Asymptotic Behavior of Global Positive Solution to an Information Diffusion Model with Random Perturbation in Social Network

2017 2nd International Conference on Wireless Communication and Network Engineering (WCNE 2017) ISBN: 978-1-60595-531-5

Asymptotic Behavior of Global Positive Solution to an Information

Diffusion Model with Random Perturbation in Social Network

Shu-yun YUAN

1,2

and Jian-bo HU

1,* 1

Equipment Management and Safety Engineering College, Air Force Engineering University, Xi’an ShaanXi 70051, China

2

Institute 706, Second Academy of China Aerospace and Industry Corporation, Beijing 100854, China

*Corresponding author

Keywords: Information diffusion model, Random perturbation, Social network, Positive solution, Lyapunov function, Stability.

Abstract. The paper explore an information diffusion models with random perturbation in social network. First, we show the models exit the unique global positive solution. By the construction of the Lyapunov function, we give the positive solution is stochastically asymptotically stable in the large around disease-free equilibrium, i.e. the conditions of the information diffusion will die out, investigate the stochastic asymptotic behavior of the positive solution around endemic equilibrium of the deterministic models, obtain the stochastic asymptotic stability condition, i.e. the conditions of the information diffusion will be persistent in social networks.

Introduction

Social media such as blogs, discussion forums, and social networking sites provide new channels for individuals to share information and express their opinions. Research on social networks has received remarkable attention in the past decade, the diffusion process on the online social network became an ongoing research topic [1-7]. Due to similar patterns in the spread of epidemics and social contagion processes, most research adopts the same theoretical principles for epidemics in describing the information diffusion. Mathematical models are used extensively in the study of information diffusion phenomena.

Jaime Mena-Lorcat, Herbert W. Hethcote [8] investigate the most basic SIRS models is as follows

( ) ( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ),

S t S t I t S t R t

I t S t I t I t

R t I t R t

λ β µ δ

β µ ε γ

γ µ δ

 _{= − − +}



= − + +

 

= − +



(1)

where the parameter _{λ β µ ε γ δ}, , , , , are positive constants, and S(t),I(t),R(t) _{denote the number of} the individuals susceptible to the disease, of infected members and of members who have been removed from the possibility of infection through full immunity, respectively. Some notable features of the model: The influx of individuals into the susceptible is given by a constant _λ; it is assumed that the natural death rates are assumed to be equal (denoted by constant _µ) and individuals in I(t) _suffer an additional death due to disease with rate constant _ε, ,_{β δ} and_γrepresent the disease transmission coefficient, the rate of losing their immunity on diseases and the rate of recovery from infection, respectively[8].

For describing the information diffusion in social network, we revise the model (1), let the constant 0

attention to the stochastic Information Diffusion model [9,10]. We assumed that stochastic perturbations were of white noise type, consider that the rate of recovery from infection coefficient γ was subject to stochastic perturbations in model (1), i.e. the fluctuations in the environment will manifest themselves mainly as fluctuations in the parameter

_γ

,

(t),

B γ →γ +σ

where B t( ) _{is standard Brownian motions with} B_{(0)=0, and with intensity of white noise σ}2_{. The}

equations with random perturbation of information diffusion be deduced:

( ) (( ( ) ( ) ( ) ( )) ,

( ) ( ( ) ( ) ( ) ( )) ( ) (t),

( ) ( ( ) ( ) ( )) ( ) (t).

dS t S t I t S t R t dt

dI t S t I t I t dt I t dB

dR t I t R t dt R t dB

λ β µ δ

β µ γ σ

γ µ δ σ

= − − +

 

= − + −

 

= − + +



(2)

Where S(t) _{denotes the number of future population who might have an interest in information at} time t_, I₍t_{)denotes the number of current population who know information at time} t_. R₍t_)denotes

the number of past population who lose interest in information at time t_{. For convenience,}

) ( ), ( ),

(t I t R t

S _{are still named the susceptible, infectious, and recovered on information diffusion in}

social network, respectively.

