# Dynamics Analysis of a Delayed Rumor Propagation Model in an Emergency-Affected Environment

Authors: Chunru Li; Zujun Ma
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

Rumors influence people’s decisions in an emergency-affected environment. How to describe the spreading mechanism is significant. In this paper, we propose a delayed rumor propagation model in emergencies. By taking the delay as the bifurcation parameter, the local stability of the boundary equilibrium point and the positive equilibrium point is
investigated and the conditions of Hopf bifurcation are obtained. Furthermore, formulas for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by applying the normal form method and center manifold
theorem. Finally, some numerical simulations are also given to illustrate our theoretical results.


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## Body

## 1. Introduction

Emergencies usually cause serious negative impacts on people’s work and life in a variety of ways. For example, the event itself may lead to financial loss or personal injuries, and also at the same time the rumor may begin to prevail and it leads to panic feelings or irrational behavior. Rumor is usually defined as the unconfirmed elaboration or annotation of the public interesting things, events, or issues that spread through various channels [1–3]. When emergencies occur, people often fail to reflect rumor’s seriousness and take active measures to deal with the related issues in time. Rumor especially the exaggerated rumor, as a product of emergencies, may bring helplessness and distress to the public, hamper the regulation of public opinions, and simultaneously destroy social harmony and stability. For instance, during the 2011 earthquake and Fukushima nuclear leak in Japan, because of the propagation of the rumors about the water pollution, many people in China blindly rush to buy salts, which greatly affects the regular social order [4]. Thus, how to effectively control and prevent rumor propagation in emergencies has become one of the most valuable research topics.Accurate analysis of the dynamic behavior of rumor propagation is significant for controlling and preventing rumor propagation in emergencies. Using mathematical models to analyze the dynamic characteristics about rumor propagation has now become a compelling topic [5–9]. Rumor propagation is very similar to the diffusion of the virus; thus, most of the existing models of rumor propagation are derived from the models of infectious diseases. The classical models (i.e., the DK model) for rumor propagation were introduced by Daley and Kendall [10]. They mainly divided people into the following three classes: ignorants (those who cannot identify the rumors), spreaders (those who spread rumors), and stiflers (those who do not believe the rumors again). Thenceforth, most of the researchers applied or improved DK model to study rumor propagation. Maki and Thompson applied a mathematical model to study rumor propagation, and they mainly focused on the theoretical analysis [11]. Z.-L. Zhang and Z.-Q. Zhang [12] supposed that the emergency state and the spreading rate of rumors were continuous functions of time. Then, they established a rumor propagation model in emergencies based on three-molecule models. Their results showed that rumor spreading at certain rate had positive effects on situation stability and especially had delaying effects on the rapid proliferation of an emergency. Based on the analysis of [12], Zhao [13] considered the impact of authorities’ media and rumor propagation on the evolution of emergencies, and they presented a new rumor propagation model with the impact of authorities’ media. Huo [14] studied a rumor transmission model with constant immigration and incubation. Their analysis results meet the actual circumstances better and more externally, and they are easier to understand. Zhao and Wang [15], based on epidemic-like models, established a ISRW dynamical model considering the medium to describe the rumor propagation in emergencies. They discussed the stability of the rumor propagation model and suggested that the government should enhance the management of Internet. Recently, Zhang et al. [16] considered an 8-state ICSAR rumor propagation model with official rumor refutation. They believed that high spread probability and low block value could lead to rumor refutation. However, this model did not consider the impact of delays on rumor propagation.In fact, delays always exist in rumor propagation. For example, when rumor susceptible people are infected by rumor spreading people, there is a time during which the infected people can have the ability to spread rumor. On the other hand, when rumor spreading people are educated by our government, there also exists a delayed time after which the rumor spreading people can really recover and will not believe the rumor. Therefore, delay is an important factor in rumor propagation.Inspired by the abovementioned work, this paper deals with a novel delayed rumor propagation model with media report, which is of the following form:(1)dSdt=A1-SS+m-βSI-μS,dIdt=βSI-μI-αIt-τ,dRdt=αIt-τ-μR,with initial conditions(2)St=S0,t∈-τ,0,It=I0,t∈-τ,0,Rt=R0,t∈-τ,0,where S(t) is the density of the rumor susceptible people at time t, I(t) is the density of the rumor infected people at time t, R(t) is the density of the rumor recovered people at time t, and the coefficients A, m, α, β, μ are positive constants. A represents the natural growth rate of the rumor susceptible people, m represents the impact of media report, α represents the impact of our government on the rumor infected people, β is the infected rate, and μ describes a rate that people remove from the system since they lose interest in rumors [17]. τ≥0 is a delay, which implies that when rumor infected people are educated by our government, there exists a delayed time after which the rumor spreading people can really recover. S0, I0, R0 are the initial densities of the rumor susceptible people, the rumor infected people, and the rumor recovered people, respectively.Accordingly, the remaining structure of this paper is arranged as follows. In Section2, we are concerned with local stability and Hopf bifurcation of the equilibrium points for our model. In Section 3, the properties of Hopf bifurcation, formula determining the direction of Hopf bifurcation, and the stability of spatially homogeneous bifurcating periodic solutions are investigated. In Section 4, numerical simulations are presented to demonstrate our theoretical results. Finally, conclusions are drawn in Section 5.


