Modeling the dynamics of information propagation in the temporal and spatial environment

Yi Zhang and Linhe Zhu

School of Mathematical Sciences,Jiangsu University,Zhenjiang,212013,China

Abstract In this paper,we try to establish a non-smooth susceptible–infected–recovered (SIR) rumor propagation model based on time and space dimensions.First of all,we prove the existence and uniqueness of the solution.Secondly,we divide the system into two parts and discuss the existence of equilibrium points for each of them.For the left part,we define R0 to study the relationship between R0 and the existence of equilibrium points.For the right part,we classify many different cases by discussing the coefficients of the equilibrium point equation.Then,on this basis,we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation.Further,we consider the effect of diffusion.We specifically analyze the stability of equilibrium points.In addition,we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points.According to the Lyapunov number,we also determine the direction of the bifurcation.When I=Ic,we discuss conditions for the existence of discontinuous Hopf bifurcation.Finally,through numerical simulations and combined with the practical meaning of the parameters,we prove the correctness of the previous theoretical theorem.

Keywords: non-smooth system,rumor propagation,Turing instability,Hopf bifurcation

1.Introduction

Rumors are unverified pieces of information that lack authenticity and have been extensively disseminated through various communication channels.In the social realm,rumors possess a significant capacity to mislead,capturing people’s attention to a greater extent than the actual events they concern.Many rumors have a great impact on social stability as they spread in a wider area,and most of these effects are negative;for example,in 2013,the salt looting that followed the nuclear plant leak in Japan was misguided by rumors.At the same time,with the development of science and technology,the rapid development of the Internet media and its widespread impact on Internet users,the influence of rumors has reached an unprecedented level.In order to predict the development of rumor propagation and control the harm of rumor propagation to society,more and more scholars have begun to be engaged in the research of spreading rumors.

Following observations and research on rumor propagation,the prevailing approach involves constructing mathematical models to simulate the process of rumor dissemination.By studying the dynamic characteristics of these mathematical models,researchers aim to predict the tendencies of rumor propagation.Before rumor propagation,infectious disease models we a hot topic of research for many scholars.It has been found that rumor propagation and infectious disease propagation have very similar transmission mechanisms.Therefore,many mathematical models of infectious disease propagation are used in the study of rumor propagation.Daley and Kendall put forward the first rumor spreading model [1,2],which was called the DK model.It has been widely investigated by many scholars.However,this simple model cannot well reflect the dynamic characteristics of rumor propagation,so many other types of rumor propagation models have been established.For example,the susceptible–infected–recovered (SIR) model [3–9],which is a common ordinary differential equation model utilized to study rumor propagation,divides a population into three categories: rumor unknowns,rumor spreaders and rumor suppressors.In recent years,people have also considered the influence of the forgetting mechanism,information transmission rate,hesitation mechanism,constraint mechanism and external control on rumor propagation.On this basis,new rumor propagation models have been established [10–12].Moreover,many rumor propagation models have been established based on complex social networks,such as in[13–16] and the susceptible–exposed–infected–removed(SEIR) model [17–20],which mainly considers the large influence of hesitating constitution mechanisms in the process of rumor spreading.In [21],Li et al proposed a partial differential equation (PDE) based on the rumor propagation model.Li et al believed that in addition to considering the diffusion in space and time dimensions,rumor also had latency,so the delay factor also needed to be discussed.On this basis,Li et al analyzed and studied the dynamic characteristics of the spatial diffusion rumor propagation model with delay.At present,research on the PDE rumor propagation model is still in the development stage and there are many problems waiting to be solved.Therefore,research on this kind of model provides a new research idea for us to analyze the rumor propagation model.

The mathematical models mentioned above are all smooth rumor propagation models.However,similar to the threshold for disease transmission in [22],there is also a threshold for rumor propagation.When a rumor’s influence is not very big,people often take a liberal approach to it,but when the tale number exceeds a predetermined threshold and the rumor’s influence increases gradually,people take the necessary steps to control the rumor.Therefore,there exists a discontinuous function in regard to tales and the threshold.The rumor spreading model also has a non-smooth characteristic.This is an important research point in this paper.

