Asymptotic Stability of a Rumor Spreading Model with Three Time Delays and Two Saturation Functions

Abstract

Time delays and saturation effects are critical elements describing complex rumor spreading behaviors. In this article, a rumor spreading model with three time delays and two saturation functions is proposed. The basic properties of the model are reported. The structure of the rumor-endemic equilibria is deduced. A criterion for the global asymptotic stability of the rumor-free equilibrium is derived. In the presence of very small delays, a criterion for the local asymptotic stability of a rumor-endemic equilibrium is provided. The influence of the delays and the saturation effects on the dynamics of the model is made clear through simulation experiments. In particular, it is found that (a) extended time delays lead to slower change in the number of spreaders or stiflers and (b) lifted saturation coefficients lead to slower change in the number of spreaders or stiflers. This work helps to deepen the understanding of complex rumor spreading phenomenon and develop effective rumor-containing schemes.

1. Introduction

Rumors are interesting yet unofficial stories or messages. A rumor might be true or invented. In the real world, rumors can spread quickly from person to person [1]. The theoretical research of rumor spreading started from the seminal work by Allport and Postman [2]. The rapid popularization of social media brought numerous benefits, such as enhanced communication, easy access to information, boost to economic development, rich entertainment options, and improved education. Meanwhile, the rapid propagation of malicious rumors is likely to cause serious consequences, such as social panic, reputation damage, and economic loss. For example, the recent rumors that AI could predict lottery numbers with 100% accuracy have seriously mislead the public; many people chose to believe the rumors and wasted their money on buying lottery tickets according to the so-called AI-recommended numbers, resulting in financial loss.

In order to mitigate the negative impact of malicious rumors, it is critical to gain insight into the laws governing rumor spreading. Inspired by the established dynamics of infectious diseases [3], Daley and Kendall established the first rumor spreading model [4,5]. The modeling and study of rumor spreading have since attracted broad attention. In most existing rumor spreading models, some are captured by the simplest fully interconnected networks [6,7,8,9], some are captured by degree sequences of medium complexity [10,11,12,13], and the remaining are characterized by the most complex adjacency matrices [14,15,16,17].

Time delay is a phenomenon that exists widely. In the real world, delays of state transition have significant meanings in many aspects, such as communication and information processing, engineering and manufacturing, biology and medicine, and daily life. In the process of rumor spreading, there exist a spectrum of delays. In order to characterize the effect of time delay on rumor spreading, researchers have advised a multitude of rumor spreading models with delay mechanisms [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. To our knowledge, most existing delayed rumor spreading models are with single delay [18,19,20,21,22,23,24,28,29,32,33,34,35,36,37,38,39]. In practice, there may exist multiple delays. Let us just name a few as follows: (i) Due to contact with spreaders, ignorant persons may experience delayed spreading. (ii) Due to inherent influence such as bad memory or lost of interests, spreaders may experience delayed stifling. (iii) Due to contact with stiflers, spreaders may experience delayed stifling. Recently, some rumor spreading models with multiple delays have been suggested [25,26,27,30,31,40,41].

The incidence rate, which refers to the frequency at which a new event occurs, is a core concept in the field of infectious disease. When it comes to rumor spreading, there may exist two types of incidence rates as follows: the rumor infection rate, which refers to the frequency at which an ignorant person gets spreading due to contact with spreaders, and the rumor disinfection rate, which refers to the frequency at which a spreader experiences stifling due to contact with stiflers. Analogously to the conventional linear infection rate, the linear rumor infection rate (respectively, linear rumor disinfection rate) is much likely to overestimate the practical rate of rumor spreading (respectively, rumor stifling). Actually, the two rates commonly exhibit the saturation effect of “first speeding up then flattening out”. Consequently, a spectrum of rumor spreading models with saturation rumor infection rate or saturation rumor disinfection rate, where the saturation function is Holling type II [42,43,44], have been suggested [18,19,20,21,22,24,25,28,29,30,33,34,35,36].

In this article, a new rumor spreading model is proposed. In this model, three time delays are introduced, and two Holling type II saturation functions are taken into account. Ref. [40] is closely related to our work. In this article, the three mentioned time delays are introduced simultaneously, where the lengths of the delays are identical, and both the rumor infection rate and the rumor disinfection rate are linear. In practice, these assumptions greatly limit the applicability of this model. In our article, the three time delays are taken into account as well. However, the lengths of the delays may be distinct, and both the rumor infection rate and the rumor disinfection rate are the more reasonable Holling type II. Consequently, our model has wider applications. Ref. [41] is closely related to our work as well. In this article, the first and third time delays are considered, where the lengths of the two delays may be distinct, the rumor infection rate is Holling type II, and the the rumor disinfection rate is linear. In our article, all the three time delays are taken into account simultaneously, and both the rumor infection rate and the rumor disinfection rate are Holling type II. Consequently, our proposed model has more extensive applications.

The remainder of the article is organized in the following fashion: The new model is formulated in Section 2. The rumor-endemic equilibria are analyzed in Section 3. The global asymptotic stability of the rumor-free equilibrium is proved in Section 4. In the presence of very small time delays and by using a recently developed technique [40], the local asymptotic stability of a rumor-endemic equilibrium is shown in Section 5. The theoretical results are validated in Section 6. The influence of the time delays and the saturation effects on the dynamics of the model is examined. This work is summarized by Section 8.

2. Delayed Rumor Spreading Model

This section is devoted to establishing a delayed rumor spreading model.

2.1. A 3D Model

For a given online social network, the whole population in the world can be divided into two classes of individuals: the insiders, i.e., the individuals inside the network, and the outsiders, i.e., the individuals outside the network. Suppose the network is open to the world, i.e., all individuals in the world can freely enter or leave the network.

Suppose there is a rumor circulating in the network. In this context, assume (i) each insider is either ignorant of the rumor or spreading the rumor or stifling the rumor, and (ii) all outsiders are ignorant. For our purpose, introduce the following notations, terminologies, and assumptions.

($A_1$)

Let $S\left(t\right)$ (respectively, $I\left(t\right)$, $R\left(t\right)$) denote the number of ignorant insiders (respectively, spreaders, stiflers) at time t.

($A_2$)

The time unit is determined according to the actual need.

($A_3$)

Outsiders enter the network at constant rate $\mu >0$. Here, the unit of measurement of $\mu$ is the number of persons per unit time. In what follows, $\mu$ is referred to as the entrance rate.

($A_4$)

Each insider leaves the network at constant rate $\delta >0$. Here, the unit of measurement of $\delta$ is the number of persons per person per unit time. In what follows, $\delta$ is referred to as the exit rate.

($A_5$)

Owing to contact with spreaders, ignorant insiders start spreading at time t at rate ${\displaystyle \frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}$, where $\beta >0$, ${\sigma }_1>0$, and ${\tau }_1\ge 0$ are constants. In what follows, $\beta$ is referred to as the infection force, ${\sigma }_1$ as the first saturation coefficient, ${\tau }_1$ as the first delay.

($A_6$)

Owing to internal influence, spreaders experience stifling at time t at rate ${\gamma }_{1}I(t-{\tau }_2)$, where ${\gamma }_1>0$ and ${\tau }_2\ge 0$ are constants. In what follows, ${\gamma }_1$ is referred to as the first disinfection force, ${\tau }_2$ as the second delay.

($A_7$)

Owing to contact with stiflers, spreaders get stifling at time t at rate ${\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}$, where ${\gamma }_2>0$, ${\sigma }_2$, ${\tau }_3\ge 0$ are constants. In what follows, ${\gamma }_2$ is referred to as the second disinfection force, ${\sigma }_2$ as the second saturation coefficient, ${\tau }_3$ the third delay.

($A_8$)

Let $\tau =max({\tau }_1,{\tau }_2,{\tau }_3)$ denote the maximum delay.

($A_9$)

Let $S\left(\theta \right)={\varphi }_0\left(\theta \right)$, $I\left(\theta \right)={\varphi }_1\left(\theta \right)$, $R\left(\theta \right)={\varphi }_2\left(\theta \right)$, $-\tau \le \theta \le 0$ denote the initial condition. Here, ${\varphi }_0$, ${\varphi }_1$, and ${\varphi }_2$ are non-negative continuous functions.

($A_{10}$)

Due to the persistence of rumor spreading, $S\left(0\right)>0$, $I\left(0\right)>0$, and $R\left(0\right)>0$.

The above notations and explanations are listed in

Table 1. A list of notations and explanations.

Notation	Explanation
$S\left(t\right)$	number of ignorant insiders
$I\left(t\right)$	number of spreaders
$R\left(t\right)$	number of stiflers
$\mu$	entrance rate
$\delta$	exit rate
$\beta$	infection force
${\gamma }_1$	first disinfection force
${\gamma }_2$	second disinfection force
${\sigma }_1$	first saturation coefficient
${\sigma }_2$	second saturation coefficient
${\tau }_1$	first delay
${\tau }_2$	second delay
${\tau }_3$	third delay
$\tau$	maximum delay
${\varphi }_0,{\varphi }_1,{\varphi }_2$	initial condition

. On this basis, a 3D rumor spreading model is formulated below.

