note that the number of new exposed individuals in the interval (t_i, t_i+1] is equal to $ρ_{t} S_{t_{i}} I_{t_{i}} Δ t = ρ S_{t_{i}} I_{t_{i}} Δ t + σ S_{t_{i}} I_{t_{i}} Δ C_{t_{i}} .$ Thus $\begin{matrix} S_{t_{i + 1}} - S_{t_{i}} = - ρ S_{t_{i}} I_{t_{i}} Δ t - σ S_{t_{i}} I_{t_{i}} Δ C_{t_{i}} . \end{matrix}$ (2) Then we have $S_{t} - S_{0} = \sum_{i = 0}^{n - 1} (S_{t_{i + 1}} - S_{t_{i}}) .$ Letting Δt → 0, the above equation can be written as an uncertain integral equation, $S_{t} - S_{0} = - \int_{0}^{t} - ρ S_{s} I_{s} d s - \int_{0}^{t} σ S_{s} I_{s} d C_{s} .$ (3) In the same way, the number of infected individuals is equal to $\begin{matrix} E_{t_{i + 1}} - E_{t_{i}} & = (ρ S_{t_{i}} I_{t_{i}} - β E_{t_{i}}) Δ t + σ S_{t_{i}} I_{t_{i}} Δ C_{t_{i}} \end{matrix}$ (4) in the interval (t_i, t_i+1] . Then we obtain $E_{t} - E_{0} = \sum_{i = 0}^{n - 1} (E_{t_{i + 1}} - E_{t_{i}}) .$ Letting Δt → 0, an uncertain integral equation can be obtained, $E_{t} - E_{0} = \int_{0}^{t} (ρ S_{s} I_{s} - β E_{s}) d s + \int_{0}^{t} σ S_{s} I_{s} d C_{s} .$ (5) The number of removed individuals in the interval (t_i, t_i+1] is equal to $\begin{matrix} I_{t_{i + 1}} - I_{t_{i}} & = (β E_{t_{i}} - φ I_{t_{i}}) Δ t . \end{matrix}$ (6) Then we have $I_{t} - I_{0} = \sum_{i = 0}^{n - 1} (I_{t_{i + 1}} - I_{t_{i}}) .$ Letting Δt → 0, an uncertain integral equation can be obtained, $I_{t} - I_{0} = \int_{0}^{t} (β E_{s} - φ I_{s}) d s .$ (7)

Since the total population in the model is constant, i.e., $S_{t_{i + 1}} + E_{t_{i + 1}} + I_{t_{i + 1}} + R_{t_{i + 1}} = S_{t_{i}} + E_{t_{i}} + I_{t_{i}} + R_{t_{i}},$ we obtain $R_{t_{i + 1}} - R_{t_{i}} = - (S_{t_{i + 1}} - S_{t_{i}}) - (E_{t_{i + 1}} - E_{t_{i}}) - (I_{t_{i + 1}} - I_{t_{i}}) .$ Substituting S_{t
_i+1} - S_{t
_i}, E_{t
_i+1} - E_{t
_i} and I_{t
_i+1} - I_{t
_i} with (2), (4) and (6) in the above equations, respectively, it is obviously that $\begin{matrix} R_{t_{i + 1}} - R_{t_{i}} = φ I_{t_{i}} Δ t . \end{matrix}$ Then we have $R_{t} - R_{0} = \sum_{i = 0}^{n - 1} (R_{t_{i + 1}} - R_{t_{i}}) .$ Letting Δt → 0, we obtain $R_{t} - R_{0} = \int_{0}^{t} φ I_{s} d s .$ (8) Based on (3), (5), (7) and (8), we obtain uncertain integral equations of SEIR rumor model as follows, ${\begin{matrix} S_{t} & = S_{0} - \int_{0}^{t} ρ S_{s} I_{s} d s - \int_{0}^{t} σ S_{s} I_{s} d C_{s} \\ E_{t} & = E_{0} + \int_{0}^{t} (ρ S_{s} I_{s} - β E_{s}) d s + \int_{0}^{t} σ S_{s} I_{s} d C_{s} \\ I_{t} & = I_{0} + \int_{0}^{t} (β E_{s} - φ I_{s}) d s \\ R_{t} & = R_{0} + \int_{0}^{t} φ I_{s} d s . \end{matrix}$ (9) Then we get a new modified model as follows, ${\begin{matrix} d S_{t} & = - ρ S_{t} I_{t} d t - σ S_{t} I_{t} d C_{t} \\ d E_{t} & = (ρ S_{t} I_{t} - β E_{t}) d t + σ S_{t} I_{t} d C_{t} \\ d I_{t} & = (β E_{t} - φ I_{t}) d t \\ d R_{t} & = φ I_{t} d t, \end{matrix}$ (10) which is called uncertain differential equations (UDE) of SEIR rumor model. Note that C_t is a Liu process representing the uncertain disturbance, σ is a constant denoting the intensity of C_t, and ρ, β and φ are constants.