The purpose of this paper is to establish a more realistic non-smooth rumor propagation model and study its dynamic characteristics on the existing basis.The structure of this paper is as follows.In the second part,we establish a mathematical model based on practical significance.In the third part,the existence of the solution of the system is proved.In the fourth part,we discuss the existence of non-spatial bifurcations,mainly including saddle-node bifurcation.In the fifth part,we calculate the Lyapunov number to discuss the stability of equilibrium,Turing instability and the existence of Hopf bifurcation.In the sixth part,we carry out a simulation based on the theory to verify the correctness of the theory.

2.Mathematical modeling

In this part,we will propose an SIR model of rumor propagation.Our model takes into account temporal and spatial differences,which will be reflected in the form of unknowns.Moreover,according to actual people’s attitudes to rumors,we can divide people into three categories: rumor unknown individual S(x,t) who has not been exposed to rumor at position x and time t,rumor propagation individual I(x,t)who comes into contact with rumors and spreads them at position x and time t,and rumor immune individual R(x,t) who is exposed to rumors but will not spread them at position x and time t.In addition,we try to use a non-smooth control function to represent the rumor control process.When rumor spreaders reach a critical value Ic(Ic＞0),then we need to strengthen the control of rumor propagation.Thus,we establish the following rumor control function H(I,Ic),namely

where c is the conversion rate of rumor spreaders to rumor suppressors.By introducing the threshold control function,we establish the following reaction–diffusion rumor propagation model

with the Neumann boundary conditions

and the initial conditions

where di(i=1,2,3)is the diffusion rate of S(x,t),I(x,t)and R(x,t),rA represents the latest increase in the number of rumor spreaders and (1-r)A represents the newly added rumor unknowns,β is the rumor transmission rate,μ is the rate at which rumors disappear naturally,nΩ ∈R is a bounded domain of rumor diffusion,and ¯Ω is the closed set of Ω.Moreover,all the parameters of system (2) are positive.

Obviously,R(x,t) is independent of the first and second equations of system (2).Therefore,we can consider the following simplified system instead of system (2)

with the Neumann boundary conditions

where ∂Ω is the boundary of Ω and the initial conditions

Next,we will focus on the complex dynamical behavior of system (5).

3.The existence of the positive equilibrium points

In this section,we will discuss the distribution of equilibrium points for system (5).

3.1.The case for 0 ≤I ≤Ic

In this case,the equilibrium points of system (5) satisfy

3.2.The case for 0 ＜Ic ＜I

In this case,the equilibrium points of system (5) satisfy

(1) Supposeb2＞ 0,then Δ2＞ 0,we have the following results.

4.Bifurcation analysis without diffusion

In this section,we will analyze the possibility of the existence of bifurcation for the non-spatial system to further explore the dynamic characteristics of system(5).The non-spatial system is as follows

where Hi(I) has been given in equation (1).

Therefore,we have the following characteristic equation

Through calculation,we obtain the equation as follows

In an attempt to analyse the stability of E*,we just need to analyse the distribution of roots for equation (15).For convenience,when 0 ≤I ≤Ic,we consider system (12) as system(12)1;when I ＞Ic,we consider system(12)as system(12)2.

From theorem 3.1,we know when 0 ≤I ≤Ic,g1′(I*)＜0 holds.System (12)1only has one positive equilibrium point,thus collisions at equilibrium points cannot occur.While,from theorem 3.2,we can find conditions that equilibrium points coexist and collide.Therefore,we only consider system (12)2.

From the second equation of system (12)2and equation (14),we have

According to theorem 3.2,we know

Fromg′2(I*)=0,we obtain A22(E*)=0.

It follows that 0,-A11(E*) are the two eigenvalues of J(E*),the eigenvectors of 0 and-A11(E*)are,respectively,as follows

Then,we do the zero transformation x=S-S*,y=I-I*.System (12)2becomes

By simplifying, system (17) can be translated into the following form

5.Local stability and bifurcation analysis with diffusion

After considering the existence of bifurcation for system(5)without diffusion, we will consider the influence of diffusion coefficients on the system according to [29].On this basis, the stability of equilibria points and the existence of bifurcations for system (5) will be discussed in detail.Firstly, we will make some conceptual preparation for the theorem proof.