Similar to [8], It is obvious that the deterministic model corresponding to the stochastic equations

(2) have a disease-free equilibrium ₀ ( ,0,0)

µ λ =

E _{, an endemic equilibrium}

) ) (

)) ( (

, ) (

)) ( )(

( , (

*

δ γ µ βµ

γ µ µ βλ γ δ

γ µ βµ

γ µ µ βλ δ µ β

γ µ

+ +

+ − +

+ + − +

+ =

E _{, and the basic reproduction number}

0

( )

R λβ

µ µ γ

= +

. If R_{0 ≤}1, the disease-free equilibrium E₀ is globally asymptotical stable, then the

information diffusion will die out. If R_{0 >}1_{, the disease-free equilibrium}

0

E _{becomes unstable and}

the endemic equilibrium E* _{is global asymptotically stable endemic equilibrium which implies the}

information diffusion always remains.

This paper is organized as follows. In Section 2, we show there is a unique positive solution of system (3). In Section 3, we deduce the condition R_0≤1_{, investigate the asymptotic behavior of the}

positive solution around disease-free equilibrium the disease-free equilibrium E_{0 of the system (2).}

In Section 4, we deduce the condition R_0>1_{, investigate the asymptotic behavior of the positive} solution, the condition for the information diffusion being persistent is given.

Next, we give some basic theory in stochastic differential equations. Throughout this paper, unless otherwise specified, let ( ,{F } , P)Ω t t_≥0 be a complete probability space with a filtration{F }t t_≥0

satisfying the usual conditions (i.e. it is right continuous and F contains all P -null sets). Let ₀

{ | 0, 1 }

d d

i

x x i d

+ = ∈ > ∀ ≤ ≤

_, d { d| 0, 1 }

i

x x i d

= ∈ ≥ ∀ ≤ ≤

_.

In general, consider the n_{-dimensional stochastic differential equation}

( ) ( ( ), ) ( ( ), ) ( )

dx t ₌ f x t t dt₊g x t t dB t ₍₅₎

with initial value ( )_{0 0} d

x t ₌x _∈R . ( )B t denotes n-dimensional standard Brownian motion defined on the above probability space. Define the differential operator L _{associated with (5)}

2

1 , 1

1

( , ) [g ( , ) g( , )] 2

d d

T i

i i i j i j

L f x t x t x t

t x x x

= =

∂ ∂ ∂

= + +

∂

∑

∂

∑

∂ ∂

If L _{acts on a function} 2,1

( d ; )

V C

+ +

∈ × , then

( ( m ), ( m ), ( m ) ( (0), (0), (0)) ( m )

EV S_{τ ∧}T I_{τ ∧}T R_{τ ∧}T _≤V S I R ₊KES _{τ ∧}T _,

where EV S( (τm∧T), (Iτm∧T), (Rτm∧T)≤V S( (0), (0), (0))I R +KT and

2

( )

xx d d

i j

V V

x x ×

∂ =

∂ ∂

. By It oˆ _’s

formula, if ( ) d

x t _∈R _{, then}

( ( ), ) ( ( ), ) x( ( ), ) ( ( ), ) ( )

dV x t t ₌LV x t t dt₊V x t t g x t t dB t _.

Consider (4), assume f(0, )t ₌0 and (0, )g t ₌0 for all t_≥t₀. So ( )x t _≡0 is a solution of (5),

called the trivial solution or equilibrium position.

Existence and Uniqueness of Positive Solution

In this section we first show that the solution of system (3) is global and positive. To get a unique global (i.e. no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [11]. However, the coefficients of system (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (3) may explode in finite time. In this section, using the Lyapunov analysis method, we show the solution of system (1) is positive and global.