## 2. Local Stability and Hopf Bifurcation

Because the first two equations in (1) are independent of R(t), we can consider the following reduced model:(3)dSdt=A1-SS+m-βSI-μS,dIdt=βSI-μI-αIt-τ,with initial conditions(4)St=S0,t∈-τ,0,It=I0,t∈-τ,0.Obviously, system (3) has the following two kinds of equilibria points:(i)
The boundary equilibrium pointE1=((-mμ+mμ(mμ+4A))/2μ,0)T.(iii)
If(H1)Amβ2-μ(μ+α)(mβ+α+μ)>0 holds, then there exists the positive equilibrium point(5)E∗=α+μβ,mAβ2-μμ+αmβ+α+μβμ+αmβ+α+μT.In this part, we will discuss the local stability and Hopf bifurcation of the boundary equilibrium pointE1 and the positive equilibrium point E∗ with the time delay τ as the bifurcation parameter.For further discussion, we firstly make the following assumptions:H2
β-mμ+mμmμ+4A-2μμ+α<0,H3
β-mμ+mμmμ+4A+2μα-μ>0,H4
μ-αβm+α+μ-2αα+μ<0.


### 2.1. Stability and Hopf Bifurcation of the Boundary Equilibrium PointE1

By a simple calculation, it is easy to obtain the characteristic equation of system (3) at the boundary equilibrium point E1 as the following form:(6)λ+μ+mm+S02λ-βS0+μ+αe-λτ=0,where S0=(-mμ+mμ(mμ+4A))/2μ.Lemma 1.
If(H2) holds, then the boundary equilibrium point E1 of system (3) with τ=0 is locally asymptotically stable.Proof.
Clearly, (6) always has a negative real root; namely, (7)λ=-μ-mm+S02.Therefore, we only consider the form of(8)λ-βS0+μ+αe-λτ=0.When τ=0, (8) can be rewritten as (9)λ-βS0+μ+α=0.Obviously, under condition (H2), (10)λ=12μβ-mμ+mμmμ+4A-2μμ+α<0.Thus, the boundary equilibrium point E1 is locally asymptotically stable as τ=0.Now we discuss the effect of the delayτ on the stability of E1. Assume that iω is a root of (8). Then ω should satisfy the following equations:(11)αcos⁡ωτ=βS0-μ,αsin⁡ωτ=ω.Taking square on both sides of the equations of (11) and summing them up, we obtain(12)ω2=α2-βS0-μ2.If (H2) and (H3) hold, then (12) has a positive real root ω+, and (13)ω+=α2-βS0-μ2.Therefore, according to (11), it is easy to show that (14)τi=1ωarccos⁡S0β-μα+2iπi=0,1,2,….Furthermore, differentiating both sides of (8) with respect to τ, we have (15)dλdτ=1α2>0.Based on the above analysis, we obtain the following result.Theorem 2.
IfH2~(H3) hold, then the boundary equilibrium point E1 of system (3) is locally asymptotically stable when τ∈[0,τ0). Moreover, it undergoes a Hopf bifurcation when τ=τi.