4 The existence and uniqueness of the solution of the model

In this section, we investigate the existence and uniqueness of the solution of the uncertain SEIR rumor model (10) with the assumption that the coefficients of the model satisfy linear growth condition and Lipschitz continuous condition. According to the actual situation, we only dealing with the solution on [0, t] for a given real number t .

Lemma 1. In the uncertain SEIR rumor model (10), C_t is a Liu process. Then the inequality $\begin{matrix} | \int_{0}^{t} σ S_{s} (γ) I_{s} (γ) d C_{s} (γ) | \leq K (γ) \int_{0}^{t} | σ S_{s} (γ) I_{s} (γ) | d s \end{matrix}$ holds, where K (γ) is the Lipschitz constant of C_t (γ) .

Proof. Let $0 = t_{0} < t_{1} < \dots < t_{n} = t, Δ = max_{1 \leq i \leq t} | t_{i} - t_{i - 1} | .$ According to the definition of uncertain integral, we can obtain $\begin{matrix} | \int_{0}^{t} σ S_{s} (γ) I_{s} (γ) d C_{s} (γ) | \\ \leq & | lim_{Δ t \to 0} \sum_{i = 0}^{n} σ S_{t_{i}} (γ) I_{t_{i}} (γ) (C_{t_{i}} (γ) - C_{t_{i - 1}} (γ)) | \\ \leq & lim_{Δ t \to 0} \sum_{i = 0}^{n} | σ S_{t_{i - 1}} (γ) I_{t_{i - 1}} (γ) | | C_{t_{i}} (γ) - C_{t_{i - 1}} (γ) | . \end{matrix}$ Each sample γ follows from the definition of Liu process. Therefore, there exists K (γ) , which is a finite number, and we call it Lipschitz constant. In this way, |C_{t
_i} (γ) - C_{t
_i-1} (γ) | ≤ K (γ) |t_i - t_i-1|, i = 1, 2, ⋯ , n . Thus, $\begin{matrix} lim_{Δ t \to 0} \sum_{i = 0}^{n} | σ S_{t_{i - 1}} (γ) I_{t_{i - 1}} (γ) | | C_{t_{i}} (γ) - C_{t_{i - 1}} (γ) | \\ \leq & K (γ) lim_{Δ t \to 0} \sum_{i = 0}^{n} | σ S_{t_{i - 1}} (γ) I_{t_{i - 1}} (γ) | | t_{i} - t_{i - 1} | \\ = & K (γ) \int_{0}^{t} | σ S_{s} (γ) I_{s} (γ) | d s . \end{matrix}$ The lemma is proved. □

Theorem 3. If the coefficients of the uncertain SEIR rumor model (10) satisfy the Lipschitz conditions $\begin{matrix} max {| - ρ (IS - \tilde{I} \tilde{S}) |, | ρ (IS - \tilde{I} \tilde{S}) - β (E - \tilde{E}) |, \\ | β (E - \tilde{E}) - φ (I - \tilde{I}) |, | φ (I - \tilde{I}) |} + | σ (IS - \tilde{I} \tilde{S}) | \\ \leq & L max {| S - \tilde{S} |, | E - \tilde{E} |, | I - \tilde{I} |, | R - \tilde{R} |}, t \geq 0 \end{matrix}$ (11) and linear growth conditions $\begin{matrix} max {| - ρ SI |, | ρ SI - β E |, | β E - φ I |, | φ I |} + | σ SI | \\ \leq & 1 + max {| S |, | E |, | I |, | R |}, t \geq 0 \end{matrix}$ (12) for any $(S, E, I, R), (\tilde{S}, \tilde{E}, \tilde{I}, \tilde{R}) \in R^{4}$ and some constants L, then the model (10) has a unique solution.

Proof. In this paper, the successive approximation method is used to prove that the uncertain SEIR rumor model (10) exists a set of solutions.