Theorem 1. There is a unique solution ( ( ), ( ), ( ))S t I t R t _{of system (2) on} t_{≥0for any initial value} 3

( (0), (0), (0))S I R +

∈ , and the solution will remain in 3+ with probability 1, namely,

3

( ( ), ( ), ( ))S t I t R t +

∈ for all t_≥0 almost surely.

Proof. _{Since the coefficients of the equation are locally Lipschitz continuous for any given initial}

value 3

( (0), (0), (0))S I R +

∈ , there is a unique local solution( ( ), ( ), ( ))S t I t R t _on t∈[0, )τe , where τe is the

explosion time (see [11]). To show this solution is global, we need to show that τe = ∞a s. . Let m0 >0

be sufficiently large so that 0 0

1 (0) [ , ]

S m

m

∈ , 0 0

1 (0) [ , ]

I m

m

∈ and 0

0

1 (0) [ , ]

R m

m

∈ . For each integer

0

m_>m _{, define the stopping time}

0 0

1 1 1

inf{ [0, ) : ( ) ( , ) or ( ) ( , ) or ( ) [ , ]}

m t e S t m I t m R t m

m m m

τ = ∈ τ ∉ ∉ ∈ ,where, we set inf∅ = ∞ (as usual ∅

denotes the empty set). According to the definition, τm is increasing as m→ ∞. Set lim m m τ∞ τ

→∞

= ,

whence τ_∞≤τma s. .If we can show that τ_∞≤ ∞a s. . thenτ ≤ ∞e a s. .and

2

( ( ), ( ), ( ))S t I t R t a s. .

+

∈ for all 0

t_{≥ .In other words, to complete the proof all we need to show is thatτ} a s_{. .}

∞≤ ∞ If this statement is false, then there exist a pair of constants T _>0 _{and ε∈(0,1) such that} P_{τ T_{} ε.}

∞≤ > Hence there is an integer m_1≥m_{0 such that for all}

1

m_≥m _{we have}

{ m }

P_{τ ≤}T _{≥ε. (6)}

Define a _C2 _-function 3

: [0, )

V

+→ +∞

by V S I R( , , ) (S a alog )S (I 1 log )I (R 1 log )R

a

= − − + − − + − − ,

where a_>0_{is real positive constants to be chosen later. The nonnegativity of this function} V S I R_{( , , )}

can be seen from u a alogu 0

a

− − ≥ for allu_>0,a_>0_.

Let m_≥m_{0 and} T _{>0 be arbitrary. Applying the It}oˆ_{’s formula, we obtain}

2 2

1 1 1

( , , ) (1 ) (1 ) ( )

2

a

dV S I R dS dI dI

S I I

= − + − + 2 2

1 1 1

(1 ) ( )

2

dR dR

R R

Where _LV: 2 _{[0, )}

+→ +∞

is defined by

2

( 2 ) ( ) a R I ( )

LV a S a a I R

S S R

λ µ µ γ δ σ µ β λ δ γ β µ µ

= + + + + + − + − − − + − −

. (8)

Let a µ

β

= such that a_β−µ=0_{.Substituting this into (8), we get}

2

( 2 ) ( ) a R I

LV a S a R

S S R

λ µ µ γ δ σ µ β λ δ γ µ

= + + + + + − + − − − − _≤(_λ+a_µ+2_{µ+γ +δ +σ}2) :₌ K_.

Substituting this into (7), we getdV_≤Kdt_+σ(_{− +}I R dB t) ( ). Integrating this from 0 to τm∧T and

dividing t on the both sides and and taking the expectation, we have

( ( m ), ( m ), ( m ) ( (0), (0), (0))

EV S_{τ ∧}T I_{τ ∧}T R_{τ ∧}T _≤V S I R ₊KT ₍₉₎

Let Ω =m {τm∧T}, for m≥m1and by (6) ,we have P(Ωm)≥ε. Note that for every ω∈ Ωm, there is at

least one of S(τm, )ω ,I(τm, )ω and R(τm, )ω _{that equals either} mor 1

m, and hence

( ( m ), ( m ), ( m ))

V S_{τ ∧ω} I _{τ ∧ω} R_{τ ∧ω min{}m _{1 log ,}m 1 _{1 log ,}m m a a_logm_,1 a a_log 1 _}

m a m mk

≥ − − − + − − − − .