### 2.2. Stability and Hopf Bifurcation of the Positive Equilibrium PointE∗

For the positive equilibrium pointE∗, the characteristic equation of system (3) is as follows(16)λ2+Amβ2μ+αmβ+α+μ-α+Amβ2mβ+α+μ2λ+Amμβ2μ+αmβ+α+μ-Amαβ2mβ+α+μ2-μμ+α+λ+Amβ2μ+αmβ+α+μ+Amβ2mβ+α+μ2αe-λτ=0.Lemma 3.
If(H1) holds, then the positive equilibrium point E∗ of system (3) with τ=0 is locally asymptotically stable.Proof.
Whenτ=0, we have(17)λ2+Amβ2mβ+α+μ1μ+α+1βm+α+μλ+1βm+α+μAmβ2-μμ+αmβ+α+μ=0.Under condition (H1), it is easy to obtain (18)Amβ2-μμ+αmβ+α+μ>0.According to the Routh-Hurwitz criteria, all the roots of (17) have negative real parts. Therefore, when τ=0, the positive equilibrium point E∗ is locally asymptotically stable.Next we discuss the effect of the delayτ on the stability of E∗. Assume that iω(ω>0) is a root of (16). Then ω should satisfy the following equation: (19)αAmβ2μ+αβm+α+μ+Amβ2βm+α+μ2cos⁡ωτ+ωsin⁡ωτ=ω2-Amμβ2μ+αmβ+α+μ-Amαβ2mβ+α+μ2-μμ+α,α-Amβ2μ+αβm+α+μ+Amβ2βm+α+μ2sin⁡ωτ+ωcos⁡ωτ=-ωAmβ2μ+αmβ+α+μ-α+Amβ2mβ+α+μ2,which implies (20)ω4+ω2A1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22=0,where (21)A1=Amβ2α+μβm+α+μ,A2=Amβ2βm+α+μ2.Let z=ω2; then(22)z2+zA1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22=0.Without loss of generality, define (23)hz=z2+zA1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22.Lemma 4.
If(H4) holds, then (20) has a unique positive real root.Proof.
Under condition(H4), a simple calculation shows that (24)A1μ-A2α-αμ-μ22-α2A1+A22=1βm+α+μAmβ2-μμ+αmβ+α+μAmβ2μ+αmβ+μ+α2μ-αβm+α+μ-2αα+μ-μμ+α<0.Thus, (22) has a unique positive real root; namely, (25)z0=12-A1+A22+2μ+αA1-μ+A1+A22-2μ+αA1-μ2-4A1μ-A2α-αμ-μ22-α2A1+A22.Furthermore, we obtain ω0=z0. This completes the proof.Now, from (19), it is easy to show that (26)τj∗=1ω0arccos⁡A1+A2-A1μ+A2α+αμ+μ2+αω02αA1+A22+ω02+2jπ.Lemma 5 (see [18]).
Letλ(τ)=ϑ(τ)±iω(τ) be the root of (16) near τ=τ0∗ satisfying ϑ(τ0∗)=0, ω(τ0∗)=ω0. Suppose that h′(ω02)>0. Then ±iω0 is a pair of simple purely imaginary roots of (16). Moreover, the following transversality condition holds: (27)dReλτdττ=τ0∗,λ=iω0>0.
Applying Lemma3–Lemma 5, we have the following result.Theorem 6.
If(H1) and (H4) hold. Further suppose that h′(ω02)>0 satisfies. Then(i)
the positive equilibrium pointE∗ of system (3) is locally asymptotically stable when τ∈[0,τ0∗);(ii)
system (3) undergoes a Hopf bifurcation at E∗ when τ=τ0∗.


## 2.1. Stability and Hopf Bifurcation of the Boundary Equilibrium PointE1

By a simple calculation, it is easy to obtain the characteristic equation of system (3) at the boundary equilibrium point E1 as the following form:(6)λ+μ+mm+S02λ-βS0+μ+αe-λτ=0,where S0=(-mμ+mμ(mμ+4A))/2μ.Lemma 1.
If(H2) holds, then the boundary equilibrium point E1 of system (3) with τ=0 is locally asymptotically stable.Proof.
Clearly, (6) always has a negative real root; namely, (7)λ=-μ-mm+S02.Therefore, we only consider the form of(8)λ-βS0+μ+αe-λτ=0.When τ=0, (8) can be rewritten as (9)λ-βS0+μ+α=0.Obviously, under condition (H2), (10)λ=12μβ-mμ+mμmμ+4A-2μμ+α<0.Thus, the boundary equilibrium point E1 is locally asymptotically stable as τ=0.Now we discuss the effect of the delayτ on the stability of E1. Assume that iω is a root of (8). Then ω should satisfy the following equations:(11)αcos⁡ωτ=βS0-μ,αsin⁡ωτ=ω.Taking square on both sides of the equations of (11) and summing them up, we obtain(12)ω2=α2-βS0-μ2.If (H2) and (H3) hold, then (12) has a positive real root ω+, and (13)ω+=α2-βS0-μ2.Therefore, according to (11), it is easy to show that (14)τi=1ωarccos⁡S0β-μα+2iπi=0,1,2,….Furthermore, differentiating both sides of (8) with respect to τ, we have (15)dλdτ=1α2>0.Based on the above analysis, we obtain the following result.Theorem 2.
IfH2~(H3) hold, then the boundary equilibrium point E1 of system (3) is locally asymptotically stable when τ∈[0,τ0). Moreover, it undergoes a Hopf bifurcation when τ=τi.