We define

$(S_{t}^{(0)} (γ), E_{t}^{(0)} (γ), I_{t}^{(0)} (γ), R_{t}^{(0)} (γ))$

= (S₀, E₀, I₀, R₀) and $\begin{matrix} S_{t}^{(n + 1)} (γ) & = S_{0} - \int_{0}^{t} ρ S_{s}^{(n)} (γ) I_{s}^{(n)} (γ) d s \\ - \int_{0}^{t} σ S_{s}^{(n)} (γ) I_{s}^{(n)} (γ) d C_{s}, \\ E_{t}^{(n + 1)} (γ) & = E_{0} + \int_{0}^{t} (ρ S_{s}^{(n)} (γ) I_{s}^{(n)} (γ) - β E_{s}^{(n)} (γ)) d s \\ + \int_{0}^{t} σ S_{s}^{(n)} (γ) I_{s}^{(n)} (γ) d C_{s}, \\ I_{t}^{(n + 1)} (γ) & = I_{0} + \int_{0}^{t} (β E_{s}^{(n)} (γ) - φ I_{s}^{(n)} (γ)) d s, \\ R_{t}^{(n + 1)} (γ) & = R_{0} + \int_{0}^{t} φ I_{s}^{(n)} (γ) d s, \end{matrix}$ and $\begin{matrix} G_{t}^{(n)} (γ) \\ = & max_{0 \leq s \leq t} {max {| S_{s}^{(n + 1)} (γ) - S_{s}^{(n)} (γ) |, \\ | E_{s}^{(n + 1)} (γ) - E_{s}^{(n)} (γ) |, | I_{s}^{(n + 1)} (γ) - I_{s}^{(n)} (γ) |, \\ | R_{s}^{(n + 1)} (γ) - R_{s}^{(n)} (γ) |}} \end{matrix}$ for any sample γ and for n = 0, 1, 2, ⋯ . We claim that $\begin{matrix} G_{t}^{(n)} (γ) \leq (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) \\ \cdot \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1}, n = 0, 1, 2, \dots, 0 \leq t \leq T, \end{matrix}$ (13) where T is a constant. The inequality is proved by using the mathematical induction as follows.

When n = 0, we can obtain $\begin{matrix} G_{t}^{(0)} (γ) \\ = & max_{0 \leq v \leq t} {max {| S_{v}^{(1)} (γ) - S_{v}^{(0)} (γ) |, | E_{v}^{(1)} (γ) - E_{v}^{(0)} (γ) |, \\ | I_{v}^{(1)} (γ) - I_{v}^{(0)} (γ) |, | R_{v}^{(1)} (γ) - R_{v}^{(0)} (γ) |}} \end{matrix}$ $\begin{matrix} = & max_{0 \leq v \leq t} {max {| - \int_{0}^{v} ρ S_{0} I_{0} d s - \int_{0}^{v} σ S_{0} I_{0} d C_{s} |, \\ | \int_{0}^{v} (ρ S_{0} I_{0} - β E_{0}) d s + \int_{0}^{v} σ S_{0} I_{0} d C_{s} |, \\ | \int_{0}^{v} (β E_{0} - φ I_{0}) d s |, | \int_{0}^{v} φ I_{0} d s |}} \\ \leq & max_{0 \leq v \leq t} {max {| - \int_{0}^{v} ρ S_{0} I_{0} d s |, | \int_{0}^{v} (ρ S_{0} I_{0} - β E_{0}) d s |, \\ | \int_{0}^{v} (β E_{0} - φ I_{0}) d s |, | \int_{0}^{v} φ I_{0} d s |}} \\ + max_{0 \leq v \leq t} | \int_{0}^{v} σ S_{0} I_{0} d C_{s} | \\ \leq & \int_{0}^{t} max {| - ρ S_{0} I_{0} |, | ρ S_{0} I_{0} - β E_{0} |, | β E_{0} - φ I_{0} |, | φ I_{0} |} d s \\ + K (γ) \int_{0}^{t} | σ S_{0} I_{0} | d s \\ \leq & (1 + K (γ)) L (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) t \end{matrix}$ which is satisfied if we take $L \geq \frac{V}{1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}},$ which V = max{ | - ρS₀I₀|, |ρS₀I₀ - βE₀|, |βE₀ - φI₀|, |φI₀| } $+ | σ S_{0} I_{0} | .$ The above results are derived by Lemma 4 and the linear growth condition.

