Spatial dynamics and optimization method for a rumor propagation model in both homogeneous and heterogeneous environment

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https://doi.org/10.1007/s11071-021-06782-9

O R I G I NA L PA P E R

Spatial dynamics and optimization method for a rumor propagation model in both homogeneous and heterogeneous environment

Linhe Zhu · Xuewei Wang · Zhengdi Zhang · Chengxia Lei

Received: 7 October 2020 / Accepted: 16 April 2021 / Published online: 17 August 2021

Abstract Considering the influence of environmen- tal capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–

removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–

susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the nec- essary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

L. Zhu (

B

⁾ ^·^{X. Wang}^·^{Z. Zhang}

School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, People’s Republic of China e-mail: zlhnuaa@126.com

C. Lei

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China

Keywords Hopf bifurcation·Stability·Reaction–

diffusion system·Optimal control·Spatial heterogeneous environment

1 Introduction

Rumor is a statement that arouses public interest and is spread without any corresponding official basis to prove it. Nowadays, with the development of science and technology, online social networks emerge in an endless stream, such as Instagram, WeChat, Twitter and Facebook, which make it more convenient for people to share information. At the same time, rumors spread faster and more diversified. Because the information spread by rumor is untrue or even harmful, rumor will often lead to economic loss, social unrest and other adverse effects. Particularly in major events, deliber- ately spreading rumors will often lead to the deterio- ration of the situation. For example, during the spread of the atypical pneumonia caused by a novel coron- avirus (COVID-19) in China, many rumors cause public panic and even lead to the phenomenon of looting shuanghuanglian, which hinders the government from controlling the disease as soon as possible. Thus, to understand the mechanism of rumor spreading and then control rumor spreading have become the concern of scholars.

Because there are many similarities between the spread of rumor and the spread of infectious diseases, many epidemic models are applied to the analysis

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of rumor spread. In 1965, DK model, the first clas- sic model of rumor spreading, was put forward by Daley and Kendal [1]. In DK model, the population are divided into three groups, ignorant (people who do not notice the rumor), spreader (people who notice and spread the rumor) and stifler (people who notice but do not spread the rumor). On this basis, Maki and Thompson assumed that when a spreader contacted with other spreader, the spreader would stop spreading rumors and proposed MK model [2]. Until today, many researchers applied or improved the classical rumor propagation model, such as SI (susceptible–infected), SIR, and applied it in rumor spreading [3–10]. In Ref. [4], Qian improved the traditional SIR model with the concept of independent spreaders in complex networks. In view of the fact that rumors are spread in mul- tilingual environment, Wang established a SIR model with cross-transmitted mechanism [5]. Zhao changed SIR model to SIHR (susceptible–infected–hibernator–

removed) model by adding the connection between ignorants and stiflers. At the same time, the influence of forgetting rate and average degree on rumor- spreading scale was discussed [6]. Yao improved SIR model to SDILR (susceptible–dangerous–infective–

latent–recovered) model by considering the filtering function of social media for rumors [10]. In addition, many workers have also established new models based on specific situations in real life [8,11–14]. For example, Tian assumed that ignorants had three different atti- tudes when facing rumors, and thus, they established a new SDILR model [8]. In Ref. [13], Liu divided the population into ignorant, lurker, spreader, removal and established the ILSR model.

With the deepening of research and the rapid development of network media, researchers find that the traditional model relying on ordinary differential equation cannot reflect the influence of space on rumor spreading. Therefore, many scholars begin to consider whether they could use partial differential equation to study the spread of rumors [15–18]. Considering the effect of space-time diffusion, Zhu proposed a rumor propagation model with uncertainty [15]. Guo discussed a kind of reaction–diffusion model with nonlocal delay effect and nonlinear boundary conditions [16]. Similarly, in the field of infectious diseases, more in-depth research is continuing [19–24].

However, the above discussions are all carried out in homogeneous environment; that is to say, the parameters corresponding to each point in the space are the

same. Obviously, such a model cannot fully reflect the actual situation of rumor propagation. Therefore, many researchers try to improve it to spatial heterogeneous environment and analyze it. In the field of infectious diseases, there have been some achieve- ments [25–28]. In Ref. [25], Lei improved SIR ordinary differential equation model, which was studied in Ref. [26], to the partial differential equation model in the heterogeneous space, and discussed the asymptotic behavior of the positive equilibrium point when the diffusion rate tended to zero. Zhang put forward a reaction–diffusion epidemic model–SVIR (susceptible–vaccinated–infected–removed) in heterogeneous environment and found that the diffusion rate of disease in heterogeneous space was higher than that in homogeneous space [27]. At present, it is rare to consider the spatial and temporal propagation model of rumors in heterogeneous environments.