## 2.2. Stability and Hopf Bifurcation of the Positive Equilibrium PointE∗

For the positive equilibrium pointE∗, the characteristic equation of system (3) is as follows(16)λ2+Amβ2μ+αmβ+α+μ-α+Amβ2mβ+α+μ2λ+Amμβ2μ+αmβ+α+μ-Amαβ2mβ+α+μ2-μμ+α+λ+Amβ2μ+αmβ+α+μ+Amβ2mβ+α+μ2αe-λτ=0.Lemma 3.
If(H1) holds, then the positive equilibrium point E∗ of system (3) with τ=0 is locally asymptotically stable.Proof.
Whenτ=0, we have(17)λ2+Amβ2mβ+α+μ1μ+α+1βm+α+μλ+1βm+α+μAmβ2-μμ+αmβ+α+μ=0.Under condition (H1), it is easy to obtain (18)Amβ2-μμ+αmβ+α+μ>0.According to the Routh-Hurwitz criteria, all the roots of (17) have negative real parts. Therefore, when τ=0, the positive equilibrium point E∗ is locally asymptotically stable.Next we discuss the effect of the delayτ on the stability of E∗. Assume that iω(ω>0) is a root of (16). Then ω should satisfy the following equation: (19)αAmβ2μ+αβm+α+μ+Amβ2βm+α+μ2cos⁡ωτ+ωsin⁡ωτ=ω2-Amμβ2μ+αmβ+α+μ-Amαβ2mβ+α+μ2-μμ+α,α-Amβ2μ+αβm+α+μ+Amβ2βm+α+μ2sin⁡ωτ+ωcos⁡ωτ=-ωAmβ2μ+αmβ+α+μ-α+Amβ2mβ+α+μ2,which implies (20)ω4+ω2A1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22=0,where (21)A1=Amβ2α+μβm+α+μ,A2=Amβ2βm+α+μ2.Let z=ω2; then(22)z2+zA1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22=0.Without loss of generality, define (23)hz=z2+zA1+A2-α2-2A1μ-A2α-αμ-μ2-α2+A1μ-A2α-αμ-μ22-α2A1+A22.Lemma 4.
If(H4) holds, then (20) has a unique positive real root.Proof.
Under condition(H4), a simple calculation shows that (24)A1μ-A2α-αμ-μ22-α2A1+A22=1βm+α+μAmβ2-μμ+αmβ+α+μAmβ2μ+αmβ+μ+α2μ-αβm+α+μ-2αα+μ-μμ+α<0.Thus, (22) has a unique positive real root; namely, (25)z0=12-A1+A22+2μ+αA1-μ+A1+A22-2μ+αA1-μ2-4A1μ-A2α-αμ-μ22-α2A1+A22.Furthermore, we obtain ω0=z0. This completes the proof.Now, from (19), it is easy to show that (26)τj∗=1ω0arccos⁡A1+A2-A1μ+A2α+αμ+μ2+αω02αA1+A22+ω02+2jπ.Lemma 5 (see [18]).
Letλ(τ)=ϑ(τ)±iω(τ) be the root of (16) near τ=τ0∗ satisfying ϑ(τ0∗)=0, ω(τ0∗)=ω0. Suppose that h′(ω02)>0. Then ±iω0 is a pair of simple purely imaginary roots of (16). Moreover, the following transversality condition holds: (27)dReλτdττ=τ0∗,λ=iω0>0.
Applying Lemma3–Lemma 5, we have the following result.Theorem 6.
If(H1) and (H4) hold. Further suppose that h′(ω02)>0 satisfies. Then(i)
the positive equilibrium pointE∗ of system (3) is locally asymptotically stable when τ∈[0,τ0∗);(ii)
system (3) undergoes a Hopf bifurcation at E∗ when τ=τ0∗.