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =\mu -{\displaystyle \frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-\delta S\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dI\left(t\right)}{dt}}& ={\displaystyle \frac{\beta S(t-{\tau }_1)I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-{\gamma }_{1}I(t-{\tau }_2)-{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}-\delta I\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dR\left(t\right)}{dt}}& ={\gamma }_{1}I(t-{\tau }_2)+{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}-\delta R\left(t\right),\hfill \\ \hfill & t\ge 0,\hfill \\ \hfill S\left(\theta \right)& ={\varphi }_0\left(\theta \right),I\left(\theta \right)={\varphi }_1\left(\theta \right),R\left(\theta \right)={\varphi }_2\left(\theta \right),-\tau \le \theta \le 0.\hfill \end{array}\right.$$

(1)

2.2. The Reduced Model

Let $N\left(t\right)=S\left(t\right)+I\left(t\right)+R\left(t\right)$. Then, ${lim}_{t\to +\infty }N\left(t\right)={\displaystyle \frac{\mu }{\delta }}$. Consequently, the hyperplane $S+I+R={\displaystyle \frac{\mu }{\delta }}$ is attracting in the first octant. It follows from [45,46] that model (1) can be reduced to a 2D rumor spreading model, which is formulated below.

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dI\left(t\right)}{dt}}& =f_1(I\left(t\right),I(t-{\tau }_1),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_1),R(t-{\tau }_3))\hfill \\ \hfill & ={\displaystyle \frac{\beta \left[\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)\right]I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-{\gamma }_{1}I(t-{\tau }_2)\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}-\delta I\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dR\left(t\right)}{dt}}& =f_2(I\left(t\right),I(t-{\tau }_3),R\left(t\right),R(t-{\tau }_3))\hfill \\ \hfill & ={\gamma }_{1}I(t-{\tau }_2)+{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}-\delta R\left(t\right),\hfill \\ \hfill & t\ge 0,\hfill \\ \hfill I\left(\theta \right)& ={\varphi }_1\left(\theta \right),R\left(\theta \right)={\varphi }_2\left(\theta \right),-\tau \le \theta \le 0.\hfill \end{array}\right.$$

(2)

This model is referred to as the delayed SIR model. Henceforth, our attention will be concentrated on this model.

2.3. Basic Properties of the Delayed SIR Model

First, we consider the existence and uniqueness of the solution to model (2). It is noticed that the functions $f_1$ and $f_2$ are continuous and locally Lipschitz. By the existence and uniqueness theorem for the solution to the delayed functional differential equation [47], there is a finite time interval $[0,T]$ in which model (2) admits a unique solution.

Second, we consider the positivity of the solution to model (2).

Lemma 1.

Let $\left(I\right(t),R(t\left)\right)$, $t\ge 0$ be a solution to model (2). Then,

$$I\left(t\right)>0,R\left(t\right)>0,t\ge 0.$$

(3)

Proof of Lemma 1.

On the contrary, suppose there is $t_1>0$ such that (a) either $I\left(t_1\right)=0$ or $R\left(t_1\right)=0$, (b) $I\left(t\right)>0$, $R\left(t\right)>0$, $0\le t<t_1$. Without loss of generality, assume $I\left(t_1\right)=0$. Let

$$\begin{array}{cc}\hfill & g(I\left(t\right),I(t-{\tau }_1),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_1),R(t-{\tau }_3)))\hfill \\ \hfill & ={\displaystyle \frac{f_1(I\left(t\right),I(t-{\tau }_1),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_1),R(t-{\tau }_3))}{I\left(t\right)}},0\le t<t_1.\hfill \end{array}$$

(4)

It follows from model (2) that

$${\displaystyle \frac{dI\left(t\right)}{dt}}=I\left(t\right)g(I\left(t\right),I(t-{\tau }_1),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_1),R(t-{\tau }_3))),0\le t<t_1.$$

(5)

So,

$$\begin{array}{cc}\hfill I\left(t\right)& =I\left(0\right)exp\left({\int }_0^{t}g(I\left(s\right),I(s-{\tau }_1),I(s-{\tau }_2),I(s-{\tau }_3),R(s-{\tau }_1),R(s-{\tau }_3)))ds\right),\hfill \\ \hfill & 0\le t<t_1.\hfill \end{array}$$

(6)

As $I\left(t\right)$ is continuous on $[0,t_1)$, there is $ϵ>0$ so that

$$I\left(t\right)\ge I\left(0\right)exp(-ϵ),0\le t<t_1.$$

(7)

So,

$$I\left(t_1\right)=\underset{t\to t_1-0}{lim}I\left(t\right)\ge I\left(0\right)exp(-ϵ).$$

(8)

As $I\left(0\right)>0$, it follows that $I\left(t_1\right)>0$. This is a contradiction. Hence, $I\left(t\right)>0$ for $t>0$. Similarly, $R\left(t\right)>0$. □

Next, consider the boundedness of the solution to model (2). It is easily verified that $N\left(t\right)\le max(N\left(0\right),{\displaystyle \frac{\mu }{\delta }})$, $t\ge 0$. In view of Lemma 1, it follows that $0\le I\left(t\right),R\left(t\right)\le {\displaystyle \frac{\mu }{\delta }}$, $t\ge 0$.

Finally, examine the extendability of the solution to model (2). In view of the boundedness of the solution, it follows from [47] that this solution is extendable to $[0,+\infty )$.

2.4. Basic Reproduction Number for the Major Model

Let

$$F(I(t-{\tau }_1),R(t-{\tau }_1))={\displaystyle \frac{\beta \left[\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)\right]I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}},$$

(9)

$$V(I\left(t\right),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_3))={\gamma }_{1}I(t-{\tau }_2)+{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}+\delta I\left(t\right).$$

(10)

Then,

$${\displaystyle \frac{\partial F(I(t-{\tau }_1),R(t-{\tau }_1))}{\partial I(t-{\tau }_1)}}{|}_{(0,0)}={\displaystyle \frac{\beta \mu }{\delta }},$$

(11)

$${\displaystyle \frac{\partial V(I\left(t\right),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_3))}{\partial I\left(t\right)}}{|}_{(0,0,0,0)}=\delta ,$$

(12)

$${\displaystyle \frac{\partial V(I\left(t\right),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_3))}{\partial I(t-{\tau }_2)}}{|}_{(0,0,0,0)}={\gamma }_1,$$

(13)

$${\displaystyle \frac{\partial V(I\left(t\right),I(t-{\tau }_2),I(t-{\tau }_3),R(t-{\tau }_3))}{\partial I(t-{\tau }_3)}}{|}_{(0,0,0,0)}=0.$$

(14)

By applying the next-generation matrix method [48,49], the basic reproduction number for model (2) equals

$$R_0={\displaystyle \frac{\beta \mu }{\delta ({\gamma }_1+\delta )}}.$$

(15)

3. Rumor-Endemic Equilibria

At any case, model (2) admits a unique rumor-free equilibrium, $E_0=(0,0)$. Now, consider the rumor-endemic equilibria. For this purpose, a set of auxiliary quantities are introduced below.

$$M_1={\displaystyle \frac{\beta \mu }{\delta }}-({\gamma }_1+\delta )=({\gamma }_1+\delta )(R_0-1),$$

(16)

$$M_2=\beta +({\gamma }_1+\delta ){\sigma }_1+{\displaystyle \frac{\beta {\gamma }_1}{\delta }}>0,$$

(17)

$$M_3=({\gamma }_1+\delta ){\sigma }_2+{\gamma }_2-{\displaystyle \frac{\beta \mu {\sigma }_2}{\delta }}={\gamma }_2+({\gamma }_1+\delta ){\sigma }_2(1-R_0),$$

(18)

$$M_4=\beta {\sigma }_2+({\gamma }_1+\delta ){\sigma }_1{\sigma }_2+{\gamma }_2{\sigma }_1+{\displaystyle \frac{\beta ({\gamma }_1{\sigma }_2+{\gamma }_2)}{\delta }}>0,$$

(19)

$$M_5={\sigma }_1{\sigma }_2{\delta }^2+(\beta {\sigma }_2+{\gamma }_1{\sigma }_1{\sigma }_2+{\gamma }_2{\sigma }_1)\delta +\beta ({\gamma }_1{\sigma }_2+{\gamma }_2)>0,$$

(20)

$$a_1={\displaystyle \frac{({\sigma }_1-2{\sigma }_2){\delta }^3+({\gamma }_1{\sigma }_1-2{\gamma }_1{\sigma }_2-2{\gamma }_2+\beta ){\delta }^2+\beta (2\mu {\sigma }_2+{\gamma }_1)\delta +\beta \mu ({\gamma }_1{\sigma }_2+{\gamma }_2)}{-\delta M_5}},$$

(21)

$$\begin{array}{cc}\hfill a_2& =-{\displaystyle \frac{\delta }{(\beta +\delta {\sigma }_1)M_5}}\times [({\sigma }_2-2{\sigma }_1){\delta }^4+({\gamma }_1{\sigma }_2-2{\gamma }_1{\sigma }_1+{\gamma }_2-2\beta ){\delta }^3\hfill \\ \hfill & +\beta (\mu {\sigma }_1-2\mu {\sigma }_2-2{\gamma }_1){\delta }^2+\beta \mu (\beta -{\gamma }_1{\sigma }_2-{\gamma }_2)\delta +{\beta }^2{\mu }^2{\sigma }_2],\hfill \end{array}$$

(22)

$$a_3={\displaystyle \frac{\delta ({\gamma }_1+\delta )(R_0-1)}{(\beta +\delta {\sigma }_1)M_5}}.$$

(23)

Theorem 1.