Assume (13) satisfies at n, then $\begin{matrix} G_{t}^{(n + 1)} (γ) \\ = & max_{0 \leq v \leq t} {max {| S_{v}^{(n + 2)} (γ) - S_{v}^{(n + 1)} (γ) |, . . \\ | E_{v}^{(n + 2)} (γ) - E_{v}^{(n + 1)} (γ) |, | I_{v}^{(n + 2)} (γ) - I_{v}^{(n + 1)} (γ) |, \\ . . | R_{v}^{(n + 2)} (γ) - R_{v}^{(n + 1)} (γ) |}} \\ = & max_{0 \leq v \leq t} {max {| - \int_{0}^{v} ρ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) d s . . . \\ - . \int_{0}^{v} σ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) d C_{s} |, \\ | \int_{0}^{v} (ρ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) - β (E_{n + 1} - E_{n})) d s . \\ + . \int_{0}^{v} σ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) d C_{s} |, \\ | \int_{0}^{v} (β (E_{n + 1} - E_{n}) - φ (I_{n + 1} - I_{n})) d s |, \\ . . | \int_{0}^{v} φ (I_{n + 1} - I_{n}) d s |}} \\ \leq & \int_{0}^{t} max {| - ρ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) . |, \\ | ρ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) - β (E_{n + 1} - E_{n}) |, \\ | β (E_{n + 1} - E_{n}) - φ (I_{n + 1} - I_{n}) |, \\ . | φ (I_{n + 1} - I_{n}) |} d s \\ + K (γ) \int_{0}^{t} | σ (S_{n + 1} I_{n + 1} - S_{n} I_{n}) | d s \\ \leq & L (1 + K (γ)) \int_{0}^{t} (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) \\ \cdot \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} s^{n + 1} d s \\ = & (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) \\ \cdot \frac{L^{n + 2} (1 + K (γ))^{n + 2}}{(n + 2)!} t^{n + 2} . \end{matrix}$ The above results are derived by Lemma 4 and Lipschitz condition. Therefore, the inequality (13) is proved.

By the Weierstrass criterion, we obtain $\begin{matrix} \sum_{n = 0}^{+ \infty} (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) \\ \cdot \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} t^{n + 1} \\ \leq & \sum_{n = 0}^{+ \infty} (1 + max {| S_{0} |, | E_{0} |, | I_{0} |, | R_{0} |}) \\ \cdot \frac{L^{n + 1} (1 + K (γ))^{n + 1}}{(n + 1)!} T^{n + 1} < + \infty, \forall γ \in Γ . \end{matrix}$ Thus $(S_{t}^{(i)} (γ), E_{t}^{(i)} (γ), I_{t}^{(i)} (γ), R_{t}^{(i)} (γ))$ converges uniformly in t ∈ [0, T] . For ∀γ ∈ Γ, t ∈ [0, T] , we obtain $\begin{matrix} S_{t} (γ) = lim_{iarrow + \infty} S_{t}^{(i)} (γ), E_{t} (γ) = lim_{iarrow + \infty} E_{t}^{(i)} (γ), \\ I_{t} (γ) = lim_{iarrow + \infty} I_{t}^{(i)} (γ), R_{t} (γ) = lim_{iarrow + \infty} R_{t}^{(i)} (γ) . \end{matrix}$ Therefore, $\begin{matrix} {\begin{matrix} S_{t} (γ) & = S_{0} - \int_{0}^{t} ρ S_{s} (γ) I_{s} (γ) d s - \int_{0}^{t} σ S_{s} (γ) I_{s} (γ) d C_{s} \\ E_{t} (γ) & = E_{0} + \int_{0}^{t} (ρ S_{s} (γ) I_{s} (γ) - β E_{s} (γ)) d s \\ + \int_{0}^{t} σ S_{s} (γ) I_{s} (γ) d C_{s} \\ I_{t} (γ) & = I_{0} + \int_{0}^{t} (β E_{s} (γ) - φ I_{s} (γ)) d s \\ R_{t} (γ) & = R_{0} + \int_{0}^{t} φ I_{s} (γ) d s \end{matrix} . \end{matrix}$ (14) is the solution of the uncertain SEIR rumor model (10) for all t ∈ [0, T] .