## 3. Properties of Hopf Bifurcation

In this section, for the positive equilibrium pointE∗ we will investigate the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions by using the theory of [19, 20].Denoteτ0∗ by τ∗ and introduce the new parameter γ=τ-τ∗. Normalizing the delay τ by the time scaling t→t/τ, then system (3) can be rewritten as(28)dUtdt=Lτ∗Ut+FUt,μ,where(29)Lγφ=μ-Amβ2βm+μ+α1βm+μ+α+1μ+αφ10-μ+αφ20Amβ2βm+μ+αμ+α-μφ10+αφ20-αφ2-τ,Fφ,γ=Lγφ+qφ,γ,qφ,γ=τ∗+γ-βφ10φ20+Amβ3βm+μ+α3φ120-Amβ4βm+α+μ4φ130βφ10φ20,for φ=(φ1,φ2)T∈C.The linearized system of (28) at (0,0) is(30)dUtdt=Lτ∗Ut.Based on the discussion in Section 2, we can easily know that for τ=τ∗ the characteristic equation of (30) has a pair of simple purely imaginary eigenvalues Λ0={iω0τ∗,-iω0τ∗}.LetC≔C([-1,0],R2). By the Riesz representation theorem, there exists a 2×2 matrix function η(θ,τ), -1≤θ≤0, whose elements are of bounded variation such that(31)Lτ∗φ=∫-10dηθ,τ∗φθ,forφ∈C.In fact, we can choose(32)ηθ,τ∗=τ∗-Amβ2βm+μ+α1βm+μ+α+1μ+α-α+μAmβ2βm+μ+αμ+α-μαδθ-τ∗000-αδθ+1,where δ(0)=1, δ(θ)=0(-1≤θ<0).LetA(τ∗) denote the infinitesimal generator of the semigroup induced by the solutions of (30) and let A∗ be the formal adjoint of A(τ∗) under the bilinear pairing(33)ψ,ϕ=ψ0,ϕ0-∫-10∫ξ=0θψξ-θdηθϕξdξ(34)=ψ0,ϕ0+τ∗∫-10ψθ+1000-αϕθdθ,for ϕ∈C, ψ∈C∗=C([0,1],R2). Then A(τ∗) and A∗ are a pair of adjoint operators. From the discussion in Section 2, we know that A(τ∗) has a pair of simple purely imaginary eigenvalues ±iω0τ∗ and they are also eigenvalues of A∗ since A(τ∗) and A∗ are a pair of adjoint operators. Let P and P∗ be the center spaces, that is, the generalized eigenspaces of A(τ∗) and A∗ associated with Λ0, respectively. Then P∗ is the adjoint space of P and dim⁡P=dim⁡P∗=2. In addition, some easy computations give the following result.Lemma 7.
Let(35)σ=-Amβ2+iω0+μmβ+α+μ2mβ+μ+α2μ+iω+αe-iω0τ∗,σ∗=Amβ2mβ+2α+μ+α+μβm+α+μ2βm+α+μAmβ2+α+μmβ+μ+α-μ-α+iω+αe-iω0τ∗.Then (36)p1θ=eiω0τ∗θ1,σT,p2θ=p1θ¯,-1≤θ≤0,is a basis of P associated with Λ0 and(37)q1s=1,σ∗e-iω0τ∗s,q2s=q1s¯,0≤s≤1,is a basis of Q associated with Λ0.
LetΦ=(Φ1,Φ2) and Ψ∗=(Ψ1∗,Ψ2∗)T, where (38)Φ1θ=p1θ+p2θ2=Reeiω0τ∗θReσeiω0τ∗θ=cos⁡ω0τ∗θ-cos⁡ω0τ∗θω0ω0-αsin⁡ω0τ∗+μμ+αcos⁡ω0τ∗-sin⁡ω0τ∗θω0μ+αcos⁡ω0τ∗-μω0-αsin⁡ω0τ∗μ+αcos⁡ω0τ∗2+ω0-αsin⁡ω0τ∗2,Φ2θ=p1θ-p2θ2i=Im⁡eiω0τ∗θIm⁡σeiω0τ∗θ=sin⁡ω0τ∗θ-sin⁡ω0τ∗θω0ω0-αsin⁡ω0τ∗+μμ+αcos⁡ω0τ∗+cos⁡ω0τ∗θω0μ+αcos⁡ω0τ∗-μω0-αsin⁡ω0τ∗μ+αcosω0τ∗2+ω0-αsinω0τ∗2for θ∈[-1,0], and (39)Ψ1∗s=q1s+q2s2=Ree-iω0τ∗sReσ∗e-iω0τ∗sT=cos⁡ω0τ∗s-B1cos⁡ω0τ∗s+B2sin⁡ω0τ∗smβ+μ+αAmβ2+μ+αmβ+μ+α-μ-α+αcos⁡ω0τ∗2+μ+α2mβ+α+μ2ω0-αsin⁡ω0τ∗2T,Ψ2∗s=q1s-q2s2i=Im⁡e-iω0τ∗sIm⁡σ∗e-iω0τ∗sT=-sin⁡ω0τ∗s-B2cos⁡ω0τ∗s-B1sin⁡ω0τ∗smβ+μ+αAmβ2+μ+αmβ+μ+α-μ-α+αcos⁡ω0τ∗2+μ+α2mβ+α+μ2ω0-αsin⁡ω0τ∗2T,for s∈[0,1], where(40)B1=Amβ2mβ+2μ+α+μ+α2mβ+α+μ2Amβ2+α+μβm+μ+α-μ-α+αcosω0τ∗+ω0μ+α∗βm+μ+α3ω0-αsinω0τ∗,B2=-μ+αβm+α+μω0-αsin⁡ω0τ∗Amβ2βm+2μ+2α+μ+α2βm+α+μ2+ω0μ+αmβ+α+μ2Amβ2+μ+αβm+μ+α-μ-α+αcos⁡ω0τ∗.From (33), we can obtain (Ψ1∗,Φ1) and (Ψ1∗,Φ2). Noting that (41)q1,p1=Ψ1∗,Φ1-Ψ2∗,Φ2+iΨ1∗,Φ2+Ψ2∗,Φ1,q1,p1=1+σσ∗-τ∗e-iω0τ∗ασσ∗≔D∗.Therefore, we can obtain (42)Ψ1∗,Φ1-Ψ2∗,Φ2=ReD∗,Ψ1∗,Φ2+Ψ2∗,Φ1=Im⁡D∗.Now, we define(Ψ∗,Φ)=(Ψj∗,Φk)(j,k=1,2) and construct a new basis ψ for Q by Ψ=(Ψ1,Ψ2)T=(Ψ∗,Φ)-1Ψ∗. Obviously, (Ψ,Φ)=I2, the second order identity matrix. In addition, define f=(ξ1,ξ2) and c·f=c1ξ1+c2ξ2 for c=(c1,c2)T, cj∈R(j=1,2). Then the center space of linear equation (30) is given by PCNC, where(43)PCNφ=ΦΨ,φ,f·f,φ∈C,and C=PCNC⊕PSC; here PSC denote the complementary subspace of PCNC.LetAτ∗ be defined by(44)Aτ∗φθ=φ˙θ+X0θLτ∗φθ-φ˙0,φ∈BC,where(45)X0θ=0,-1≤θ<0,I,θ=0.Then Aτ∗ is the infinitesimal generator induced by the solution of (30) and (28) can be rewritten as the following operator differential equation:(46)U˙t=Aτ∗Ut+X0FUt,μ.Using the decompositionC=PCNC⊕PSC and (43), the solution of (46) can be rewritten as(47)Ut=Φx1tx2t·f+hx1,x2,γ,where(48)x1tx2t=Ψ,Ut,f,and h(x1,x2,γ)∈Psc with h(0,0,0)=Dh(0,0,0)=0. In particular, the solution of (28) on the center manifold is given by (49)Ut∗=Φx1tx2t·f+hx1,x2,0.Settingz=x1-ix2 and noticing that p1=Φ1+iΦ2, then (49) can be rewritten as(50)Ut∗=12Φz+z¯iz-z¯·f+wz,z¯=12p1z+p1z¯·f+Wz,z¯,where W(z,z¯)=h((z+z¯)/2,-(z-z¯)/2i,0). Moreover, by [17], z satisfies (51)z˙=iω0τ∗z+gz,z¯,where(52)gz,z¯=Ψ10-iΨ20FUt∗,0,f.Let(53)Wz,z¯=W20z22+W11zz¯+W02z¯22+⋯,(54)gz,z¯=g20z22+g11zz¯+g02z¯22+⋯.According to (50) we have(55)FUt∗,0,f=τ∗4-βσ+mAβ3mβ+μ+α3z2+-βσ+2mAβ3mβ+μ+α3-βσ¯zz¯+mAβ3mβ+μ+α3-βσ¯z¯2βσz2+βσ+σ¯zz¯+βσ¯z¯2+τ∗4×-β2W1120+W2020+σ¯W2010+2σW1110+2mAβ3mβ+μ+α32W1110+W2010-3mAβ42mβ+α+μ4,1β2W1120+W2020+σ¯W2010+2σW1110,1z2z¯+⋯,where (56)Wijnθ,1=1π∫0πWijnθxdx,i+j=2,n=1,2,3.Let(ψ1,ψ2)=Ψ1(0)-iΨ2(0). Therefore, we obtain (57)g20=τ∗2-βσ+mAβ3mβ+μ+α3ψ1+βσψ2,g11=τ∗4-βσ-βσ¯+2mAβ3mβ+μ+α3ψ1+βσ+σ¯ψ2,g02=τ∗2-βσ¯+mAβ3mβ+μ+α3ψ1+βσ¯ψ2,g21=τ∗2-β2W1120+W2020+σ¯W2010+2σW1110+2mAβ3mβ+μ+α32W1110+W2010-3mAβ42mβ+α+μ4,1ψ1+β2W1120+W2020+σ¯W2010+2σW1110,1ψ2.SinceW20(θ), W11(θ) for θ∈[-1,0] appear in g21, we still need to