$E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2) if and only if the following conditions hold.

(C1) 

$I^{*}>0$.

(C2) 

$I^{*3}+a_1{I}^{*2}+a_2{I}^{*}+a_3=0$.

(C3) 

$R^{*}={\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}>0$.

Proof of Theorem 1.

Necessity. Assume $E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2). Then, $I^{*}>0$, $R^{*}>0$,

$${\displaystyle \frac{\beta \left(\frac{\mu }{\delta }-I^{*}-R^{*}\right)}{1+{\sigma }_1{I}^{*}}}-{\gamma }_1-{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}-\delta =0,$$

(24)

and

$${\gamma }_1{I}^{*}+{\displaystyle \frac{{\gamma }_2{I}^{*}{R}^{*}}{1+{\sigma }_2{R}^{*}}}-\delta R^{*}=0.$$

(25)

Equation (24) yields

$$R^{*2}={\displaystyle \frac{{\gamma }_1{I}^{*}-\delta R^{*}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}{R}^{*}}{\delta {\sigma }_2}}.$$

(26)

Equation (23) leads to

$$\begin{array}{cc}\hfill R^{*2}& ={\displaystyle \frac{1}{\beta {\sigma }_2}}\{\left[{\displaystyle \frac{\beta \mu }{\delta }}-({\gamma }_1+\delta )\right]-[\beta +({\gamma }_1+\delta ){\sigma }_1]I^{*}-\left[\beta +({\gamma }_1+\delta ){\sigma }_2+{\gamma }_2-{\displaystyle \frac{\beta \mu {\sigma }_2}{\delta }}\right]R^{*}\hfill \\ \hfill & -[\beta {\sigma }_2+({\gamma }_1+\delta ){\sigma }_1{\sigma }_2+{\gamma }_2{\sigma }_1]I^{*}{R}^{*}\}.\hfill \end{array}$$

(27)

Combining Equations (25) and (26) yields

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\gamma }_1{I}^{*}-\delta R^{*}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}{R}^{*}}{\delta {\sigma }_2}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\beta {\sigma }_2}}\{\left[{\displaystyle \frac{\beta \mu }{\delta }}-({\gamma }_1+\delta )\right]-[\beta +({\gamma }_1+\delta ){\sigma }_1]I^{*}-\left[\beta +({\gamma }_1+\delta ){\sigma }_2+{\gamma }_2-{\displaystyle \frac{\beta \mu {\sigma }_2}{\delta }}\right]R^{*}\hfill \\ \hfill & -[\beta {\sigma }_2+({\gamma }_1+\delta ){\sigma }_1{\sigma }_2+{\gamma }_2{\sigma }_1]I^{*}{R}^{*}\}.\hfill \end{array}$$

(28)

Solving for $R^{*}$ yields

$$\begin{array}{cc}\hfill R^{*}& ={\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}={\displaystyle \frac{({\gamma }_1+\delta )(R_0-1)-M_2{I}^{*}}{{\gamma }_2+({\gamma }_1+\delta ){\sigma }_2(1-R_0)+M_4{I}^{*}}}>0.\hfill \end{array}$$

(29)

Substituting Equation (28) into Equation (24) yields

$${\gamma }_1{I}^{*}-\delta {\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}{\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}-\delta {\sigma }_2{\displaystyle \frac{{(M_1-M_2{I}^{*})}^2}{{(M_3+M_4{I}^{*})}^2}}=0.$$

(30)

So,

$$\begin{array}{cc}\hfill & {\gamma }_1{I}^{*}{(M_3+M_4{I}^{*})}^2-(M_1-M_2{I}^{*})(M_3+M_4{I}^{*})\hfill \\ \hfill & +({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}(M_1-M_2{I}^{*})(M_3+M_4{I}^{*})-\delta {\sigma }_2({(M_1-M_2{I}^{*})}^2=0.\hfill \end{array}$$

(31)

Combining like terms yields the monic cubic equation

$$I^{*3}+a_1{I}^{*2}+a_2{I}^{*}+a_3=0.$$

(32)

The necessity is proved.

Sufficiency. Suppose conditions (C1)–(C3) hold. Then,

$$\begin{array}{cc}\hfill & {\gamma }_1{I}^{*}+{\displaystyle \frac{{\gamma }_2{I}^{*}{R}^{*}}{1+{\sigma }_2{R}^{*}}}-\delta R^{*}\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_1{I}^{*}-\delta R^{*}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}{R}^{*}-\delta {\sigma }_2{R}^{*2}}{1+{\sigma }_2{R}^{*}}}\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_1{I}^{*}-\delta \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}\frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}-\delta {\sigma }_2\frac{{(M_1-M_2{I}^{*})}^2}{{(M_3+M_4{I}^{*})}^2}}{1+{\sigma }_2{R}^{*}}}\hfill \\ \hfill & ={\displaystyle \frac{M_4[{\gamma }_1{M}_4-({\gamma }_1{\sigma }_2+{\gamma }_2)M_2](I^{*3}+a_1{I}^{*2}+a_2{I}^{*}+a_3)}{(1+{\sigma }_2{R}^{*}){(M_3+M_4{I}^{*})}^2}}=0.\hfill \end{array}$$

(33)

Additionally,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\beta \left(\frac{\mu }{\delta }-I^{*}-R^{*}\right)}{1+{\sigma }_1{I}^{*}}}-{\gamma }_1-{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}-\delta \hfill \\ \hfill & ={\displaystyle \frac{1}{(1+{\sigma }_1{I}^{*})(1+{\sigma }_2{R}^{*})}}\{\left[{\displaystyle \frac{\beta \mu }{\delta }}-({\gamma }_1+\delta )\right]-[\beta +({\gamma }_1+\delta ){\sigma }_1]I^{*}\hfill \\ \hfill & -\left[\beta +({\gamma }_1+\delta ){\sigma }_2+{\gamma }_2-{\displaystyle \frac{\beta \mu {\sigma }_2}{\delta }}\right]R^{*}-[\beta {\sigma }_2+({\gamma }_1+\delta ){\sigma }_1{\sigma }_2+{\gamma }_2{\sigma }_1]I^{*}{R}^{*}\hfill \\ \hfill & -\beta {\sigma }_2{R}^{*2}\}.\hfill \end{array}$$

(34)

Condition (C3) (i.e., Equation (28)) implies Equation (27). This plus Equation (32) implies

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\beta \left(\frac{\mu }{\delta }-I^{*}-R^{*}\right)}{1+{\sigma }_1{I}^{*}}}-{\gamma }_1-{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}-\delta \hfill \\ \hfill & ={\displaystyle \frac{\beta [{\gamma }_1{I}^{*}-\delta R^{*}+({\gamma }_1{\sigma }_2+{\gamma }_2)I^{*}{R}^{*}-\delta {\sigma }_2{R}^{*2}]}{\delta (1+{\sigma }_1{I}^{*})(1+{\sigma }_2{R}^{*})}}=0.\hfill \end{array}$$

(35)

Hence, $(I^{*},R^{*})$ is a rumor-endemic equilibrium. □

Remark 1.

It is seen from Theorem 1 that the I-component of the rumor-endemic equilibrium is determined by the cubic Equation (C2), which in turn is determined by the following three quantities: $a_1$, $a_2$, and $a_3$, which are in turn determined by the auxiliary quantity $M_5$.

Remark 2.

It is seen from Theorem 1 that the P-component of the rumor-endemic equilibrium is determined by Equation (C3), which in turn is determined by the I-component of the rumor-endemic equilibrium and the following four quantities: $M_1$, $M_2$, $M_3$, $M_4$.

Theorem 1 assumes five lemmas, which are given below.

Lemma 2.

Suppose model (2) admits a rumor-endemic equilibrium. Then, $R_0>1$.

Proof of Lemma 2.

Let $E^{*}=(I^{*},R^{*})$ be a rumor-endemic equilibrium. It follows from Theorem 1 that

$$R^{*}={\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}={\displaystyle \frac{({\gamma }_1+\delta )(R_0-1)-M_2{I}^{*}}{{\gamma }_2+({\gamma }_1+\delta ){\sigma }_2(1-R_0)+M_4{I}^{*}}}.$$

(36)

On the contrary, suppose $R_0\le 1$. Then, $R^{*}<0$. This is a contradiction. Hence, $R_0>1$. □

This lemma shows that there is no backward bifurcation.

Lemma 3.

Suppose model (2) admits a rumor-endemic equilibrium. Then, $a_3>0$.

Proof of Lemma 3.

The assertion follows from Equation (22) and Lemma 2. □

Lemma 4.

Suppose $E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2). Suppose

$$R_0\le 1+{\displaystyle \frac{{\gamma }_2}{({\gamma }_1+\delta ){\sigma }_1}}.$$

(37)

Then, $I^{*}<{\displaystyle \frac{M_1}{M_2}}$.

Proof of Lemma 4.

It follows from Lemma 1 that $R_0>1$. Hence, $M_1>0$. Additionally, it follows from Equation (28) that $M_3>0$. The assertion follows. □

Lemma 5.

Suppose $E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2). Suppose

$$R_0>1+{\displaystyle \frac{{\gamma }_2}{({\gamma }_1+\delta ){\sigma }_1}}.$$

(38)

Then, $-{\displaystyle \frac{M_3}{M_4}}<I^{*}<{\displaystyle \frac{M_1}{M_2}}$.