Next, we will prove the uniqueness of the solution based on the given conditions. Suppose that (S_t, E_t, I_t, R_t) and $(S_{t}^{*}, E_{t}^{*}, I_{t}^{*}, R_{t}^{*})$ are two sets of solutions to the uncertain SEIR rumor model (10) with the same initial value (S₀, E₀, I₀, R₀) . For each γ ∈ Γ, we obtain $\begin{matrix} max {| S_{t} (γ) - S_{t}^{*} (γ) |, | E_{t} (γ) - E_{t}^{*} (γ) |, . \\ . | I_{t} (γ) - I_{t}^{*} (γ) |, | R_{t} (γ) - R_{t}^{*} (γ) |} \\ = & max {| - \int_{0}^{v} ρ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) d u . . \\ - . \int_{0}^{v} σ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) d C_{u} |, \\ | \int_{0}^{v} (ρ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) - β (E_{u} - E_{u}^{*})) d u . \\ + . \int_{0}^{v} σ (S_{u} I_{u} - S_{s}^{*} I_{s}^{*}) d C_{u} |, \\ | \int_{0}^{v} (β (E_{u} - E_{u}^{*}) - φ (I_{u} - I_{u}^{*})) d u |, \\ . | \int_{0}^{v} φ (I_{u} - I_{u}^{*}) d u |} \\ \leq & \int_{0}^{t} max {| - ρ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) |, . \\ | ρ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) - β (E_{u} - E_{u}^{*}) |, \\ | β (E_{u} - E_{u}^{*}) - φ (I_{u} - I_{u}^{*}) |, . | φ (I_{u} - I_{u}^{*}) |} d u \\ + K (γ) \int_{0}^{t} | σ (S_{u} I_{u} - S_{u}^{*} I_{u}^{*}) | d u \\ \leq & L (1 + K (γ)) \int_{0}^{t} max {| S_{u} (γ) - S_{u}^{*} (γ) |, . \\ | E_{u} (γ) - E_{u}^{*} (γ) |, | I_{u} (γ) - I_{u}^{*} (γ) |, \\ . | R_{u} (γ) - R_{u}^{*} (γ) |} d u . \end{matrix}$ The above conclusions are derived by Lemma 4 and Lipschitz condition. Therefore, $\begin{matrix} max {| S_{t} (γ) - S_{t}^{*} (γ) |, | E_{t} (γ) - E_{t}^{*} (γ) |, . \\ . | I_{t} (γ) - I_{t}^{*} (γ) |, | R_{t} (γ) - R_{t}^{*} (γ) |} \\ \leq & 0 \cdot \exp (L (1 + K (γ)) t \cdot 0) = 0 \end{matrix}$ can be obtained by the Gronwall inequality. It is equivalent to $(S_{t}, E_{t}, I_{t}, R_{t}) = (S_{t}^{*}, E_{t}^{*}, I_{t}^{*}, R_{t}^{*})$ almost surely.

The existence and uniqueness of the solution of the uncertain SEIR rumor model (10) is proved. □

5 The stability of the model

In this section, we study the stability of the uncertain SEIR rumor model (10) and here we look at the stability in mean.

Theorem 4. If the coefficient functions satisfy the strong Lipschitz conditions $\begin{matrix} max {| - ρ (IS - \tilde{I} \tilde{S}) |, | ρ (IS - \tilde{I} \tilde{S}) - β (E - \tilde{E}) |, . \\ . | β (E - \tilde{E}) - φ (I - \tilde{I}) |, | φ (I - \tilde{I}) |} \\ \leq & L_{1 t} max {| S - \tilde{S} |, | E - \tilde{E} |, | I - \tilde{I} |, | R - \tilde{R} |}, t \geq 0, \end{matrix}$ and $\begin{matrix} | σ (IS - \tilde{I} \tilde{S}) | \\ \leq & L_{2 t} max {| S - \tilde{S} |, | E - \tilde{E} |, | I - \tilde{I} |, | R - \tilde{R} |}, t \geq 0 \end{matrix}$ then the uncertain SEIR rumor model (10) is stable in mean for any $(S, E, I, R), (\tilde{S}, \tilde{E}, \tilde{I}, \tilde{R}) \in R^{4}$ , where L_1t and L_2t are two functions satisfying $\int_{0}^{+ \infty} L_{1 s} d s < + \infty and \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3}} .$