compute them. It follows easily from (53) that (58)W˙z,z¯=W20zz˙+W11z˙z¯+zz˙¯+W02z¯z˙¯+⋯,Aτ∗W=Aτ∗W20z22+Aτ∗W11zz¯+Aτ∗W02z¯22+⋯.In addition, by [17], W(z(t),z¯(t)) satisfies(59)W˙=Aτ∗W+Hz,z¯,where(60)Hz,z¯=H20z22+H11zz¯+H02z¯22+⋯=X0FUt∗,0-ΦΨ,X0FUt∗,0,f·f,with Hij∈PSC, i+j=2.Thus, from (51) and (58)–(60), we can obtain (61)2iω0τ∗-Aτ∗W20=H20,-Aτ∗W11=H11.Noticing thatAτ∗ has only two eigenvalues ±iω0τ∗ with zero real parts, therefore, (59) has unique solution Wij(i+j=2) in PSC given by(62)W20=2iω0τ∗-Aτ∗-1H20,W11=-Aτ∗-1H11.From (60), we know that, for -1≤θ<0,(63)Hz,z¯=-ΦθΨ0FUt∗,0,f0·f0,=-p1θ+p2θ2,p1θ-p2θ2iΨ10Ψ20×FUt∗,0,f0·f0,=-12p1θΨ10-iΨ20+p2θΨ10+iΨ20×FUt∗,0,f0·f0,=-14g20p1θ+g¯02p2θz2·f0-12g11p1θ+g¯11p2θzz¯·f0+⋯.Therefore, for-1≤θ<0, (64)H20θ=-12g20p1θ+g¯02p2θ·f,H11θ=-12g11p1θ+g¯11p2θ·f,Hz,z¯0=FUt∗,0-ΦΨ,FUt∗,0,f·f,H200=τ∗2-βσ+mAβ3mβ+μ+α3βσ-12g20p10+g¯02p20·f,H110=τ∗4-βσ-βσ¯+2mAβ3mβ+μ+α3βσ+σ¯-12g11p10+g¯11p20·f.By the definition ofAτ∗, we get from (62) that(65)W˙20θ=2iω0τ∗W20θ+12g20p1θ+g¯02p2θ·f,-1≤θ<0.Noting that p1(θ)=p1(0)eiω0τ∗, -1≤θ≤0, (66)W20θ=i2g20ω0τ∗p1θ+g¯023ω0τ∗p2θ·f0+Ee2iω0τ∗θ,(67)E=W200-i2g20ω0τ∗p10+g¯023ω0τ∗p20·f.Using the definition ofAτ∗ and combining (62) and (67), we get(68)2iω0τ∗ig202ω0τ∗p10·f+ig¯026ω0τ∗p20·f+E-Lτ∗ig202ω0τ∗p1θ·f+ig¯026ω0τ∗p2θ·f+Ee2iω0τ∗θ=τ∗2-βσ+mAβ3mβ+α+μ3βσ-12g20p10+g¯02p20·f.From (69)Lτ∗p1θ·f=iω0τ∗p10·f,Lτ∗p2θ·f=-iω0τ∗p20·f,we get (70)2iω0τ∗E-Lτ∗Ee2iω0τ∗θ=τ∗2-βσ+mAβ3mβ+μ+α3βσ.From the above expression, we can see easily that (71)E=122iω0+Amβ2βm+μ+α1βm+μ+α+1μ+αα+μ-Amβ2βm+μ+αμ+α+μ2iω0-α+αe-2iω0τ∗-1×-βσ+mAβ3mβ+μ+α3βσ.By a similar way, we have (72)W˙11θ=12g11p1θ+g¯11p2θ·f,-1≤θ<0,W11θ=i2ω0τ∗g11p1θ-g¯11p2θ·f+F.Similar to the above, we can obtain(73)F=14Amβ2βm+μ+α1βm+μ+α+1μ+αα+μ-Amβ2βm+μ+αμ+α+μ0-1×-βσ-βσ¯+2mAβ3mβ+μ+α3βσ+βσ¯.Theorem 8.
System (3) has the followingPoincaré normal form: (74)ξ˙=iω0τ∗ξ+c10ξξ2+oξ5,where (75)c10=i2ω0τ∗g20g11-2g112-g0223+g212,so we can compute the following result: (76)σ2=-Rec10Reλ′τ∗,β2=2Rec10,T2=-Im⁡c10+σ2Im⁡λ′τ∗ω0τ∗,which determine the properties of bifurcating periodic solutions at the critical values τ∗; that is, σ2 determines the directions of the Hopf bifurcation: if σ2>0(σ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τ∗; β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions on the center manifold are stable (unstable), if β2<0(β2>0); and T2 determines the period of the bifurcating periodic solutions: the periodic increase (decrease), if T2>0(T2<0).