Proof of Lemma 5.

It follows from Lemma 2 that $R_0>1$. So, $M_1>0$. Additionally, it follows from Equation (35) that $M_3<0$. Hence, either

$$-{\displaystyle \frac{M_3}{M_4}}<I^{*}<{\displaystyle \frac{M_1}{M_2}}$$

(39)

or

$${\displaystyle \frac{M_1}{M_2}}<I^{*}<-{\displaystyle \frac{M_3}{M_4}}.$$

(40)

Notice that

$$M_1{M}_4+M_2{M}_3={\displaystyle \frac{\beta {\gamma }_2\mu (\beta +\delta {\sigma }_1)}{{\delta }^2}}>0.$$

(41)

It follows that

$$-{\displaystyle \frac{M_3}{M_4}}<I^{*}<{\displaystyle \frac{M_1}{M_2}}.$$

(42)

□

Lemma 6.

Suppose $E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2). Then, $max(0,-{\displaystyle \frac{M_3}{M_4}})<I^{*}<{\displaystyle \frac{M_1}{M_2}}$.

Proof of Lemma 6.

The assertion follows by combining Lemma 4 with Lemma 5. □

Theorem 2.

$E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium of model (2) if and only if the following conditions hold.

(C1) 

$R_0>1$.

(C2) 

$max(0,-{\displaystyle \frac{M_3}{M_4}})<I^{*}<{\displaystyle \frac{M_1}{M_2}}$.

(C3) 

$I^{*3}+a_1{I}^{*2}+a_2{I}^{*}+a_3=0$.

(C4) 

$R^{*}={\displaystyle \frac{M_1-M_2{I}^{*}}{M_3+M_4{I}^{*}}}$.

Proof of Theorem 2.

The assertion follows from Theorem 1, Lemma 2, and Lemma 6. □

Lemma 7.

Let

$$A={\displaystyle \frac{a_1{a}_2}{6}}-{\displaystyle \frac{a_1^3}{27}}-{\displaystyle \frac{a_3}{2}},B={\displaystyle \frac{a_2}{3}}-{\displaystyle \frac{a_1^2}{9}},\Delta =A^2+B^3.$$

(43)

Then, the following assertions hold.

(A1) 

The cubic equation

$$x^3+a_1{x}^2+a_{2}x+a_3=0$$

(44)

admits the following three roots:

$$x_1=-{\displaystyle \frac{a_1}{3}}+\sqrt[3]{A+\sqrt{\Delta }}+\sqrt[3]{A-\sqrt{\Delta }}.$$

(45)

$$x_2=-{\displaystyle \frac{a_1}{3}}+{\displaystyle \frac{-1-i\sqrt{3}}{2}}\sqrt[3]{A+\sqrt{\Delta }}+{\displaystyle \frac{-1+i\sqrt{3}}{2}}\sqrt[3]{A-\sqrt{\Delta }}.$$

(46)

$$x_3=-{\displaystyle \frac{a_1}{3}}+{\displaystyle \frac{-1+i\sqrt{3}}{2}}\sqrt[3]{A+\sqrt{\Delta }}+{\displaystyle \frac{-1-i\sqrt{3}}{2}}\sqrt[3]{A-\sqrt{\Delta }}.$$

(47)

(A2) 

If $\Delta >0$, then Equation (43) admits a real root, $x_1$, and a pair of conjugate complex roots.

(A3) 

If $\Delta =0$, $A\ne 0$, then Equation (43) admits a simple real root, $x_1=-{\displaystyle \frac{a_1}{3}}+2\sqrt[3]{A}$, and a double real root, $x_2=x_3=-{\displaystyle \frac{a_1}{3}}-\sqrt[3]{A}$.

(A4) 

If $\Delta =0$, $A=0$, then Equation (43) admits a triple real root, $x_1=x_2=x_3=-{\displaystyle \frac{a_1}{3}}$.

(A5) 

If $\Delta <0$, then Equation (43) admits three simple real roots, $x_1$, $x_2$, and $x_3$.

Proof of Lemma 7.

The assertions follow from Cardano’s formula. □

Theorem 3.

For $k=1,2,3$, let

$$y_k={\displaystyle \frac{M_1-M_2{x}_k}{M_3+M_4{x}_k}}.$$

(48)

If $max(0,-{\displaystyle \frac{M_3}{M_4}})<x_k<{\displaystyle \frac{M_1}{M_2}}$, then $(x_k,y_k)$ is a rumor-endemic equilibrium.

Proof of Theorem 3.

The assertion follows from Theorem 2 and Lemma 7. □

By Theorem 3, all the rumor-endemic equilibria of model (2) can be determined.

Theorem 4.

The following assertions hold.

(A1) 

If $\Delta >0$, then model (2) admits no rumor-endemic equilibrium.

(A2) 

If $\Delta =0$, $A\ne 0$, then model (2) admits no more than two rumor-endemic equilibria.

(A3) 

If $\Delta =0$, $A=0$, then model (2) admits no more than one rumor-endemic equilibrium.

(A4) 

If $\Delta <0$, then model (2) admits no more than two rumor-endemic equilibria.

Proof of Theorem 4.

In the case where $\Delta >0$, it follows from Lemma 3 and Vieta’s theorem that Equation (43) admits no positive root. Hence, model (2) admits no rumor-endemic equilibrium. The remaining assertions follow from Assertions (A3)–(A5) of Lemma 7 and Vieta’s theorem. □

Example 1.

Consider model (2) with $\mu =50$, $\delta =2.3$, $\beta =0.7$, ${\gamma }_1=0.16$, ${\gamma }_2=1.6$, ${\sigma }_1=0.32$, and ${\sigma }_2=0.75$. Then, $\Delta =-3.73\times {10}^5<0$. Hence, Equation (45) admits three simple real roots as follows: −9.011, 0.021, and 22.404. In this case, $-{\displaystyle \frac{M_3}{M_4}}=3.705$ and ${\displaystyle \frac{M_1}{M_2}}=8.306$. By Theorem 2, model (4) has no rumor-endemic equilibrium.

Example 2.

Consider model (2) with $\mu =50$, $\delta =1$, $\beta =0.1$, ${\gamma }_1=0.05$, ${\gamma }_2=1$, ${\sigma }_1=1$, ${\sigma }_2=0.1$. Then, $\Delta =-6.512<0$. Hence, Equation (45) admits three simple real roots: −0.835, 0.837, 4.051. In this case, $-{\displaystyle \frac{M_3}{M_4}}=-0.498$, ${\displaystyle \frac{M_1}{M_2}}=3.420$. By Theorem 2, model (4) has the sole rumor-endemic equilibrium: $(0.873,1.765)$.

Example 3.

Consider model (2) with $\mu =50$, $\delta =1.1$, $\beta =0.1$, ${\gamma }_1=0.07$, ${\gamma }_2=3.62$, ${\sigma }_1=3.1$, and ${\sigma }_2=0.1$. Then, $\Delta =-1.866\times {10}^{-4}<0$. Hence, Equation (45) admits three simple real roots as follows: −0.217, 0.515, and 0.722. In this case, $-{\displaystyle \frac{M_3}{M_4}}=-0.275$ and ${\displaystyle \frac{M_1}{M_2}}=0.904$. By Theorem 2, model (4) has a pair of rumor-endemic equilibria as follows: $(0.515,0.154)$ and $(0.722,0.057)$.

4. Dynamics of the Rumor-Free Equilibrium

This section is devoted to the asymptotic stability of the rumor-free equilibrium of model (2).

4.1. Local Asymptotic Stability

It follows from [47] that the linearized system of model (2) at $E_0=(0,0)$ is

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dx\left(t\right)}{dt}}& =-\delta x\left(t\right)+{\displaystyle \frac{\beta \mu }{\delta }}x(t-{\tau }_1)-{\gamma }_{1}x(t-{\tau }_2),t\ge 0,\hfill \\ \hfill {\displaystyle \frac{dy\left(t\right)}{dt}}& ={\gamma }_{1}x(t-{\tau }_2)-\delta y\left(t\right),t\ge 0,\hfill \\ \hfill x\left(\theta \right)& ={\varphi }_1\left(\theta \right),y\left(\theta \right)={\varphi }_2\left(\theta \right),-\tau \le \theta \le 0.\hfill \end{array}\right.$$

(49)

The associated characteristic equation is

$$(\lambda +\delta )\left(\lambda +\delta -{\displaystyle \frac{\beta \mu }{\delta }}{e}^{-\lambda {\tau }_1}+{\gamma }_1{e}^{-\lambda {\tau }_2}\right)=0.$$

(50)

Let

$$P\left(\lambda \right)=\lambda +\delta -{\displaystyle \frac{\beta \mu }{\delta }}{e}^{-\lambda {\tau }_1}+{\gamma }_1{e}^{-\lambda {\tau }_2}.$$

(51)

Theorem 5.

The following assertions hold.

(A1) 

Suppose ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1\le \delta$. Then, $E_0$ is locally asymptotically stable.

(A2) 

Suppose $R_0>1$. Then, $E_0$ is unstable.

Proof of Theorem 5.

Equation (49) admits the negative root ${\lambda }_1=-\delta$.