In addition to the dynamic behavior of the model, the optimal control of the model is also an enduring topic.

Scholars in various fields have studied the optimal control problem for different models [29–33]. Gashirai applied the optimal control theory to the FMD trans- mission model and analyzed how to effectively control the disease [31]. Based on MK rumor model, Kand- hway discussed how to spread as much information as possible with limited funding [32]. In Ref. [33], Huo proved the existence of the optimal control of rumor propagation model with media reports by using the Pontryagin’s maximum principle. It can be seen that the study of optimal control has high practical value, in particular, the optimal control of spatiotemporal propagation model.

In the past, the growth rate of population was often recorded as a constant in the establishment of rumor propagation model or epidemic model. However, some researchers suggest that the capacity of rumor spreading system is not infinite due to the restriction of technology and the capacity of the real world for human beings is limited. Therefore, it is more realistic to express the population growth with the logistic equation, which is influenced by the internal growth rate and certain carrying capacity [34–36]. At the same time, many practical factors, such as people’s thinking and judgment based on their own experience and knowl- edge when they first contact with rumors, and people don’t necessarily review the news as soon as they receive it, will make the rumor communication system need to be added with time delay [12]. Similarly, there

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is usually a time interval between people’s exposure to rumors and spreading them. Based on the above two aspects, we will consider the influence of time delay and logistic equation when establishing rumor propagation model.

In this paper, we will improve the traditional SIR rumor propagation model to the following form

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

∂S

∂t −dSS =r(x)S

1− S K(x)

−β(x)S I−μ1(x)S+γ (x)I, t >0, x∈,

∂I

∂t −dII =β(x)S I −δ(x)I −μ2(x)I−γ (x)I, t >0, x∈,

∂R

∂t −dRR=δ(x)I−μ3(x)R, t >0, x∈,

∂S

∂ν = ∂I

∂ν =∂R

∂ν =0, t >0, x∈∂,

S(0,x)=S0(x)≥0,I(0,x)=I0(x)≥,≡0,R(0,x)=R0(x)≥0, x∈.

(1.1)

where S(t,x) denotes the rumor–susceptible who have not been exposed to rumors and may become a rumor–infector, I(t,x)denotes the rumor–infector who believe and spread rumors with a certain probability, and R(t,x)denotes the removal who already know the rumor is false and will not pay any more attention to it. We assume that the network exit rates of all three groups areμ1(x), μ2(x)andμ3(x). The diffusion coefficients dS, dI and dR represent the migra- tion rate of rumor–susceptible individuals, the migra- tion rate of rumor–infector individuals and the rate of movement of removal, respectively.r(x)represents the internal growth rate of population, andK(x)represents the maximum capacity of network environment. The probability that a rumor–susceptible becomes a rumor–

infector due to contact with rumor is expressed asβ(x). γ (x)andδ(x)indicate the probability that the rumor–

infector becomes a rumor–susceptible due to forgetting and a remover due to identifying the false information, respectively. The habitat ⊂ R^N (N ≥ 1)is a bounded area, and its boundary∂is smooth. The Neumann boundary condition means that there is no

population flow on the boundary∂. We assume that μ1(x),μ2(x),μ3(x),r(x),K(x),β(x),γ (x)andδ(x) are positive and H¨older continuous functions on.

Since the first two equations of system (1.1) are independent ofR(t,x), we can simplify it to

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂S

∂t −dSS=r(x)S

1− S K(x)

−β(x)S I −μ1(x)S+γ (x)I,t >0,x∈,

∂I

∂t −dII =β(x)S I −δ(x)I−μ2(x)I−γ (x)I, t >0,x∈,

∂S(t,x)

∂ν = ∂I(t,x)

∂ν =0, t >0,x∈∂,

S(0,x)=S0(x)≥0,I(0,x)=I0(x)≥,≡0, x∈.

(1.2)

In the following, we will study the dynamic behavior of system (1.2). Obviously, according to Ref. [37],

by applying the strong maximum principle and Hopf lemma for the elliptic equations corresponding to system (1.2), we know thatS(x)≥0 andI(x)≥0 for all x∈ ¯.

The innovation of this paper includes the following three points. Firstly, considering the difference of loca- tion parameters, we establish a heterogeneous spatial model, which is different from the traditional rumor- spreading process in a homogeneous environment [17].