Proof. Suppose that S_t, E_t, I_t and R_t are the solutions of the model (10) with initial values S₀, E₀, I₀ and R₀ respectively. And assume that ${\tilde{S}}_{t}, {\tilde{E}}_{t}, {\tilde{I}}_{t}, {\tilde{R}}_{t}$ are another set of solutions of the model with initial values $\tilde{S_{0}}, \tilde{E_{0}}, \tilde{I_{0}}$ and $\tilde{R_{0}}$ respectively. Letting $F_{t} = max {| S_{t} (γ) - \tilde{S_{t}} (γ) |, | E_{t} (γ) - \tilde{E_{t}} (γ) |, .$ $. | I_{t} (γ) - \tilde{I_{t}} (γ) |, | R_{t} (γ) - \tilde{R_{t}} (γ) |}, t \geq 0 .$ Then by the strong Lipschitz conditions, we have $\begin{matrix} F_{t} = max {| S_{t} (γ) - \tilde{S_{t}} (γ) |, | E_{t} (γ) - \tilde{E_{t}} (γ) |, . \\ . | I_{t} (γ) - \tilde{I_{t}} (γ) |, | R_{t} (γ) - \tilde{R_{t}} (γ) |} \\ = & max {| S_{0} - \tilde{S_{0}} + \int_{0}^{t} - ρ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) d s . . \\ - . \int_{0}^{t} σ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) d C_{s} |, \\ | E_{0} - \tilde{E_{0}} + \int_{0}^{t} (ρ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) - β (E_{s} - \tilde{E_{s}})) d s . \\ + . \int_{0}^{t} σ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) d C_{s} |, \\ | I_{0} - \tilde{I_{0}} + \int_{0}^{t} (β (E_{s} - \tilde{E_{s}}) - φ (I_{s} - \tilde{I_{s}})) d s |, \\ . | R_{0} - \tilde{R_{0}} + \int_{0}^{t} φ (I_{s} - \tilde{I_{s}}) d s |} \\ \leq & max {| S_{0} - \tilde{S_{0}} |, | E_{0} - \tilde{E_{0}} |, | I_{0} - \tilde{I_{0}} |, | R_{0} - \tilde{R_{0}} |} \\ + \int_{0}^{t} max {| - ρ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) |, . \\ | ρ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) - β (E_{s} - \tilde{E_{s}}) |, \\ | β (E_{s} - \tilde{E_{s}}) - φ (I_{s} - \tilde{I_{s}}) |, . | φ (I_{s} - \tilde{I_{s}}) |} d s \\ + K (γ) \int_{0}^{t} | σ (S_{s} I_{s} - \tilde{S_{s}} \tilde{I_{s}}) | d s \\ \leq & F_{0} + \int_{0}^{t} L_{1 s} F_{s} d s + K (γ) \int_{0}^{t} L_{2 s} F_{s} d s \\ = & F_{0} + \int_{0}^{t} (L_{1 s} + L_{2 s} K (γ)) F_{s} d s \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ) , 0 ≤ s ≤ t . For any t ≥ 0, we can obtain $\begin{matrix} F_{t} \leq & F_{0} exp (\int_{0}^{t} L_{1 s} d s) exp (K (γ) \int_{0}^{t} L_{2 s} d s) \\ \leq & F_{0} exp (\int_{0}^{+ \infty} L_{1 s} d s) exp (K (γ) \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ by the Gronwall’s inequality. Therefore, we can obtain $\begin{matrix} sup_{t \geq 0} {F_{t}} \\ = & sup_{t \geq 0} {max {| S_{t} - \tilde{S_{t}} |, | E_{t} - \tilde{E_{t}} |, | I_{t} - \tilde{I_{t}} |, | R_{t} - \tilde{R_{t}} |}} \\ \leq & max {| S_{0} - \tilde{S_{0}} |, | E_{0} - \tilde{E_{0}} |, | I_{0} - \tilde{I_{0}} |, | R_{0} - \tilde{R_{0}} |} \\ \cdot exp (\int_{0}^{+ \infty} L_{1 s} d s) exp (K \int_{0}^{+ \infty} L_{2 s} d s) \\ = & F_{0} \cdot exp (\int_{0}^{+ \infty} L_{1 s} d s) exp (K \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ (15) almost surely, where K is a nonnegative uncertain variable. According to Theorem 2 we have $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \geq 2 (1 + exp (\frac{- π x}{\sqrt{3}}))^{- 1} - 1 . \end{matrix}$ And then for inequality (1)5, we take the expected value of both sides of it and get $\begin{matrix} E [sup_{t \geq 0} {F_{t}}] \\ \leq & F_{0} exp (\int_{0}^{+ \infty} L_{1 s} d s) E [exp (K \int_{0}^{+ \infty} L_{2 s} d s)] . \end{matrix}$ For $\begin{matrix} \int_{0}^{+ \infty} L_{1 s} d s < + \infty, \end{matrix}$ we have $\begin{matrix} E [exp (\int_{0}^{+ \infty} L_{1 s} d s)] < + \infty . \end{matrix}$ Since $\begin{matrix} \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3}}, \end{matrix}$ it is conveniently called m . Then the derivation goes as follows. $\begin{matrix} E [exp (Km)] \\ = & \int_{0}^{+ \infty} M {exp (Km) \geq w} d w \\ \leq & 1 + \int_{1}^{+ \infty} M {exp (Km) \geq w} d w \\ = & 1 + m \int_{0}^{+ \infty} exp (vm) M {K \geq v} d v \\ \leq & 1 + m \int_{0}^{+ \infty} exp (vm) \\ \cdot {1 - (2 (1 + exp (\frac{- π v}{\sqrt{3}}))^{- 1} - 1)} d v \\ = & 1 + 2 m \int_{0}^{+ \infty} exp (vm) (1 + exp (\frac{- π v}{\sqrt{3}}))^{- 1} d v \\ = & 1 + 2 \int_{1}^{+ \infty} {1 + w^{π / \sqrt{3} m}}^{- 1} d w < + \infty . \end{matrix}$ So we have $\begin{matrix} lim_{F_{0} arrow 0} E [sup_{t \geq 0} {F_{t}}] \\ \leq & lim_{F_{0} arrow 0} F_{0} \\ \cdot exp (\int_{0}^{+ \infty} L_{1 s} d s) E [exp (K \int_{0}^{+ \infty} L_{2 s} d s)] \\ = & 0 . \end{matrix}$ Hence, according to the definition of stability in mean of uncertain differential equation, we can get that under strong Lipschitz condition, the uncertain SEIR rumor model (10) is stable in mean. □