It then follows from (9) that

( (0), (0), (0))

V S I R ₊KT _{[1 ( ) ( ( ), ( ), ( ))]}

m m m m

E _ωV S_{τ ω} I_{τ ω} R_{τ ω}

Ω

≥ ∧ ∧ ∧

1 1 1

min{m 1 log ,m 1 log ,m m a alogm, a alog }

m a m mk

ε

≥ − − − + − − − −

, where 1Ωmis the indicator function of Ωm. Letting

m_{→ ∞, leads to the contradiction}

( (0), (0), (0))

V S I R KT

∞ > + = ∞.

So we must therefore have _τ∞= ∞. Namely,( ( ), ( ), ( ))S t I t R t 3

+

∈ for all t_≥0 almost surely.

Then,for the system (3) has unique solution ( ( ), ( ), ( ))S t I t R t _on t_{≥0 for any initial value} 3

( (0), (0), (0))S I R

+

∈ , and the solution will remain in 3+ with probability 1.This finishes the proof .

Asymptotic Behavior Around E₀

If R_0≤1_{, then}

0

E _{is globally stable, which means the information will die out after some period of}

time. Hence, it is interesting to study the disease-free equilibrium for controlling information diffusion. In this section,we show the average oscillation around E_{0 in time to exhibit whether the}

information will die out.

Theorem 2. If

(

)

0 1

R βλ

µ µ γ

= ≤ +

and the condition is satisfied 2 2 ( )

min{2 , }

2 µ µ δ

σ µ

µ γ

+ <

+

, then the

solution ( ( ), ( ), ( ))S t I t R t _{of system (3) with initial value} 3

( (0), (0), (0))S I R

+

∈ is stochastically asymptotically stable in the large.

Proof. _Let u S λ

µ

= − ,v₌I _and w₌R_,then u_∈R_,v_>0,w_>0and du t_{( ) ( µ}u _βuv _β λv _δw dt_{) ,}

µ

= − − − +

( ) [ ( ) ] ( )

dv t _βuv _{µ γ β} λ v dt _σvdB t

µ

= − + − − , dw t( )=[γv−(µ+δ) ]w dt+σwdB t( ).

Define a _C2

− function V: 3 [0, ) +→ +∞

by 2 2

1 2

( , , ) ( )

V u v w ₌ u_{+ +}v w ₊c v₊c w _{, where}

1 0

c _{> and}

2 0

Using Itoˆ_{’s formula, we compute}

1 3

((2( ) ) 2( (1 ) ) ) ( )

dV ₌LVdt_−σ u_{+ +}v w ₊c v₊ u_{+ +}v ₊c w w dB t _{, (10)}

where

2 2 2

2( )( ) ( )

LV ₌ u_{+ +}v w _−µu_−µv_−µw ₊ v ₊w _σ 2 2

1 2 2

+ [c _βuv (_{µ γ β} λ) ]v 2c w[_γv (_{µ δ}) ]w c_σ w

µ

− + − + − + +

2 2 2 2 2 2

2 1 2 1 2

0

1

2 (u v ) 2[ c ( )]w (c 4 )uv 2(c 2 )vw 4 uw c ( 1)v (v (1 c)w). R

λ

µ µ µ δ β µ γ µ µ β σ µ

= − − − + + + − + − − − − + + +

We choose c₁ 4µ

β

= and 2

2 =

c µ

γ such that 1 4 0

c_{β− µ=} and c_2γ−2_µ=0. We can obtain

2 2 2 2 2 2 2

1 0

2[ 2 ( )] 1 2

2 2 4 ( 1)

LV u v w uw c v v w

R

µγ µ µ δ λ µ γ

µ µ µ β σ σ

γ µ γ

+ + +

= − − − − − − + + .