Secondly, we study the spatial nonhomogeneous bifurcation caused by time delay. To some extent, our results have improved the phenomenon of space homogeneous bifurcation [18]. Finally, through numerical simulations, our work describes the propagation process from the perspective of heterogeneous space and finds out the influence of the difference of spatial diffusion ability on information propagation in heterogeneous environment. The numerical results confirm the necessity of the study of spatial propagation.

The paper is organized as follows. In Sect. 2, we establish a spatially heterogeneous model with logistic growth and discuss its consistent persistence and the asymptotic behavior of equilibrium point. In Sect.3, we degenerate the model into a spatial homogeneous

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model and study the local and global stability. In addition, we use the maximum principle to study the optimal control of the model. In Sect.4, considering the effect of time delay in real life, we research Hopf bifurcation of our model with time delay [38]. In order to verify our conclusions and discuss the effect of diffusion coefficients, we give some simulation results in Sect.5. A brief conclusion is given in Sect.6.

2 Spatially heterogeneous model

In this section, we always assume thatr(x) > μ1(x) in. For any given nonnegative and continuous initial value(S0,I0), system (1.2) has unique classical solution(S(t,x),I(t,x))for allt >0 andx ∈ by the standard theory for parabolic equations. In addition, (S(t,x),I(t,x)) > (0,0)fort >0,x ∈ provided thatI0(x)≥,≡0.

2.1 Uniform persistence

In this subsection, we will study the uniform persistence of the solution for system (1.2). For convenience, we define

f^∗=max

x∈ f(x), f_∗=min

x∈ f(x),

where f(x) = μ1(x), μ2(x), μ3(x),r(x),K(x), β(x), γ (x) andδ(x)forx∈.

Proposition 2.1 There exists a positive constant Cˆ depending on initial data such that the solution(S,I) of (1.2) satisfies

||S(·,x)||L∞()+ ||I(·,x)||L∞()≤ ˆC, t≥0.

(2.1) Furthermore, there exists a positive constantC inde-˜ pendent of initial data such that

||S(·,x)||L^∞()+ ||I(·,x)||L^∞()≤ ˜C, t≥T, (2.2) for some large time T >0.

Proof By the first equation of (1.2), it can be calculated directly that

St −dsS=r(x)S

1− S K(x)

−β(x)S I

−μ1(x)S+γ (x)I

≤ r^∗K^∗

4 +γ (x)I− [β(x)I +μ1(x)]S

≤ [β(x)I +μ1(x)]

max

γ^∗ β∗, r^∗K^∗

4(μ1)∗ −S

for allx∈,t>0.

DenoteM1=max γ^∗

β_∗, r^∗K^∗

4(μ1)_∗ , then the following parabolic problem

⎧⎪

⎨

⎪⎩

wt−d_Sw=[β(x)I+μ1(x)](M₁−w),t>0, x∈,

∂w

∂ν =0, t>0, x∈∂,

w(0,x)=S₀(x), x∈

(2.3) has a unique solutionw. We apply the comparison principle to conclude that

S(t,x)≤w(t,x)≤max{M1,max

x∈S0(x)}, t

>0, x∈. (2.4)

It is easy to see thatv(t)= M1+max

x∈e^−(μ¹⁾^∗^t is the supersolution of (2.3). Thus, we have

S(t,x)≤w(t,x)≤v(t), t >0, x∈. Therefore, we obtain

tlim→∞sup

x∈

S(t,x)≤M1.

We can find a largeT1>0 such that

S(t,x)≤2M1, t>T1, x∈. (2.5)

Setμ=min{(μ1)_∗, (μ2)_∗+δ_∗}, we define W(t)=

[S(t,x)+I(t,x)]dx,

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then dW(t)

dt =

r(x)S

1− S

K(x)

−μ2(x)I

−μ1(x)S−δ(x)I

dx

≤

r^∗K^∗

4 dx−μW(t)

≤ r^∗K^∗||

4 −μW(t).

Thus, we have

W(t)≤W(0)e^−μ^t+r^∗K^∗||

4μ (1−e^−μ^t),∀t ≥0.

That is,

[S(t,x)+I(t,x)]dx

≤

[S0(x)+I0(x)] dxe^−μ^t +r^∗K^∗||

4μ

1−e^−μ^t

,∀t≥0. (2.6)

In view of (2.4) and the last two equations of system (1.1), we use Theorem 1.1 of [39] (or Lemma 2.1 in [40]) to deduce that there exists a positive constantM2

depending on the initial data such that

I(t,x)≤M2fort ≥0,x ∈. (2.7) By (2.6), we have

lim sup

t→∞

I(t,x)≤lim sup

t→∞ W(t)≤ r^∗K^∗||

4μ . We apply Lemma 2.1 of [40] to conclude that there exists a constant M3 > 0 independent of initial data such that

I(t,x)≤M3fort ≥T2,x∈ (2.8) for someT2>T1.