Definition 1.AssumeE*=(S*,I*)is any equilibrium point of system (5).Denote

where i=1, 2.By calculating, we obtain the following equation

Next, we need to discuss three cases: (1) 0 ≤I ＜Ic;(2)0 ＜Ic＜I;(3) I=Ic.

Case 1:0 ≤I ＜Ic

Theorem 5.1.The positive equilibrium E2of system (5) is locally asymptotically stable.

Proof.Whatever R0is, we findg′1(I2)＜0always holds.Moreover, according to the same method in theorem 4.1, we have

Hence, equation (20) has two negative roots, which means that E2of system (5) is locally asymptotically stable.

Theorem 5.2.Assume that(H)1holds.Then,for system (5),we have the following results.

(i) According to theorems 3.2 and 4.1,we have

Therefore,equation (20) has at least one positive root without diffusion,which means E4is unstable.

(ii) Firstly,we denote

LettingΔ1be the discriminant of f1=0 with respect to γ,we obtain that

Since A22(E3)＞0,we have

Then we obtain

Moreover,itis obvious that(βI3+μ)2＞0,G(I3)2＞ 0.Hence,according to the signs of coefficients of f1,we can easily find that f1=0 has two positive roots

(iii) We define

From theorem 3.2,we haveg′2(I5)＜0,which implies that the solution for investigating the locally stability of E5is the same as that of E3.So we omit the proof here.

Next,we will study the Tuning instability of the positive equilibria E3and E5.We only discuss the Tuning instability of E5,because E3and E5have similar dynamic characteristics.The Turing instability occurs when the positive equilibria point satisfies two conditions that are linearly stable in the absence of diffusion and are linearly unstable in the presence of diffusion.

Theorem 5.3.Assume(H1) andA11(E5)＞ 0hold.Ifγ＜γ3,then E5is Turing unstable.

Thus equation (20) has two negative roots.Hence,E5in linearly stable in the absence of diffusion.

After passing some hours of the night, not without considerable fear and trembling, he noticed a light shining at a little distance, and hoping it might proceed from some house where he could find a better shelter than in the top of the tree, he cautiously descended5 and went towards the light

Next,we will investigate the stability of E5for equation (20) in the presence of diffusion.We can easily obtain that

According to theorem 4.2,we obtain that

Proof.For system (5),under the condition(H)1,the positive equilibrium point E5exists.First,we will prove the existence of Hopf bifurcation at E5.We assume there existsβ* ＞0,whenβ=β*,A11(E5)=0holds.Then for i0=0,we have

Moreover,combining with the proof of theorem 5.2,whenγ＞γ3andA11(E5)=0hold,for any i∈N0andi≠ 0,we have

Assume there exists a unique pair of eigenvalues δ(β)±iω(β) near the imaginary.Taking the derivative of equation (20) with respect to β,referring to the methods of[27],then we obtain

Thus,there exists β=β*such that

Second,we will investigate the direction of spatially homogeneous Hopf bifurcation at E5.We perform the Taylor expansion of system(12)at the equilibrium point E5,then we have

Thus,if σ ＞0,then the direction of Hopf bifurcation is supercritical,that is to say,the periodic solutions are unstable;if σ ＜0,then the direction of Hopf bifurcation is subcritical,that is to say,the periodic solutions are locally asymptotically stable.

Case 3:I=Ic

In this section,we need to investigate the stability ofE′,whereE′=(Sc,Ic)is the positive equilibrium at I=Ic.Since system (5) is not smooth at I=Ic,it is difficult to determine the stability ofE′.Next,we consider the following two systems

Fig.1.The equilibrium point E2 is locally asymptotically stable.

Fig.2.The equilibrium point E5 is locally asymptotically stable.

Fig.3.The influence of rumor propagation rate β on rumor spreaders.

Obviously,when I=Ic,system (25) is equal to system (26),which means that system(25)and system(26)share the same positive equilibriumE′.Moreover,from theorem 5.1,we know that system(25)is always stable.Thus,if we can prove system (26) is unstable,then the discontinuous Hopf bifurcation may occur at I=Ic.Moreover,we noteJ-(E′)andJ+(E′)be the left and right Jacobian of system (5) atE′,respectively,where

Next,we introduce the approximately smooth system as follows

where 0 ≤α ≤1,then the Jacobian matrix atE′ is

Then we have the characteristic equation of system(27)atE′as follows

Fig.4.The equilibrium point E5 is unstable.