From R0 ≤1andv>0,we have 1 0

1

( 1) 0

c v

R

λ β

µ

− − ≤ . Moreover, using Cauchy inequality such that

2 2

2

2λ u λ

µ≤ +µ and

2 2

2uw u _ρw

ρ

− ≤ + , set ρ=γ µ δ γ

+ + _{, hence,}

2 2 2 2 2 2 2

2 ( ) 2 ( ) 2

2

LV µ µ δ u _µv µ µ δ w _σ v µ γ_σ w

γ µ δ γ γ

+ + +

= − − − + + + +

2 2 2 2 2

2 ( ) 2 ( ) 2

(2 ) ( ) 0

u v w

µ µ δ µ µ δ µ γ

µ σ σ

γ µ δ γ γ

+ + +

= − − − − − ≤ + +

.

Consequently, LV _{is negative-definite, This combined with the definite of} V _{gives that the}

solution ( ( ), ( ), ( ))S t I t R t _{of system (3) with initial value} 3

( (0), (0), (0))S I R

+

∈ is stochastically asymptotically stable in the large. This finishes the proof of Theorem 2.

Asymptotic Behavior Around _E*

In this section, we show the average oscillation around E* _{in time to exhibit whether the information}

will be persistent.

Theorem 3. If

(

)

0 1

R βλ

µ µ γ

= > +

, and the condition is satisfied 2 2 ( )

min{2 , }

2 µ µ δ

σ µ

µ γ

+ <

+

, then the

solution ( ( ), ( ), ( ))S t I t R t _{of system (2) with initial value} 3

( (0), (0), (0))S I R

+

∈ has the property

* 2 * 2 * 2

2 2

2 3

0

1 2 2 ( )

limsup [( ( ) ) ( ( ) ) ( ( ) ) ] .

2 2 ( ) (2 )

t

t

K

E S r S I r I R r R dr

t M

σ

µ µ µ δ

µ σ µ µ δ µ γ σ

→∞

+

− + − + − ≤

− + − +

∫

Where * * * *

( , , )

E ₌ S I R _{is the endemic equilibrium of system (1), and}

2 2

( ) 1 2 ( ) (2 )

min{ , , }

2 2

M µ µ δ _{µ σ} µ µ δ µ γ σ

γ µ δ γ

+ + − +

= −

+ + ,

2 2

2 2

* * * 2

2 2

2 ( )(2 )

2 2 [2 ( ) (2 ) ]

K I R I

σ

µσ µ µ δ µ γ σ µ

σ

µ σ γ µ µ δ µ γ σ β

+ +

= + +

− + − +

.

Proof. _{Define a C}2

−function V: 3 [0, ) +→ +∞

* * * 2 * * * *

1 1

( ) ( ) ( log ) ( )

2 2

I

V x S S I I R R a I I I p R R

I

= − + − + − + − − + − ,

where a_>0,p_>0 _{are real positive constants to be chosen later.} Applying the Itoˆ_{’s formula, we have}

* * * * * * *

( ) [ ( ) ( )] ( )+ [ (1 )( )] ( )

dV x ₌LVdt₊ a I₋I _−σI S₋S _{+ −}I I ₊R₋R dB t _σR S₋S _{+ −}I I _{+ +}p R₋R dB t ₍₁₁₎

where

*

* * * 2 2 2

* 2 * 2 2

* 2 * 2 * 2 * *

* * * * 2 2

1

( )( ) ( ) (1 )[ ( ) ]

2

1 1

( )[ ( ) ]

2 2

( ) ( ) [ ( )]( ) ( 2 )( )( )