SetCˆ =max{max{M1,max

S0(x)},M2}andC˜ = max{2M1,M3}. Due to (2.4) and (2.7), we obtain (2.1).

It follows from (2.5) and (2.8) that (2.2) holds.

DefineR0as follows R0= sup₀_=ϕ∈_H¹₍₎

β(x)S˜ϕ²dx

d_I|∇ϕ|²+(δ(x)+μ2(x)+γ (x))ϕ²dx , whereS˜is the unique positive solution of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−dSu=r(x)u

1− u K(x)

−μ1(x)u,t>0,x∈,

∂u

∂ν =0, t>0,x∈∂,

u(0,x)=S₀(x), x∈. (2.9) Lemma 2.1 The following properties of R0hold.

(i) R0 is a monotone decreasing function of dI. R0 → max

x∈

β(x)S˜(x)

δ(x)+μ2(x)+γ (x) as dI → 0, R0 → max

x∈

β(x)S˜(x)dx

[δ(x)+μ2(x)+γ (x)]dx asdI → ∞;

(ii) 1−R0has the same sign asλ1, whereλ1is the prin- cipal eigenvalue of the following eigenvalue prob- lem

⎧⎨

⎩

dI+ [β(x)S(x)˜ −(δ(x)+μ2(x)+γ (x))]+λ=0,x∈,

∂

∂ν =0, x∈∂. (2.10)

Theorem 2.1 For the given initial data(S0,I0), let (S,I)be the unique solution of (1.2). If R0>1, then system (1.2) is uniformly persistent: There exists a con- stant >0such that

lim inf

t→∞ S(t,x)≥, lim inf

t→∞ I(t,x)

≥uniformly for x ∈. (2.11) In particular, this implies that system (1.2) admits at least one positive equilibrium solution.

Proof Let X₊:=C

,R²₊ , X⁰₊

:= {ς=(u, v)∈X₊:v≡0}

and∂X⁰₊:=X₊\X⁰₊= {ς ∈X₊:v≡0}.

For a givenς ∈X₊, system (1.2) generates a semi- flow, denoted by(t), and

[(t)ς](x)=(S(t,x, ς),I(t,x, ς)),

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where(S(t,x, ς),I(t,x, ς))is the unique solution to system (1.2) with(S0,I0)=ς.

Next, we will claim that (S,˜ 0) attracts ς = (S0,0) ∈ ∂X⁰₊. As I0 ≡ 0, the unique solution (S(t,x, ς),I(t,x, ς))satisfies I(t,x, ς) ≡ 0 for all t≥0 andSsolves

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

St−dSS=r(x)S

1− S K(x)

−μ1(x)S,t>0, x∈,

∂S

∂ν =0, t>0, x∈∂,

S(0,x)=S0(x), x∈.

(2.12) It can be easily proved that

tlim→∞S(t,x)= ˜S(x)uniformly on. This proves our claim.

AsR0>1, it follows from Lemma2.1thatλ1<0.

Then, we can conclude that there exists a constant 0<

0<−λ1such that lim sup

t→∞ ||(t)ς−ς0|| ≥0

for anyς∈X⁰₊andς0=(S,˜ 0).

Suppose that lim sup_t_→∞||(t)ς −ς0||< 0for someς ∈ X⁰₊. For the given0, we can findT > 0 such that

S˜(x)−0<S(t,x) <S˜(x)+0, 0<I < 0

fort ≥T andx∈. Then, we have

⎧⎪

⎨

⎪⎩

∂I

∂t ≥dII +β(x)[ ˜S(x)−0] −(δ(x)+μ2(x)+γ (x))I,t >T, x∈,

∂I

∂ν =0, t >T, x∈∂.

We apply the comparison principle to deduce thatI is an upper solution of the following parabolic problem

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∂v

∂t =dIv+β(x)[ ˜S(x)−0] −(δ(x)+μ2(x)+γ (x))v,t >T, x∈,

∂v

∂ν =0, t >T, x∈∂,

v(T,x)=I(T,x) >0, x∈.