Fig.5.The Turing instability of E5.

Fig.6.The change of stabilities for the equilibrium point E5.

It is obvious that when α=0,equation (28) has two negative roots,which means system (27) is stable;when α=1 and (H1) hold,equation (28) has at least one positive root,which means system (27) is unstable.Thus,the discontinuous Hopf bifurcation will occur.Moreover,there exists a α*,0 ≤α*≤1,when α=α*,equation(28)has a pair of pure imaginary roots λ=±iω(ω ＞0).

Next,we need to find out α*and ω*satisfying our conditions.Fori=0,=0,substituting iω(ω ＞0) into equation(28)and separating the real and imaginary parts,we obtain that

For ω ＞0,then we haveC22(E′)＞0and equation (29) can be reduced to

For i∈N0and i ≠0,whenC11(E′)＞0andC22(E′)＞ 0hold,we have that

Hence,for i∈N0,when 0 ≤α ≤α*,system(27)is stable;when α ＞α*,system (27) is unstable;thus,the spatially homogeneous Hopf bifurcation atE′ for system (27) occurs.In other words,the discontinuous Hopf bifurcation occurs atE′ for system(5)when α crosses through α*,so we have the following results.

Theorem 5.5.When(H1),C11(E′) ＞ 0andC22(E′)＞ 0hold,system(5)undergoes a discontinuous Hopf bifurcation atE′.

Proof.From Theorem 5.1,we can obtain that the positive equilibriumE′=E2of system (5) is locally asymptotically stable;when(H1) holds,system(5)has a positive equilibriumE′=E4andE′ is unstable.Thus,system (5) undergoes a discontinuous Hopf bifurcation atE′.

6.Simulation

In this section,we will conduct numerical simulations according to the theories of equilibrium point and some bifurcations.We divide the argument into 0 ≤I ＜Icand 0 ＜Ic＜I.

6.1.Case 1: 0 ≤I ＜Ic

When 0 ≤I ＜Ic,we mainly verify the stability of equilibrium point E2.According to theorem 5.1,the equilibrium point E2is always asymptotically stable.Let the parameters of system(5) be A=0.8,r=0.3,μ=0.4,β=0.1,c=0.4,Ic=1 and let d1,d2vary in [0,1],we assume d1=0.01,d2=0.02.By calculating,we get the equilibrium point E2of system (5) is(1.1561,0.8439).It is obvious that E2is locally asymptotically stable as shown in figure 1.

6.2.Case 2: I ＞Ic

When I ＞Ic,we need to verify the dynamic characteristics of the right-semi system.First,we take the parameter of the system (5) with A=0.5,r=0.1,μ=0.1,β=0.4,c=0.6,Ic=0.4,then we obtain b2=0.0790 ＞0,b3=0.0250 ＞0,Δ2=0.0514 ＞0,which satisfies the existence condition of E5.According to equation (11),we have three solutions,one of them is E5=(0.8743,1.0367).Next,we need to change the value of diffusion coefficients to prove the stability of equilibria.When we let d1=0.01,d2=0.03,we find A11(E5)=0.7361 ＞0 and γ=3 ＞γ3=0.0004.According to theorem 5.2,E5is locally asymptotically stable as shown in figure 2.

Moreover,we consider if the rumor spreading rate β changes,whether the positive equilibrium point will change.Fixing A=0.5,r=0.01,μ=0.1,c=0.55,Ic=0.001,d1=0.01,d2=0.02 and only changing β from 0.3 to 0.7.Then as vividly shown in figure 3,we find as the rate of rumor spreading rate β increases,the number of people who spread rumors has also increased,and the stability of the component I for the equilibrium point E5increases,which is in contradiction with the actual phenomenon.

Fig.7.The Hopf bifurcation at E5.

Accordingly,we can study the instability of E5.Fixing A=0.3,r=0.001,μ=0.05,β=0.3,c=0.8,Ic=0.01,d1=0.01,d2=0.02.By simple calculation,we can find A11(E5)=-0.0035 ＜0,then the equilibrium point E5=(0.2007,2.7194) is unstable according to theorem 5.2,which is shown in figure 4.