In the case where ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1\le \delta$, it is easily verified that $R_0<1$ and $P\left(0\right)=({\gamma }_1+\delta )(1-R_0)>0$. Furthermore, $\lambda >0$ implies

$$P\left(\lambda \right)=\lambda +\delta -{\displaystyle \frac{\beta \mu }{\delta }}{e}^{-\lambda {\tau }_1}+{\gamma }_1{e}^{-\lambda {\tau }_2}\ge \lambda +\delta -{\displaystyle \frac{\beta \mu }{\delta }}>\delta -{\displaystyle \frac{\beta \mu }{\delta }}\ge 0.$$

(52)

So, $P\left(\lambda \right)$ has no non-negative zero.

Now, suppose $P\left(\lambda \right)$ has a zero with a non-negative real part. It follows from [50] that $P\left(\lambda \right)$ has a pair of conjugate purely imaginary zeros, $\pm i\omega$ ($\omega >0$). Hence,

$$P\left(i\omega \right)=i\omega +\delta -{\displaystyle \frac{\beta \mu }{\delta }}{e}^{-i\omega {\tau }_1}+{\gamma }_1{e}^{-i\omega {\tau }_2}=0.$$

(53)

Now, let us use the technique developed in [51]. Separating the real and imaginary parts yields

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{\beta \mu }{\delta }}cos\left(\omega {\tau }_1\right)-{\gamma }_{1}cos\left(\omega {\tau }_2\right)& =\delta ,\hfill \\ \hfill {\displaystyle \frac{\beta \mu }{\delta }}sin\left(\omega {\tau }_1\right)-{\gamma }_{1}sin\left(\omega {\tau }_2\right)& =-\omega .\hfill \end{array}\right.$$

(54)

Squaring the two equations and simplifying, it follows that

$${\displaystyle \frac{{\beta }^2{\mu }^2}{{\delta }^2}}+{\gamma }_1^2-{\displaystyle \frac{2\beta {\gamma }_1\mu }{\delta }}cos\left(\omega ({\tau }_1-{\tau }_2)\right)={\delta }^2+{\omega }^2.$$

(55)

So,

$${\displaystyle \frac{{\beta }^2{\mu }^2}{{\delta }^2}}+{\gamma }_1^2-{\delta }^2-{\displaystyle \frac{2\beta {\gamma }_1\mu }{\delta }}cos\left(\omega ({\tau }_1-{\tau }_2)\right)={\omega }^2>0.$$

(56)

As

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\beta }^2{\mu }^2}{{\delta }^2}}+{\gamma }_1^2-{\delta }^2-{\displaystyle \frac{2\beta {\gamma }_1\mu }{\delta }}cos\left(\omega ({\tau }_1-{\tau }_2)\right)\hfill \\ \hfill & \le {\displaystyle \frac{{\beta }^2{\mu }^2}{{\delta }^2}}+{\gamma }_1^2-{\delta }^2+{\displaystyle \frac{2\beta {\gamma }_1\mu }{\delta }}\hfill \\ \hfill & ={({\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1)}^2-{\delta }^2,\hfill \end{array}$$

(57)

it follows that ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1>\delta$. This is a contradiction.

According to the Lyapunov stability theorem [52], $E_0$ is locally asymptotically stable.

In the case where $R_0>1$, it follows that $P\left(0\right)=({\gamma }_1+\delta )(1-R_0)<0$ and $P(+\infty )=+\infty$. So, $P\left(\lambda \right)$ has a positive zero. Hence, $E_0$ is unstable. □

4.2. Global Asymptotic Stability

Theorem 6.

Suppose $\beta \mu \le {\delta }^2$. Then, the rumor-free equilibrium is globally attracting.

Proof of Theorem 6.

Let

$$U\left(t\right)=I\left(t\right)+R\left(t\right)+\delta {\int }_{t-{\tau }_1}^{t}I\left(s\right)ds.$$

(58)

Then, $U\left(t\right)$ is positive definite. It follows that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{dU\left(t\right)}{dt}}={\displaystyle \frac{dI\left(t\right)}{dt}}+{\displaystyle \frac{dR\left(t\right)}{dt}}+\delta I\left(t\right)-\delta I(t-{\tau }_1)\hfill \\ \hfill & ={\displaystyle \frac{\beta \left[\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)\right]I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-{\gamma }_{1}I(t-{\tau }_2)-{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}\hfill \\ \hfill & -\delta I\left(t\right)+{\gamma }_{1}I(t-{\tau }_2)+{\displaystyle \frac{{\gamma }_{2}I(t-{\tau }_3)R(t-{\tau }_3)}{1+{\sigma }_{2}R(t-{\tau }_3)}}-\delta R\left(t\right)+\delta I\left(t\right)-\delta I(t-{\tau }_1)\hfill \\ \hfill & ={\displaystyle \frac{\beta \left[\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)\right]I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-\delta I(t-{\tau }_1)-\delta R\left(t\right)\hfill \\ \hfill & \le {\displaystyle \frac{\beta \left[\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)\right]I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}-\delta I(t-{\tau }_1)\hfill \\ \hfill & ={\displaystyle \frac{\beta \left\{\frac{\mu }{\delta }-I(t-{\tau }_1)-R(t-{\tau }_1)-\delta [1+{\sigma }_{1}I(t-{\tau }_1)]\right\}I(t-{\tau }_1)}{1+{\sigma }_{1}I(t-{\tau }_1)}}\hfill \\ \hfill & \le {\displaystyle \frac{\left(\frac{\beta \mu }{\delta }-\delta \right)I\left(t\right)}{1+{\sigma }_{1}I(t-{\tau }_1)}}\le 0.\hfill \end{array}$$

(59)

Moreover, ${\displaystyle \frac{dU\left(t\right)}{dt}}=0$ if $I\left(t\right)=R\left(t\right)=0$, and vice versa. The global attractivity of $E_0$ follows from LaSalle’s invariance principle [52]. □

Theorem 7.

Suppose ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1\le \delta$. Then, the rumor-free equilibrium is globally asymptotically stable.

Proof of Theorem 7.

The claim follows by combining Theorems 5 and 6. □

5. Dynamics of a Rumor-Endemic Equilibrium

It follows from Theorem 3 that all rumor-endemic equilibria of model (2) can be determined. This section is dedicated to the asymptotic stability of a rumor-endemic equilibrium.

Suppose $E^{*}=(I^{*},R^{*})$ is a rumor-endemic equilibrium. It follows from [47] that the linearized system of model (2) at $E^{*}$ is

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dx\left(t\right)}{dt}}& =-\delta x\left(t\right)+{\displaystyle \frac{\beta (\frac{\mu }{\delta }-2I^{*}-R^{*}-{\sigma }_1{I}^{*2})}{{(1+{\sigma }_1{I}^{*})}^2}}x(t-{\tau }_1)-{\gamma }_{1}x(t-{\tau }_2)\hfill \\ \hfill & -{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}x(t-{\tau }_3)-{\displaystyle \frac{\beta I^{*}}{1+{\sigma }_1{I}^{*}}}y(t-{\tau }_1)-{\displaystyle \frac{{\gamma }_2{I}^{*}}{{(1+{\sigma }_2{R}^{*})}^2}}y(t-{\tau }_3),\hfill \\ \hfill {\displaystyle \frac{dy\left(t\right)}{dt}}& ={\gamma }_{1}x(t-{\tau }_2)+{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}x(t-{\tau }_3)-\delta y\left(t\right)+{\displaystyle \frac{{\gamma }_2{I}^{*}}{{(1+{\sigma }_2{R}^{*})}^2}}y(t-{\tau }_3),\hfill \\ \hfill & t\ge 0.\hfill \end{array}\right.$$

(60)

The associated characteristic equation is

$$\begin{array}{cc}\hfill Q\left(\lambda \right)& =\left(\lambda +\delta -{\displaystyle \frac{\beta (\frac{\mu }{\delta }-2I^{*}-R^{*}-{\sigma }_1{I}^{*2})}{{(1+{\sigma }_1{I}^{*})}^2}}{e}^{-\lambda {\tau }_1}+{\gamma }_1{e}^{-\lambda {\tau }_2}+{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}{e}^{-\lambda {\tau }_3}\right)\hfill \\ \hfill & \times \left(\lambda +\delta -{\displaystyle \frac{{\gamma }_2{I}^{*}}{{(1+{\sigma }_2{R}^{*})}^2}}{e}^{-\lambda {\tau }_3}\right)\hfill \\ \hfill & +\left({\displaystyle \frac{\beta I^{*}}{1+{\sigma }_1{I}^{*}}}{e}^{-\lambda {\tau }_1}+{\displaystyle \frac{{\gamma }_2{I}^{*}}{{(1+{\sigma }_2{R}^{*})}^2}}{e}^{-\lambda {\tau }_3}\right)\left({\gamma }_1{e}^{-\lambda {\tau }_2}+{\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}{e}^{-\lambda {\tau }_3}\right)\hfill \\ \hfill & ={\lambda }^2+b_1\lambda +b_2+(b_3\lambda +b_4)e^{-\lambda {\tau }_1}+(b_5\lambda +b_6)e^{-\lambda {\tau }_2}+(b_7\lambda +b_8)e^{-\lambda {\tau }_3}\hfill \\ \hfill & +b_9{e}^{-\lambda ({\tau }_1+{\tau }_2)}+b_{10}{e}^{-\lambda ({\tau }_1+{\tau }_3)}=0,\hfill \end{array}$$