(2.13)

Denote by1the eigenfunction corresponding toλ1

which is positive on. Then, we can find a small positive constantσsuch thatσe^−(λ¹⁺⁰⁾^T1(x)≤I(T,x)

on. In view of the comparison principle, we obtain that σe^−(λ¹⁺⁰⁾^t1(x) is a lower solution of (2.13).

Thus,

I(t,x)≥σe^−(λ¹⁺⁰⁾^t1(x)→ ∞ uniformly onast → ∞.

This contradicts Proposition2.1.

Similar to [41], we can apply [42,61] to find that there is a real numberδ0>0, which is independent of initial data(S0,I0,R0), satisfying

lim inf

t→∞ min

x∈I(t,x, ς)≥δ0, ∀ς∈X⁰₊. (2.14) In other words, there exists a largeT3>0 such that I(t,x, ς)≥δ0/2, t ≥T3, x∈

for allς∈X⁰₊.

By (2.5) and (2.8), there is a sufficiently largeT4>

max{T1,T2,T3}, S_t−dSS=r(x)S− r(x)

K(x)S²−β(x)S I−μ1(x)S+γ (x)I

≥γ∗δ0

2 −β^∗M₃S−2r^∗

K_∗M₂S−(μ1)^∗S fort >T4,x ∈ .By the comparison principle, we can get

lim inf

t→∞ S(t,x, ς)≥ ^γ^∗^δ

0

2

β^∗M3+2r^∗

K_∗M2+(μ1)^∗ δ1uniformly forx∈. (2.15)

Let=min{δ0, δ1}. As (2.14) and (2.15), we obtain the assertion (2.11). In addition, this implies that system

(1.2) admits at least one positive equilibrium solution

E(S(x),I(x)).

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Remark 2.1 The uniform persistence of system (1.2) means that rumors persist for a long time. In addition, under the condition ofS0(x) > 0 andI0(x) >0, the rumor continues uniformly fort≥0 andx∈.

2.2 Asymptotic behavior ofE

In this subsection, we will discuss the asymptotic behavior of the positive equilibriumEbased onRe f s.

[25,43,44].

Corresponding to (1.2), the equilibrium problem satisfies the following elliptic system:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

−dSS =r(x)S

1− S K(x)

−β(x)S I−μ1(x)S+γ (x)I,x∈,

−dII =β(x)S I −δ(x)I−μ2(x)I −γ (x)I, x∈,

∂S

∂ν = ∂I

∂ν =0, x∈∂.

(2.16)

If(S,I)is the solution of (2.16),S ≥0 andI ≥≡ 0, then we use the strong maximum principle and Hopf lemma for system (2.16) to conclude thatS(x) >0 and I(x) >0 for allx∈ ¯.

Now, we recall some known facts, see, for instance, [45] and [46].

Lemma 2.2 Assume thatω∈C²()and∂_νω=0on

∂, then the following assertions hold.

(1) If ω has a local maximum at x1 ∈ , then ω(x1)=0andω(x1)≤0.

(2) If ω has a local maximum at x2 ∈ , then ω(x2)=0andω(x2)≥0.

Lemma 2.3 Letω ∈ C²()∩C¹() be a positive solution to the elliptic equation

ω+c(x)ω=0, in, ∂νω=0, on∂,

where c∈C(). Then, there exists a positive constant M which depends only on C where||c||_∞ ≤ C such that

max ω≤ Mmin

ω.

First, we will study the dynamic behavior of system (1.2) as dS →0. Let dS→0; we apply ([47], Lemma 3.2) to deduce that the unique solutionS˜of (2.9) satisfies

S(x)˜ → K(x)(r(x)−μ1(x))

r(x) uniformly on, then the principal eigenvalueλ1of problem (2.10) converges to the principal eigenvalueλ⁰₁of the following eigenvalue problem

⎧⎪

⎨

⎪⎩

dIφ+

β(x)K(x)(r(x)−μ1(x))

r(x) −(δ(x)+μ2(x)+γ (x))

φ+λφ=0,x∈,

∂φ

∂ν =0, x∈∂.