6 Numerical algorithm and some examples

So far, we have proposed an UDE rumor model (10) based on the ODE model (1), and numbers of theoretical analyses have been made on the uncertain SEIR rumor model. However, analytic solutions for general uncertain differential equations are extremely hard to find. Therefore, we provide a numerical algorithm based on Euler method [31] to solve the rumor model (10) and compare it with the ODE model (1) in this section.

Algorithm: For the model (10), obtaining spectrums of α-paths of S_t, E_t, I_t and R_t are the core content of this numerical algorithm.

Step1: Fix α on (0, 1). Given the initial values S₀, E₀, I₀ and R₀ and all parameters N, ρ, β, φ and σ .

Step2: Based on the recursion formula, we solve $d S_{t}^{α}, d E_{t}^{α}, d I_{t}^{α}$ and $d R_{t}^{α}$ and obtain the α-path $S_{t}^{α}, E_{t}^{α}, I_{t}^{α}$ and $R_{t}^{α},$ respectively. The results are as follows: ${\begin{matrix} E_{i + 1}^{α} & = E_{i}^{α} + (ρ S_{i}^{α} I_{i}^{α} - β E_{i}^{α} + | σ S_{i}^{α} I_{i}^{α} | Φ^{- 1} (α)) h \\ I_{i + 1}^{α} & = I_{i}^{α} + (β E_{i}^{α} - φ I_{i}^{α}) h \\ R_{i + 1}^{α} & = R_{i}^{α} + φ I_{i}^{α} h \\ S_{i + 1}^{α} & = N - E_{i + 1}^{1 - α} - I_{i + 1}^{1 - α} - R_{i + 1}^{1 - α} \end{matrix} .$ (16) where h is the step length, $E_{i + 1}^{1 - α}, I_{i + 1}^{1 - α}$ and $R_{i + 1}^{1 - α}$ satisfies the first three equations of the above model (16), Φ^-1 (α) is defined in Definition 2.

Step3: The α-path $S_{t}^{α}, E_{t}^{α}, I_{t}^{α}$ and $R_{t}^{α}$ are obtained.

Example 1. This example discusses the trajectories of S_t, E_t, I_t and R_t in the UDE model (10). The parameter values are set as follows. $\begin{matrix} N = 1, ρ = 0.35, β = 0.2, φ = 0.05, S_{0} = 0.9995, \\ E_{0} = 0, I_{0} = 0.0005, R_{0} = 0 . \end{matrix}$ (17)

The variation trends of the four states are shown in Fig. 1, and there are obvious differences in their paths, which can be explained as follows:

The percentage of S dropped rapidly in the first 50 hours and stabilized at 0 before the 75th hour, which meant that almost every state was informed of the rumor.

The proportion of E and I reached a peak of 0.21 and 0.47 before the 47th hour and the 56th hour respectively, indicating that only a small number of people paid attention to the rumor at the beginning, and then numbers of people quickly participated in the communication of the rumor, from which it could be seen that the rumor is characterized by explosive propagation. And then E goes down very quickly, and it stabilizes at the minimum of 0 until the 78th hour, which means that there is only I and R in the system.