(61)

where

$$b_1=2\delta >0,b_2={\delta }^2>0,$$

(62)

$$b_3={\displaystyle \frac{-\beta (\frac{\mu }{\delta }-2I^{*}-R^{*}-{\sigma }_1{I}^{*2})}{{(1+{\sigma }_1{I}^{*})}^2}},$$

(63)

$$b_4={\displaystyle \frac{-\beta \delta (\frac{\mu }{\delta }-2I^{*}-R^{*}-{\sigma }_1{I}^{*2})}{{(1+{\sigma }_1{I}^{*})}^2}}=\delta b_3,$$

(64)

$$b_5={\gamma }_1>0,b_6={\gamma }_1\delta >0,$$

(65)

$$b_7={\displaystyle \frac{{\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}-{\displaystyle \frac{{\gamma }_2{I}^{*}}{{(1+{\sigma }_2{R}^{*})}^2}},$$

(66)

$$b_8={\displaystyle \frac{-{\gamma }_2\delta I^{*}}{{(1+{\sigma }_2{R}^{*})}^2}}+{\displaystyle \frac{\delta {\gamma }_2{R}^{*}}{1+{\sigma }_2{R}^{*}}}=-\delta b_7,$$

(67)

$$b_9={\displaystyle \frac{\beta {\gamma }_1{I}^{*}}{1+{\sigma }_1{I}^{*}}}>0,$$

(68)

$$b_{10}={\displaystyle \frac{\beta {\gamma }_2{I}^{*}(\frac{\mu }{\delta }-2I^{*}-R^{*}-{\sigma }_1{I}^{*2})}{{(1+{\sigma }_1{I}^{*})}^2{(1+{\sigma }_2{R}^{*})}^2}}+{\displaystyle \frac{\beta {\gamma }_2{I}^{*}{R}^{*}}{(1+{\sigma }_1{I}^{*})(1+{\sigma }_2{R}^{*})}},$$

(69)

Theorem 8.

Consider model (2) with very small time delays. Let

$$s_1=b_1^2-2b_2-{(b_3+b_5+b_7)}^2,$$

(70)

$$s_2=b_2^2-{(b_4+b_6+b_8+b_9+b_{10})}^2,$$

(71)

$${\omega }_0=\sqrt{{\displaystyle \frac{-s_1\pm \sqrt{s_1^2-4s_2}}{2}}}.$$

(72)

Suppose the following conditions hold.

(C1) 

$b_1+min(b_3,0)+min(b_7,0)\ge 0$.

(C2) 

$b_2+min(b_4,0)+min(b_8,0)+min(b_{10},0)>0$.

(C3) 

$\left|Q\right(i{\omega }_0\left)\right|$ is not very small.

Then, $E^{*}$ is locally asymptotically stable.

Proof of Theorem 8.

For $\lambda \ge 0$, it follows that

$$\begin{array}{cc}\hfill Q\left(\lambda \right)& \ge {\lambda }^2+(b_1+min(b_3,0)+min(b_7,0))\lambda \hfill \\ \hfill & +(b_2+min(b_4,0)+min(b_8,0)+min(b_{10},0))\hfill \\ \hfill & >0.\hfill \end{array}$$

(73)

So, $Q\left(\lambda \right)$ admits no non-negative zero.

Suppose $Q\left(\lambda \right)$ admits a zero with a non-negative real part. It follows from [50] that $P\left(\lambda \right)$ has a pair of conjugate purely imaginary zeros, $\pm i\omega$ ($\omega >0$). Hence,

$$\begin{array}{cc}\hfill & Q\left(i\omega \right)={\left(i\omega \right)}^2+i{b}_1\omega +b_2+(i{b}_3\omega +b_4)e^{-i\omega {\tau }_1}+(i{b}_5\omega +b_6)e^{-i\omega {\tau }_2}\hfill \\ \hfill & +(i{b}_7\omega +b_8)e^{-i\omega {\tau }_3}+b_9{e}^{-i\omega ({\tau }_1+{\tau }_2)}+b_{10}{e}^{-i\omega ({\tau }_1+{\tau }_3)}\hfill \\ \hfill & =-{\omega }^2+i{b}_1\omega +b_2+(i{b}_3\omega +b_4)(cos\left(\omega {\tau }_1\right)-isin\left(\omega {\tau }_1\right))\hfill \\ \hfill & +(i{b}_5\omega +b_6)(cos\left(\omega {\tau }_2\right)-isin\left(\omega {\tau }_2\right))+(i{b}_7\omega +b_8)(cos\left(\omega {\tau }_3\right)-isin\left(\omega {\tau }_3\right))\hfill \\ \hfill & +b_9(cos\left(\omega ({\tau }_1+{\tau }_2)\right)-isin\left(\omega ({\tau }_1+{\tau }_2)\right))\hfill \\ \hfill & +b_{10}(cos\left(\omega ({\tau }_1+{\tau }_3)\right)-isin\left(\omega ({\tau }_1+{\tau }_3)\right))\hfill \\ \hfill & =[-{\omega }^2+b_2+b_3\omega sin\left(\omega {\tau }_1\right)+b_{4}cos\left(\omega {\tau }_1\right)+b_5\omega sin\left(\omega {\tau }_2\right)+b_{6}cos\left(\omega {\tau }_2\right)\hfill \\ \hfill & +b_7\omega sin\left(\omega {\tau }_3\right)+b_{8}cos\left(\omega {\tau }_3\right)+b_{9}cos\left(\omega ({\tau }_1+{\tau }_2)\right)+b_{10}cos\left(\omega ({\tau }_1+{\tau }_3)\right)]\hfill \\ \hfill & +i[b_1\omega +b_3\omega cos\left(\omega {\tau }_1\right)-b_{4}sin\left(\omega {\tau }_1\right)+b_5\omega cos\left(\omega {\tau }_2\right)-b_{6}sin\left(\omega {\tau }_2\right)\hfill \\ \hfill & +b_7\omega cos\left(\omega {\tau }_3\right)-b_{8}sin\left(\omega {\tau }_3\right)-b_{9}sin\left(\omega ({\tau }_1+{\tau }_2)\right)-b_{10}sin\left(\omega ({\tau }_1+{\tau }_3)\right)]\hfill \\ \hfill & =0.\hfill \end{array}$$

(74)

Again, separating the real and imaginary parts, it follows that

$$\left\{\begin{array}{c}{b}_3\omega sin\left(\omega {\tau }_1\right)+b_{4}cos\left(\omega {\tau }_1\right)+b_5\omega sin\left(\omega {\tau }_2\right)+b_{6}cos\left(\omega {\tau }_2\right)+b_7\omega sin\left(\omega {\tau }_3\right)\hfill \\ +b_{8}cos\left(\omega {\tau }_3\right)+b_{9}cos\left(\omega ({\tau }_1+{\tau }_2)\right)+b_{10}cos\left(\omega ({\tau }_1+{\tau }_3)\right)\hfill \\ ={\omega }^2-b_2,\hfill \\ b_3\omega cos\left(\omega {\tau }_1\right)-b_{4}sin\left(\omega {\tau }_1\right)+b_5\omega cos\left(\omega {\tau }_2\right)-b_{6}sin\left(\omega {\tau }_2\right)+b_7\omega cos\left(\omega {\tau }_3\right)\hfill \\ -b_{8}sin\left(\omega {\tau }_3\right)-b_{9}sin\left(\omega ({\tau }_1+{\tau }_2)\right)-b_{10}sin\left(\omega ({\tau }_1+{\tau }_3)\right)\hfill \\ =-b_1\omega .\hfill \end{array}\right.$$

(75)

Squaring both sides of the two equations and simplifying, it follows that

$$\begin{array}{cc}\hfill & b_3^2{\omega }^2+b_4^2+b_5^2{\omega }^2+b_6^2+b_7^2{\omega }^2+b_8^2+b_9^2+b_{10}^2+2(b_6{b}_9+b_8{b}_{10})cos\left(\omega {\tau }_1\right)\hfill \\ \hfill & -2(b_5{b}_9+b_7{b}_{10})\omega sin\left(\omega {\tau }_1\right)+2b_4{b}_{9}cos\left(\omega {\tau }_2\right)-2b_3{b}_9\omega sin\left(\omega {\tau }_2\right)\hfill \\ \hfill & -2b_3{b}_{10}\omega sin\left(\omega {\tau }_3\right)+2b_4{b}_{10}cos\left(\omega {\tau }_3\right)+2(b_3{b}_5{\omega }^2+b_4{b}_6)cos\left(\omega ({\tau }_1-{\tau }_2)\right)\hfill \\ \hfill & +2(b_3{b}_6-b_4{b}_5)\omega sin\left(\omega ({\tau }_1-{\tau }_2)\right)+2(b_3{b}_7{\omega }^2+b_4{b}_8)cos\left(\omega ({\tau }_1-{\tau }_3)\right)\hfill \\ \hfill & +2(b_3{b}_8-b_4{b}_7)\omega sin\left(\omega ({\tau }_1-{\tau }_3)\right)+2(b_5{b}_7{\omega }^2+b_6{b}_8+b_9{b}_{10})cos\left(\omega ({\tau }_2-{\tau }_3)\right)\hfill \\ \hfill & +2(b_5{b}_8-b_6{b}_7)\omega sin\left(\omega ({\tau }_2-{\tau }_3)\right)+2b_8{b}_{9}cos\left(\omega ({\tau }_1+{\tau }_2-{\tau }_3)\right)\hfill \\ \hfill & -2b_7{b}_9\omega sin\left(\omega ({\tau }_1+{\tau }_2-{\tau }_3)\right)+2b_6{b}_{10}cos\left(\omega ({\tau }_1+{\tau }_3-{\tau }_2)\right)\hfill \\ \hfill & -2b_5{b}_{10}\omega sin\left(\omega ({\tau }_1+{\tau }_3-{\tau }_2)\right)\hfill \\ \hfill & ={({\omega }^2-b_2)}^2+b_1^2{\omega }^2.\hfill \end{array}$$