(2.17)

Theorem 2.2 For fixed dI > 0, and assume that λ⁰₁<0. LetdS→0. Then, any positive solution(S,I) of (2.16) satisfies (up to a subsequence of dS → 0) (S,I)→(S⁰,I⁰)uniformly on, where

S⁰=H(x,I⁰):= K(x)(r(x)−μ1(x)−β(x)I⁰(x))+

K²(x)(β(x)I⁰(x)+μ1(x)−r(x))²+4r(x)γ (x)I⁰(x)K(x) 2r(x)

and I⁰is a positive solution of

⎧⎨

⎩

−dII⁰=β(x)H(x,I⁰)I⁰−δ(x)I⁰−μ2(x)I⁰−γ (x)I⁰,x∈,

∂I⁰

∂ν =0, x∈∂, (2.18)

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Proof Step 1:There exists a constantC>0, independent of 0<dS≤1, such that

1

C ≤S(x), I(x)≤C. (2.19)

Let S(x1) = max

x∈S(x) for some x1 ∈ .p. As Lemma2.2, we haveS(x1)≤0. For the first equation of system (2.16), we can know

r(x1)S(x1)

1− S(x1) K(x1)

−β (x1)S(x1)I(x1)

−μ1(x1)S(x1)+γ (x1)I(x1)≥0.

It follows that r(x1)K(x1)

4 ≥ μ1(x1)S(x1)+ [β(x1)S(x1)

−γ (x1)]I(x1).

Ifβ(x1)S(x1)−γ (x1) <0, we obtain S(x)≤max

x∈S(x)=S(x1) < γ (x1) β(x1) ≤ γ^∗

β∗. Ifβ(x1)S(x1)−γ (x1)≥0, we obtain S(x)≤max

x∈S(x)=S(x1)≤r(x1)K(x1)

4·μ1(x1) ≤ r^∗K^∗ 4(μ1)_∗. Therefore, we can get

S(x)≤max γ^∗

β_∗, r^∗K^∗

4(μ1)_∗ C1. (2.20) DefineW =dSS+dII, then we obtain

−W = −dSS−dII =r(x)S

1− S K(x)

−μ1(x)S−μ2(x)I −δ(x)I.

LetW(x2)=max

x∈W(x)forx2∈, we use Lemma2.2 to getW2(x0)≤0. Thus,

μ (S(x2)+I(x2))≤μ1(x2)S(x2)+μ2(x2)I +δ (x2)I(x2)

≤r(x2)S(x2)

1− S(x2) K(x2)

forμ=min{(μ1)∗, (μ2)∗+δ∗}. By the further calculation, we can get

S(x2)+I(x2)≤ r(x2)K(x2)

4μ ≤ r^∗K^∗ 4μ C.˜ Furthermore,

W(x)≤W(x2)≤ max{dS,dI}[S(x2)

+I(x2)]≤max{dS,dI} ˜C. Then,

dImax

x∈I(x)≤max

x∈W(x)=W(x2)≤max{dS,dI} ˜C. That is,

max

x∈I(x)≤ max{dS,dI} dI

C˜ C2.

Now, we will give the positive lower bound for the component I. In view of (2.20), we use the Harnack inequality (Lemma2.3)

−dII= [β(x)S−δ(x)−μ2(x)−γ (x)]I, x∈,

∂I

∂ν =0, x∈∂

(2.21) to obtain

max

x∈I(x)≤ Mmin

I(x), (2.22)

where the constantM >0 independent of dS. Suppose that I has no positive lower bound, we can find a sequence {dSn} satisfying dSn → 0 as n → ∞ and the corresponding positive solution sequence(Sn,In)of (2.16) with dS = dSn, such that min_x_∈In(x)→0 asn → ∞. In view of (2.22), we deduce that

In→0 uniformly on, asn→ ∞.

By an analogous consideration (Lemma 3.2 [47]), we have

Sn(x) → K(x)(r(x)−μ1(x))

r(x) uniformly on, asn

→ ∞.

On the other hand, asλⁿ₁(forn≥1) is the principle eigenvalue of

dI+[β(x)Sn−δ(x)−μ2(x)−γ (x)]+λ=0,x∈,

∂ν=0, x∈∂.

(2.23) By (2.21), we haveλⁿ₁ = 0. It is easy to see that the sequence of{λⁿ₁}converges toλ⁰₁<0, which is contra- diction withλ⁰₁=0. So we can draw a conclusion that Ihas a positive lower boundC3, which not depends on 0<dS ≤1.

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SetS(x3)=min

x∈S(x)forx3∈, thenS(x3)≥0 by Lemma2.2. We use the first equation of (2.16) to get

γ (x3)I(x3)≤ r(x3)S²(x3)

K(x3) +β (x3)S(x3)I(x3) +μ1(x3)S(x3) ,

that is to say, r^∗

K_∗S²(x₃)+

β^∗C₂+(μ1)^∗

S(x₃)≥γ (x₃)I(x₃)≥γ_∗C₃. It is not difficult to find

S(x)≥min

x∈S(x)≥ −K_∗

C2β^∗+(μ1)^∗ +

K_∗²

C2β^∗+(μ1)^∗2

+4K_∗γ∗r^∗C3

2r^∗ C4. (2.24)

In conclusion, we have proved that (2.19) holds.