Next,we think about the Turing instability of E5for system (5).Letting A=0.5,r=0.01,μ=0.1,β=0.3,c=0.59,Ic=0.01,d1=3.05,d2=0.1,then we have the equilibrium point E5=(2.2460,0.4013).We calculate A11(E5)=0.0081 ＞0,γ=0.0328 ＜γ3=0.2040.Thus E5is Turing unstable according to theorem 5.3.Observing the figure below,we can see when t=100,the (figure 5(b)) is dominated by fuzzy bars.As time t gets bigger by 200,the main pattern of the figure is dotted bars.Then,the strips are separated into blue dots and stabilize in a stable form when t=300.From figure 5(d),we can find three different values of S(x,t) asMAX(S),AVE(S),MIN(S)stabilize at about 2.25 when t ＜40,and then bifurcate from t=40 and stabilize at a certain value around t=180,respectively.Turing instability suggests that diffusion will cause an imbalance in the system,which is in line with the practical implications of rumor propagation;when the effects of catalysing rumor outbreaks are not aligned with curbing the spread of rumors,rumors will become uncontrollable.

Finally,we study the occurrence of Hopf bifurcation at E5by changing the value of the parameter c.Taking A=0.5,r=0.01,μ=0.1,β=0.3,Ic=0.001,d1=0.01,d2=0.02 at the same time,changing c from 0.5 to 0.9,then we find the stability of equilibrium point E5switches.When c=0.5,figure 6 shows that E5is locally asymptotically stable,while c changes to 0.6,E5becomes unstable,and it stabilizes again when c=0.8.

We now study a concrete example.Taking A=0.5,r=0.01,μ=0.1,β=0.3,c=0.6,Ic=0.01,d1=0.01,d2=0.02,then we obtain E5=(2.2764,0.3915),A11(E5)=0.0082 ＞0,and γ=2 ＞γ3=0.1986.According to theorem 5.2,E5is locally asymptotically stable,which is shown in figure 7(a) and (b).Then change the value of parameter Ic=0.001,we have E5=(2.3478,0.3695),A11(E5)=-0.0097 ＜0,from theorem 5.2,we know E5is unstable,which is shown in figure 7(c) and (d).In short,Hopf bifurcation will occur at E5when Icchanges from 0.001 to 0.01.

7.Conclusion

In this article,we observe the actual relationship between I and the threshold Icto establish a non-smooth rumor spreading model based on the time and space dimensions.Contrary to the smooth rumor propagation model of time delay,the non-smooth system is obviously a new addition to rumor propagation model collection.First of all,we prove the existence and uniqueness of the solution for system (5)according to the existence theorem of solutions.Secondly,we divide the system into two parts according to H(I,Ic)function and discuss the existence of the equilibrium points.For the left half of system (5),we define R0to study the relationship between R0and equilibrium points.Then we find that there is always an equilibrium point for the left half of the system.For the right half of system (5),we classify many different cases by discussing the coefficients of the equilibrium point equation.Then,on this basis,we perform a bifurcation analysis for the non-spatial system (12) and find conditions that make saddle-knot bifurcation exist.Further,we consider the effect of diffusion.According to the size relationship between I and Ic,we make specifically analyze the stability of equilibrium points for spatial system (5).In addition,we analyze the Turing instability and Hopf bifurcation occurring at equilibrium point E5.According to Lyapunov number,we determine the direction of the bifurcation.When I=Ic,we analyze and discuss conditions for the existence of discontinuous Hopf bifurcation.Finally,through numerical simulations and combined with the practical meaning of the parameters,we proved the correctness of the previous theoretical theorem.

In real life,rumor propagation is influenced by rumor control means and rumor propagation channels,and the nonsmooth rumor propagation system can reflect this characteristic of rumor propagation.In addition,rumor propagation will change with time and the scope of rumor propagation.Therefore,the PDE model has more practical significance[24–26].

Acknowledgments

This research is partly supported by the National Natural Science Foundation of China (Grant No.12002135),China Postdoctoral Science Foundation(Grand No.2023M731382),and the Young Science and Technology Talents Lifting Project of Jiangsu Association for Science and Technology.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.