(76)

The assumptions that ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$ are very small lead to

$$\begin{array}{cc}\hfill sin\left(\omega {\tau }_1\right)& \approx 0,sin\left(\omega {\tau }_2\right)\approx 0,sin\left(\omega {\tau }_3\right)\approx 0,\hfill \\ \hfill cos\left(\omega {\tau }_1\right)& \approx 1,cos\left(\omega {\tau }_2\right)\approx 1,cos\left(\omega {\tau }_1\right)\approx 1.\hfill \end{array}$$

(77)

So,

$$\begin{array}{cc}\hfill & b_3^2{\omega }^2+b_4^2+b_5^2{\omega }^2+b_6^2+b_7^2{\omega }^2+b_8^2+b_9^2+b_{10}^2+2(b_3{b}_5+b_3{b}_7+b_5{b}_7){\omega }^2\hfill \\ \hfill & +2(b_4{b}_6+b_4{b}_8+b_4{b}_9+b_4{b}_{10}+b_6{b}_8+b_6{b}_9+b_6{b}_{10}+b_8{b}_9+b_8{b}_{10}+b_9{b}_{10})\hfill \\ \hfill & ={(b_3+b_5+b_7)}^2{\omega }^2+{(b_4+b_6+b_8+b_9+b_{10})}^2\hfill \\ \hfill & \approx {({\omega }^2-b_2)}^2+b_1^2{\omega }^2.\hfill \end{array}$$

(78)

Thus,

$$R\left({\omega }^2\right)={\omega }^4+s_1{\omega }^2+s_2\approx 0.$$

(79)

Hence,

$$\omega \approx =\pm {\omega }_0=\pm \sqrt{{\displaystyle \frac{-s_1\pm \sqrt{s_1^2-4s_2}}{2}}}.$$

(80)

Consequently, $\left|Q\right(i{\omega }_0\left)\right|\approx 0$. This is a contradiction.

The assertion follows from Lyapunov’s stability theorem [52]. □

Remark 3.

Model (2) assumes a nonlinear infection rate and a nonlinear disinfection rate. To date, there is no report in the literature on the global stability of a rumor-endemic equilibrium of delayed rumor spreading models of this type [18]. This is our next work.

Remark 4.

The Hopf bifurcation analysis under model (2) is similar to that presented in [21] and is hence omitted here.

6. Simulation Experiments

In the preceding sections, some theoretical results on model (4) were reported. This section is devoted to examining the dynamics of model (2) through simulation experiments.

6.1. Asymptotic Stability of the Rumor-Free Equilibrium

In this subsection, the asymptotic stability of the rumor-free equilibrium is examined through simulation experiments.

Experiment 1.

Consider model (2) with $\mu =50$, $\delta =1$, $\beta =0.015$, ${\gamma }_1=0.06$, ${\gamma }_2=0.002$, ${\sigma }_1=0.6$, ${\sigma }_2=0.8$, ${\tau }_1=0.001$, ${\tau }_2=0.001$, and ${\tau }_3=0.001$. It is easily verified that ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1\le \delta$. The local asymptotic stability of $E_0$ follows from Theorem 5.

For $-\tau \le \theta \le 0$, let $(I_1\left(\theta \right),R_1\left(\theta \right))\equiv (10,40)$, $(I_2\left(\theta \right),R_2\left(\theta \right))\equiv (20,30)$, $(I_3\left(\theta \right),R_3\left(\theta \right))\equiv (30,20)$, $(I_4\left(\theta \right),R_4\left(\theta \right))\equiv (40,10)$. The time plots for $I_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 1a. The time plots for $R_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 1b. The phase portrait is plotted in Figure 1c. The local asymptotic stability of $E_0$ is observed.

Experiment 2.

Consider model (2) with $\mu =50$, $\delta =1$, $\beta =0.1$, ${\gamma }_1=0.06$, ${\gamma }_2=0.002$, ${\sigma }_1=0.6$, ${\sigma }_2=0.8$, ${\tau }_1=0.001$, ${\tau }_2=0.001$, and ${\tau }_3=0.001$. It is easily verified that $R_0>1$. Hence, the unstability of $E_0$ follows from Theorem 5.

For $-\tau \le \theta \le 0$, let $(I_1\left(\theta \right),R_1\left(\theta \right))\equiv (10,40)$, $(I_2\left(\theta \right),R_2\left(\theta \right))\equiv (20,30)$, $(I_3\left(\theta \right),R_3\left(\theta \right))\equiv (30,20)$, $(I_4\left(\theta \right),R_4\left(\theta \right))\equiv (40,10)$. The time plots for $I_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 2a. The time plots for $R_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 2b. The phase portrait is plotted in Figure 2c. The unstability of $E_0$ is observed.

Experiment 3.

Consider model (2) with $\mu =50$, $\delta =1$, $\beta =0.01$, ${\gamma }_1=0.08$, ${\gamma }_2=0.005$, ${\sigma }_1=0.8$, ${\sigma }_2=1$, ${\tau }_1=0.001$, ${\tau }_2=0.001$, and ${\tau }_3=0.001$. It is easily verified that ${\displaystyle \frac{\beta \mu }{\delta }}+{\gamma }_1\le \delta$. Hence, the global asymptotic stability of $E_0$ follows from Theorem 5.

For $-\tau \le \theta \le 0$, let $(I_1\left(\theta \right),R_1\left(\theta \right))\equiv (10,40)$, $(I_2\left(\theta \right),R_2\left(\theta \right))\equiv (20,30)$, $(I_3\left(\theta \right),R_3\left(\theta \right))\equiv (30,20)$, $(I_4\left(\theta \right),R_4\left(\theta \right))\equiv (40,10)$. The time plots for $I_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 3a. The time plots for $R_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 3b. The phase portrait is plotted in Figure 3c. The global asymptotic stability of $E_0$ is observed.

6.2. Asymptotic Stability of a Rumor-Endemic Equilibrium

In this subsection, the asymptotic stability of a rumor-endemic equilibrium is inspected through simulation experiments.

Experiment 4.

Consider model (2) with $\mu =100$, $\delta =2$, $\beta =0.09$, ${\gamma }_1=0.07$, ${\gamma }_2=0.5$, ${\sigma }_1=0.6$, ${\sigma }_2=0.1$, ${\tau }_1=0.001$, ${\tau }_2=0.001$, and ${\tau }_3=0.001$. Then, $E^{*}=(1.43,0.58)$ is the sole rumor-endemic equilibrium. It is easily verified that $b_1+min(b_3,0)+min(b_7,0)\ge 0$, $b_2+min(b_4,0)+min(b_8,0)+min(b_{10},0)>0$, $\left|Q\right(i{\omega }_0\left)\right|=2.61$ is not very small. The local asymptotic stability of $E^{*}$ follows from Theorem 8.

For $-\tau \le \theta \le 0$, let $(I_1\left(\theta \right),R_1\left(\theta \right))\equiv (10,40)$, $(I_2\left(\theta \right),R_2\left(\theta \right))\equiv (20,30)$, $(I_3\left(\theta \right),R_3\left(\theta \right))\equiv (30,20)$, $(I_4\left(\theta \right),R_4\left(\theta \right))\equiv (40,10)$. The time plots for $I_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 4a. The time plots for $R_k\left(t\right)$, $k\in \{1,2,3,4\}$, are displayed in Figure 4b. The phase portrait is plotted in Figure 4c. The local asymptotic stability of $E^{*}$ is observed.

7. Further Discussions

This section is dedicated to examining the influence of the three delays and the two saturation functions on the dynamics of model (4).

7.1. Influence of the Time Delays

In this subsection, the influence of the three delays is examined through simulation experiments.

Experiment 5.

Let ${\Gamma }_1=\{0,0.2,\dots ,0.8\}$. Consider five models (2) with $\mu =500$, $\delta =10$, $\beta =0.2$, ${\gamma }_1=0.25$, ${\gamma }_2=0.3$, ${\sigma }_1=0.1$, ${\sigma }_2=0.1$, ${\tau }_1\in {\Gamma }_1$, ${\tau }_2=0.2$, and ${\tau }_3=0.2$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\tau }_1\in {\Gamma }_1$, the time plot for the number of spreaders is displayed in Figure 5. In this case, it is observed that the number of spreaders is approximately decreasing. Moreover, it is observed that the extended ${\tau }_1$ leads to lower rate of change and more violent oscillation in the number of spreaders.

Experiment 6.