I stabilized at a minimum of 0 before the 155th hour, indicating that people were no longer paying attention to the rumor.

The proportion of R increased rapidly from 0 to 0.97 after 24 hours, and then stabilized at a maximum of 1, indicating that the discussion about the rumor gradually calmed down.

Fig. 1

The path $S_{t}^{α},$ $E_{t}^{α},$ $I_{t}^{α}$ and $R_{t}^{α}$ with σ = 0.05, α = 0.9 . The parameters are taken as (17).

Example 2. In the ODE model (1) and the UDE model (10), the parameters are taken as (17). By Algorithm, for every α we can obtain the corresponding α-path $S_{t}^{α},$ $E_{t}^{α},$ $I_{t}^{α}$ and $R_{t}^{α}$ . Fig. 2 reveals the trajectories of them when α = 0.1, 0.2, . . . , 0.9 .

Fig. 2

Trajectories of α-path $S_{t}^{α},$ $E_{t}^{α},$ $I_{t}^{α}$ and $R_{t}^{α}$ with σ = 0.05.

With the change of α, the solution of the model (10) is always asymptotically stable near the rumor-free equilibrium (0, 0, 0, 1) . which means that the rumor dies out.

When σ is fixed and α take different values, the solution of the UDE (10) will always fluctuate around the ODE (1). In addition, the larger the α is, the faster the rumor model (10) tends to stabilized.

According to the numerical algorithm, when α reaches 0.5, the uncertain disturbance has no effect on the model, and then the UDE model (10) is equivalent to the ODE model (1).

From simulation studies it can be concluded that the uncertain differential equation can reveal the real process of rumor propagation, which is of more practical significance.

Example 3. In order to study the influence of the intensity of noise σ on the rumor model in depth, we set the same parameter values as those in (17) except for σ. The simulation results are shown in Fig. 3.

When the intensity σ is set to 0, the UDE model (10) is equivalent to the ODE model (1).

The lower the intensity σ is, the longer it takes for each state to reach the peak.

The lower the intensity σ is, the longer the rumor lasts. As the intensity σ increases, it takes less time for the rumor to die out.

Fig. 3

Trajectories of the ODE model (1) and UDE model (10) with α = 0.9 ; σ₁ = 0, σ₂ = 0.05, σ₃ = 0.2, σ₄ = 0.5 .

According to the simulation results, although we have intervened on the contact rate ρ in the system, it still takes a long time for the rumor to disappear, which indicates that when other coefficients are determined, our intervention on rumor propagation must be intensified. Therefore, we may as well assume that within a certain range, when the intensity σ is large enough, the rumor will be controlled and die out more quickly.

Example 4. This example discusses the dynamic changes of the UDE model (10) with the same parameters values as shown in (17) but with different ρ.

As is shown in Fig. 4, we learn that:

The larger the ρ is, the earlier the rumor spread, the greater the peak value of E is, and the shorter the duration of the rumor is.

A small increase in ρ leads to an increase in the peak value of E, which means ρ is crucial for rumor propagation. Therefore, in order to effectively suppress the widespread and rapid spread of rumors, we should reduce ρ in various ways.

Fig. 4

Trajectories of α-path $E_{t}^{α}$ with σ = 0.05, α = 0.9 .

7 Conclusions

The uncertain differential equation was applied to formulate the dynamics transmission of rumor, in which the uncertain noise was described by Liu process in this paper. The modeling process was introduced firstly, and then the uniqueness of the solution for the rumor model is proved. Besides, this paper presents a sufficient condition for the uncertain SEIR rumor model (10) being stable in mean. Finally, a numerical method of uncertain differential equation is applied to calculate the α-path of uncertain SEIR rumor model and some examples are given to compare the uncertain SEIR rumor model under different parameters. It can be seen from the simulation results that the proposed model can explain the trend of rumor propagation correctly and describe the rumor propagation accurately. The solution and uncertain parameter estimation for the uncertain SEIR rumor model are very important, yet not involved in this paper. These consideration will be taken in our future work.

Footnotes

Acknowledgments

This work was supported by The Natural Science Foundation of Xinjiang (Grants No. 2020D01C017), National Natural Science Foundation of China –Joint Key Program of Xinjiang (Grants No. U1703262) and National Natural Science Foundation of China (Grants No. 12061072).

Qianqian Liu: Theoretical analysis, Numerical simulation, Writing- original draft. Gang Shi and Yuhong Sheng: Investigation, Writing - review and editing, Funding acquision.

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