1

( 2 )( )( ) 2 ( )( ) (

2

I

LV S S I I R R S R I R a SI I

I

aI p R R I R p R

S S I I p R R S S I I

p I I R R S S R R I R

λ µ µ µ σ β µ γ

σ γ µ δ σ

µ µ µ µ δ αβ µ

γ µ µ σ

= − + − + − − − − + + + − − +

+ + − − + +

= − − − − − + + − + − − −

+ − − − − − − + + 2) 1 * 2 1 2 2

2 2

aI _σ p_σ R

+ +

Let a 2µ

β

= such that _αβ−2_µ=0, choose p 2µ

γ

= such that p_{γ −}2_µ=0_{. This implies}

* 2 * 2 * 2

* * 2 2 2 * 2 2 2

( ) ( ) [ ( )]( )

1 2 2

2 ( )( ) ( )

2 2 2

LV S S I I p R R

S S R R I R I p R

µ µ µ µ δ

µ µ

µ σ σ σ

β γ

= − − − − − + + −

− − − + + + +

Set _ρ=γ µ δ γ

+ + _{, such that 2(S S}*_{)(R R}*₎ γ _{(S S}* 2_{) +}γ µ δ _{(R R}* 2₎

γ µ δ γ

+ +

− − − ≤ − −

+ + ,Then we compute

2

* 2 * 2 * 2 * 2 2 2 2 2 2

2

* 2 2 * 2

2

2 2

* 2 *

2 2

( ) ( ) 1

( ) ( ) ( ) ( )

2

( ) 1 2 2 ( ) (2 )

( ) ( )( ) (

2 2 2

2 ( ) 2 ( )(2 )

)

2 ( ) (2 ) 2 2 [2 (

LV S S I I I R R I R R

S S I I R

R I

µ µ δ µ µ µ δ µ

µ σ σ σ

γ µ δ β γ γ

µ µ δ µ µ µ δ µ γ σ

µ σ

γ µ δ µ σ γ

µ µ δ µσ µ µ δ µ γ σ

µ µ δ µ γ σ µ σ γ µ µ δ

+ + ≤ − − − + − − + + + + + + + − + = − − − − − − + + − + + + − + + + − + − + 2

* * 2

2

) (2 ) ]

R µI_σ

µ γ σ +β

− +

From the condition 2 2 ( )

min{2 , }

2 µ µ δ

σ µ

µ γ

+ <

+ , we have

2

1 0 2

µ− σ > and

2

2 ( ) (2 )

0 2

µ µ δ µ γ σ γ

+ − +

> , then

we have

* 2 * 2 * 2 * 2 2 2 2 2 2

* 2 2 * 2

2

2

* 2 2

( ) ( ) 1

( ) ( ) ( ) ( )

2

( ) 1 2

( ) ( )( )

2 2

(2 ( ) (2 ) ) 2 ( )

( )

2 2 ( ) (2 )

LV S S I I I R R I R R

S S I I

R R K

σ

µ µ δ µ µ µ δ µ

µ σ σ σ

γ µ δ β γ γ

µ µ δ µ

µ σ

γ µ δ µ σ

µ µ δ µ γ σ µ µ δ

γ µ µ δ µ γ σ

+ + ≤ − − − + − − + + + + + + = − − − − − + + − + − + + − − + + − +

Substituting this into (11), we get

* * * *

( ) [ ( ) ( )] ( )

dV x _≤LVdt₊ a I₋I _−σI S₋S _{+ −}I I ₊R₋R dB t * * *

+_σR S[ ₋S _{+ −}I I ₊(1₊ p)(R₋R )]dB t( )

Integrating the (11) from 0 to t _{and taking the expectation, we have}

* 2 * 2 * 2

2 2

0

1 2 2 ( )

limsup [( ( ) ) ( ( ) ) ( ( ) ) ] .

2 2 ( ) (2 )

t

t

K

E S r S I r I R r R dr

t M

σ

µ µ µ δ

µ σ µ µ δ µ γ σ

→∞

+

− + − + − ≤

− + − +

∫

Acknowledgement

This research was financially supported by the national key basic research program (No.2014CB744900) and National basic research program of Philosophy and Social Science (No.17GGL270).

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