Step: 2We study the convergence of Ias dS →0.

Iis the solution of the following system −dII+

δ(x)+μ2(x)+γ (x)

I =β(x)S I,x∈,

∂vI =0, x∈∂.

(2.25)

In view of (2.19)and the standardL^ptheory ([48]), we obtain that

I W^2,p()≤CβS I L^p()≤C.

Then, we apply the Sobolev embedding theorem (see, e.g.,Re f s.[25,43,44]) to obtain

I _C1+α()≤C.

Hence, there exists a subsequence of dS → 0 rep- resented by dn :=dSn, satisfying dn →0 asn → ∞.

And the corresponding positive solution (Sn,In) of (2.16) with dS=dSn, which satisfies that

In→ I⁰inC¹

asn → ∞, (2.26)

whereI⁰>0.

Step: 3We will prove the convergence ofS. For any >0, we use (2.26) to find a constantN >0 such that 0<I⁰−≤In≤ I⁰+on

forn≥ N.

For fixedn≥N, we know thatSnis the supersolution of

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

−dnU =r(x)U

1− U

K(x)

−β(x)(I⁰+)U−μ1(x)U+γ (x)(I⁰−),x∈,

= r(x)[F⁺(x,I⁰(x))−U][U−F⁻(x,I⁰(x))]

K(x) , x∈,

∂vU=0, x∈∂,

(2.27)

and a subsolution of

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

−dnV =r(x)V

1− V K(x)

−β(x)(I⁰−)V−μ1(x)V +γ (x)(I⁰+),x∈,

= r(x)[G⁺(x,I⁰(x))−V][V −G⁻(x,I⁰(x))]

K(x) , x∈,

∂vV =0, x∈∂,

(2.28)

where

F^± x,I⁰(x)

= K(x)

r(x)−β(x)(I⁰(x)+)−μ1(x)

±

K²(x)[r(x)−β(x)(I⁰(x)+)−μ1(x)]²+4K(x)r(x)γ (x)(I⁰(x)−) 2r(x)

and

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G^±

x,I⁰(x)

= K(x)

r(x)−β(x)(I⁰(x)−)−μ1(x)

±

K²(x)[r(x)−β(x)(I⁰(x)−)−μ1(x)]²+4K(x)r(x)γ (x)(I⁰(x)+)

2r(x) .

By the proof of [25,43], we can conclude that system (2.27) and system (2.28) have a unique positive solution Un and Vn, respectively. Then, we use the proof of Lemma 2.4 (in Ref. [49]) to conclude that

Un→ F⁺

x,I⁰(x)

, Vn →G⁺ x,I⁰(x)

uniformly on, asn →0.

In fact,Snis a supersolution for the problem (2.27) and is a subsolution for the problem (2.28), and we have Un≤ Sn≤Vnonforn >N. Thus,

F⁺

x,I⁰(x)

≤lim inf

n→∞ Sn(x)≤lim sup

n→∞ Sn(x)

≤G⁺

x,I⁰(x)

uniformly on. As >0 can be choose arbitrarily small, it is easy to show that

Sn→S⁰uniformlyon,asn→ ∞, where

S⁰=H(x,I⁰(x))= K(x)[r(x)−β(x)I⁰(x)−μ1(x)] +

K²(x)[r(x)−β(x)I⁰(x)−μ1(x)]²+4K(x)r(x)γ (x)I⁰(x)

2r(x) .

This completes the proof.

Theorem 2.3 For fixed dS > 0, assume that

x ∈ β(x)S(x) > δ(x)˜ +μ2(x)+γ (x) is nonempty.

LetdI →0. Then, any positive solution(S,I)of (2.16) satisfies (up to a subsequence ofdI →0) that S→S^∗ in L¹()with||S^∗||L¹()positive,

Idx → I^∗with I^∗is a positive constant.

3 Spatially homogeneous model

In this section, we assume that all the parameters dS,dI,r,K, δ, μ1, μ2, β, γ in system (1.2) are positive constant.

3.1 The existence of equilibrium points

Firstly, we study the existence of rumor-eliminating equilibrium pointE0and rumor-spreading equilibrium

point E_∗. Through a simple calculation, we can get E0=(S0,0)=

K(1−μ1

r ),0

. Apparently, when μ1<r, rumor-eliminating equilibrium pointE0exists.