## 4. Numerical Solutions

Using MATLAB, numerical simulations are provided to substantiate the theoretical results established in the previous sections of this paper.Example 1.
ChooseA=0.5, m=0.9, β=0.7, α=0.4, μ=0.35, S0=0.75, I0=0.1, and R0=0.1 in system (1). A simple calculation shows that the boundary equilibrium point E1=(0.7600,0,0)T. Obviously, the conditions of Theorem 2 satisfy. It is easy to obtain that τ0=3.06. According to Theorem 2, the boundary equilibrium point E1 is locally asymptotically stable for τ=2.1,2.7,2.9<τ0, as shown in Figure 1. That is to say, at last there are not any rumor infected people and rumor recovered people except rumor susceptible people, and the rumor propagation is controlled effectively. Furthermore, from Figure 1 we can find that as τ increasing, system (1) takes more time to converge to the boundary equilibrium point E1. When τ=3.08>τ0, a spatially homogeneous periodic solution emerges from the boundary equilibrium point E1 (see Figure 2). That is, the boundary equilibrium point E1 is unstable, which seriously disrupts the normal social order.Figure 1
The boundary equilibrium pointE1 is locally asymptotically stable as τ=2.1,2.7,2.9<τ0. (a) The susceptible users; (b) the infected users; (c) the rational users.
(a)
(b)
(c)Figure 2
The periodic oscillation derived from Hopf bifurcation around the boundary equilibrium pointE1 occurs as τ=3.08>τ0. (a) The susceptible users; (b) the infected users; (c) the rational users.
(a)
(b)
(c)Example 2.
Consider system (1) with A=0.2, m=0.8, β=0.5, α=0.2, μ=0.1, S0=0.61, I0=0.18, and R0=0.36. It is easy to obtain that the positive equilibrium point of system (1) is E∗=(0.6000,0.1810,0.3619)T. Obviously, the conditions of Theorem 6 satisfy. By a simple calculation, we can obtain that τ0∗=4.2707. According to Theorem 6, the positive equilibrium point E∗ is locally asymptotically stable for τ=3<τ0∗, as shown in Figure 3. That is to say, the densities of the rumor susceptible people, the rumor infected people, and the rumor recovered people are controlled in fixed values, which is beneficial in controlling the rumor propagation. When τ=4.28>τ0∗, a spatially homogeneous periodic solution emerges from the positive equilibrium point E∗ (see Figure 4), which can lead to social instability and disrupt social harmony.Figure 3
The positive equilibrium pointE∗ is locally asymptotically stable as τ=3<τ0∗. (a) The susceptible users; (b) the infected users; (c) the rational users.
(a)
(b)
(c)Figure 4
The periodic oscillation derived from Hopf bifurcation around the positive equilibrium pointE∗ occurs as τ=4.28>τ0∗. (a) The susceptible users; (b) the infected users; (c) the rational users.
(a)
(b)
(c)


## 5. Conclusion

In this paper, we introduce a delay into rumor propagation in an emergency-affected environment and establish a delayed SIR rumor propagation model. The theoretical analysis and numerical simulation reveal that the discrete delay is responsible for the stability switch of the rumor propagation model, and a Hopf bifurcation occurs as the delay increases through a certain threshold. Furthermore, when the system is stable, asτ increasing, the system will take more time to converge to the equilibrium state. In future, we will further discuss rumor spreads in space and establish a temporal-spatial rumor propagation model based on the reaction diffusion equation.



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*Source: 269561-2015-10-04.xml*