Let ${\Gamma }_1=\{0,0.2,\dots ,0.8\}$. Consider five models (2) with $\mu =500$, $\delta =10$, $\beta =0.5$, ${\gamma }_1=0.25$, ${\gamma }_2=0.3$, ${\sigma }_1=0.1$, ${\sigma }_2=0.1$, ${\tau }_1\in {\Gamma }_1$, ${\tau }_2=0.2$, and ${\tau }_3=0.2$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\tau }_1\in {\Gamma }_1$, the time plot for the number of spreaders is displayed in Figure 6. In this case, it is observed that the number of spreaders is approximately increasing. Again, it is observed that the extended ${\tau }_1$ leads to lower rate of change and more violent oscillation in the number of spreaders.

Based on 100 similar experiments, the following conclusions are drawn.

(a)

The extended first delay leads to a lower rate of change in the number of spreaders. This is because the extension takes an ignorant person longer to facilitate spreading and hence leads to a slower change in the number of spreaders.

(b)

The extended first delay leads to more violent oscillation in the number of spreaders. This is a feature of delayed dynamical systems.

Experiment 7.

Let ${\Gamma }_2\in \{0,0.5,\dots ,2.0\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.3$, ${\gamma }_2=0.5$, ${\sigma }_1=0.2$, ${\sigma }_2=0.6$, ${\tau }_1=0$, ${\tau }_2\in {\Gamma }_2$, and ${\tau }_3=0$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 200$, $R\left(\theta \right))\equiv 300$.

For ${\tau }_2\in {\Gamma }_2$, the time plot for the number of stiflers is displayed in Figure 7. In this case, it is observed that the number of stiflers is approximately decreasing. Moreover, it is observed that extended ${\tau }_2$ leads to lower rate of change and more violent oscillation in the number of stiflers.

Experiment 8.

Let ${\Gamma }_2\in \{0,0.5,\dots ,2.0\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.3$, ${\gamma }_2=0.5$, ${\sigma }_1=0.2$, ${\sigma }_2=0.6$, ${\tau }_1=0$, ${\tau }_2\in {\Gamma }_2$, and ${\tau }_3=0$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\tau }_2\in {\Gamma }_2$, the time plot for the number of stiflers is displayed in Figure 8. In this case, it is observed that the number of stiflers is not approximately decreasing. Again, it is observed that extended of ${\tau }_2$ leads to lower rate of change and more violent oscillation in the number of stiflers.

Based on 100 similar experiments, the following conclusions are drawn.

(a)

The extended second delay leads to a lower rate of change in the number of stiflers. This is because the extension takes a spreader longer to experience stifling and hence leads to slower change in the number of stiflers.

(b)

The extended second delay leads to more violent oscillation in the number of stiflers. This is a feature of delayed dynamical systems.

Experiment 9.

Let ${\Gamma }_3=\{0,0.4,\dots ,1.6\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.3$, ${\gamma }_2=0.35$, ${\sigma }_1=0.2$, ${\sigma }_2=0.6$, ${\tau }_1=0$, ${\tau }_2=00$, and ${\tau }_3\in {\Gamma }_3$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 200$, $R\left(\theta \right))\equiv 300$.

For ${\tau }_3\in {\Gamma }_3$, the time plot for the number of stiflers is displayed in Figure 9. In this case, it is observed that the number of stiflers is approximately decreasing. Moreover, it is observed that extended ${\tau }_3$ leads to a lower rate of change and more violent oscillation in the number of stiflers.

Experiment 10.

Let ${\Gamma }_3=\{0,0.4,\dots ,1.6\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.3$, ${\gamma }_2=0.5$, ${\sigma }_1=0.2$, ${\sigma }_2=0.6$, ${\tau }_1=0$, ${\tau }_2=0$, and ${\tau }_3\in {\Gamma }_3$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\tau }_3\in {\Gamma }_3$, the time plot for the number of stiflers is displayed in Figure 10. In this case, it is observed that the number of stiflers is not approximately decreasing. Again, it is observed that extended ${\tau }_3$ leads to a lower rate of change and more violent oscillation in the number of stiflers.

Based on 100 similar experiments, the following conclusions are drawn.

(a)

The extended third delay leads to a lower rate of change in the number of stiflers. This is because the extension takes a spreader longer to experience stifling and hence leads to slower change in the number of stiflers.

(b)

The extended third delay leads to more violent oscillation in the number of stiflers. This is a feature of delayed dynamical systems.

7.2. Influence of the Saturation Coefficients

Experiment 11.

Let ${\Sigma }_1=\{0,0.2,\dots ,0.8\}$. Consider five models (2) with $\mu =500$, $\delta =10$, $\beta =0.2$, ${\gamma }_1=0.25$, ${\gamma }_2=0.3$, ${\sigma }_1\in {\Sigma }_1$, ${\sigma }_2=0.5$, ${\tau }_1=0.01$, ${\tau }_2=0.01$, and ${\tau }_3=0.01$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\sigma }_1\in {\Sigma }_1$, the time plot for the number of spreaders is displayed in Figure 11. In this case, it is observed that the number of spreaders is decreasing. Moreover, it is observed that the lifted ${\sigma }_1$ leads to a faster decrease in the number of spreaders.

Based on 100 similar experiments, it is concluded that, in the case where the number of spreaders is decreasing, with the increase of the first saturation coefficient, the number of spreaders decreases more rapidly. This is because the increase of the first saturation coefficient takes an ignorant person longer to facilitate spreading.

Experiment 12.

Let ${\Sigma }_1=\{0,0.2,\dots ,0.8\}$. Consider five models (2) with $\mu =500$, $\delta =10$, $\beta =1.5$, ${\gamma }_1=0.25$, ${\gamma }_2=0.3$, ${\sigma }_1\in {\Sigma }_1$, ${\sigma }_2=0.5$, ${\tau }_1=0.01$, ${\tau }_2=0.01$, ${\tau }_3=0.01$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\sigma }_1\in {\Sigma }_1$, the time plot for the number of spreaders is displayed in Figure 12. In this case, it is observed that the number of spreaders is increasing. Moreover, it is observed that the lifted ${\sigma }_1$ leads to slower increase in the number of spreaders.

Based on 100 similar experiments, it is concluded that, in the case where the number of spreaders is increasing, with the increase of the first saturation coefficient, the number of spreaders increases more slowly. This is because the increase of the first saturation coefficient takes an ignorant person longer to facilitate spreading.

Experiment 13.

Let ${\Sigma }_2=\{0,1,\dots ,4\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.1$, ${\gamma }_2=0.5$, ${\sigma }_1=2$, ${\sigma }_2\in {\Sigma }_2$, ${\tau }_1=0.1$, ${\tau }_2=0.1$, and ${\tau }_3=0.1$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 40$, $R\left(\theta \right))\equiv 300$.

For ${\sigma }_2\in {\Sigma }_2$, the time plot for the number of and stiflers is displayed in Figure 13. In this case, it is observed that the number of stiflers is decreasing. Moreover, it is observed that the lifted ${\sigma }_2$ leads to faster decrease in the number of stiflers.

Based on 100 similar experiments, it is concluded that, in the case where the number of stiflers is decreasing, with the increase of the second saturation coefficient, the number of stiflers decreases more rapidly. This is because the increase of the second saturation coefficient takes a spreader longer to experience stifling.

Experiment 14.

Let ${\Sigma }_2=\{0,1,\dots ,4\}$. Consider five models (2) with $\mu =500$, $\delta =1$, $\beta =0.2$, ${\gamma }_1=0.3$, ${\gamma }_2=0.5$, ${\sigma }_1=2$, ${\sigma }_2\in {\Sigma }_2$, ${\tau }_1=0.01$, ${\tau }_2=0.01$, and ${\tau }_3=0.01$. For $-\tau \le \theta \le 0$, let $I\left(\theta \right)\equiv 5$, $R\left(\theta \right))\equiv 5$.

For ${\sigma }_2\in {\Sigma }_2$, the time plot for the number of stiflers is displayed in Figure 14. In this case, it is observed that the number of stiflers is increasing. Moreover, it is observed that the lifted ${\sigma }_2$ leads to a slower increase in the number of stiflers.

Based on 100 similar experiments, it is concluded that, in the case where the number of stiflers is increasing, with the increase of the second saturation coefficient, the number of stiflers increases more slowly. This is because the increase of the second saturation coefficient takes a spreader longer to experience stifling.

8. Conclusions

A rumor spreading model with three time delays and two saturation effects has been suggested. The structure of the rumor-endemic equilibria is analyzed. The asymptotic stability of the rumor-free and rumor-endemic equilibria has been investigated. The influence of the delays and the saturation effects on rumor spreading has been inspected.

Several relevant issues are yet to be addressed. First, it is urgent yet challenging to gather delayed rumor spreading-related data [53,54]. Second, most existing delayed rumor spreading models are Holling type II models. Recently, a delayed rumor spreading model with the Crowley–Martin saturation effect [55,56] was reported [32]. For the purpose of better understanding the effect of saturation on rumor spreading, it is critical to explore rumor spreading models with more complex saturation functions [55,56,57,58,59,60]. Third, although numerous optimal control-related issues have been addressed, less work on optimal impulsive control-related issues has been reported in the literature [61,62,63,64]. Due to the intermittence of rumor spreading, it is of practical importance to study rumor spreading models in the framework of optimal impulsive control theory. Next, rumor spread and rumor refutation form a non-cooperative game should be studied [65]. Consequently, it is essential to establish game theory-based rumor spreading models. Finally, it is crucial to develop a machine intelligence-based approach for fitting a complex rumor spreading model [66]. This is our next work.