Next, we use the next generation matrix to calculate the basic reproduction number R0[50] and try to find the relationship betweenR0and the existence of rumor- spreading equilibrium pointE_∗.

Theorem 3.1 If R0>1, system(1.2)has the rumor- spreading equilibrium point E_∗ = (S_∗,I_∗) = δ+μ2+γ

β ,r(δ+μ2+γ )²

β²(δ+μ2)K (R0−1)

.

Proof Let the equations of system (1.2) be equal to zero, we get

E_∗=(S_∗,I_∗)=

δ+μ2+γ

β ,δ+μ2+γ

β (δ+μ2) (r−μ1)

−r(δ+μ2+γ )² Kβ²(δ+μ2)

.

According to system (1.2), we can obtain the following two matrices

F=βS and

V=δ+μ2+γ.

Through a direct calculation, we can get that R0=FV⁻¹(E0)= βK(r−μ1)

r(δ+μ2+γ ) andE_∗=(S_∗,I_∗)=

δ+μ2+γ

β ,r(δ+μ2+γ )² β²(δ+μ2)K (R0−1)

. So we can get the conclusion that there

existsE_∗whenR0>1.

Since the stability of the equilibrium point is directly related to the spread and control of rumors, we first discuss the local stability and global stability of the equilibrium pointsE0andE_∗through the linearization technique, Hurwitz theorem and Lyapunov function.

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3.2 The local and global stability analysis ofE0and E_∗

In this section, we will analyze the local stability and global stability ofE0andE_∗.

Theorem 3.2 If 0 < R0 < 1 holds, the rumor- eliminating equilibrium point E0is locally asymptoti- cally stable.

Proof The Jacobian matrix of system (1.2) atE0is

J(E0)=

⎛

⎝ μ1−r −βK(1−μ1

r )+γ 0 βK(1−μ1

r )−δ−μ2−γ

⎞

⎠,

(3.1) and the characteristic equation becomes

λ+dSπ²k²

L² −μ1+r

[λ +dIπ²k²

L² −βK

1−μ1

r

+δ+μ2+γ

=0.

Clearly, λ1= −dSπ²k²

L² +μ1−r, λ2= −dIπ²k²

L² +βK

1−μ1

r

−δ−μ2−γ = −dIπ²k² L² +(δ+μ2+γ ) (R0−1);

assume 0< R0<1, thenλ1 <0 andλ2<0. There- fore,E0is locally asymptotically stable.

Theorem 3.3 If 2 > R0 > 1 holds, the rumor- spreading equilibrium point E_∗ is locally asymptoti- cally stable.

Proof The Jacobian matrix of system (1.2) atE_∗is J(E_∗)

=

⎛

⎝ r−2r S_∗

K −βI_∗−μ1 −βS_∗+γ βI_∗ βS_∗−δ−μ2−γ

⎞

⎠, (3.2)

and the characteristic equation is

λ²+

dSπ²k²

L² −βS_∗+δ+μ2+γ+dIπ²k² L² −r +2r S_∗

K +βI_∗+μ1

λ+dIdSπ⁴k⁴ L⁴

−dSπ²k² L² βS_∗ +(δ+μ2+γ )dSπ²k²

L² −rdIπ²k² L² −rβS_∗

−r(δ+μ2+γ ) +2r S_∗

K dIπ²k²

L² −2r S_∗ K βS_∗ +2r S_∗

K (δ+μ2+γ )+dIπ²k² L² βI_∗ +βI_∗(δ+μ2)

+μ1dIπ²k²

L² −μ1βS_∗ +μ1(δ+μ2+γ )=0.

(3.3) Assumeλ1 andλ2 are the roots of the characteristic equation, then

λ1+λ2= −dSπ²k²

L² −dIπ²k² L² +βS_∗

−(δ+μ2+γ )+r−2r S_∗

K −βI_∗−μ1

= −dSπ²k²

L² −dIπ²k²

L² +(r−μ1)

− 2r

Kβ (δ+μ2+γ )

−r(δ+μ2+γ )²

Kβ (δ+μ2) (R0−1)

= −dSπ²k²

L² −dIπ²k² L² +r(δ+μ2+γ )

Kβ (R0−2)

−r(δ+μ2+γ )²

Kβ (δ+μ2) (R0−1) , (3.4) λ1λ2=dIπ²k²

L²

dSπ²k² L² −r +2r S_∗

K +βI_∗+μ1

+βI_∗(δ+μ2)