# The Stability of SI Epidemic Model in Complex Networks with Stochastic Perturbation

Authors: Jinqing Zhao; Maoxing Liu; Wanwan Wang; Panzu Yang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We investigate a stochastic SI epidemic model in the complex networks. We show that
this model has a unique global positive solution. Then we consider the asymptotic behavior of the model
around the disease-free equilibrium and show that the solution will oscillate around the disease-free
equilibrium of deterministic system whenR0≤1. Furthermore, we derive that the disease will be persistent
when R0>1. Finally, a series of numerical simulations are presented to illustrate our mathematical
findings. A new result is given such that, when R0≤1, with the increase of noise intensity the solution of
stochastic system converging to the disease-free equilibrium is faster than that of the deterministic
system.


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## Body

## 1. Introduction

Epidemiology is the science to study the distribution of disease and influencing factors, so as to explore the etiology, clarify the popular rule of the disease, and formulate the countermeasures and measures for preventing, controlling, and eliminating the disease. Many mathematical models of diseases spreading help us to understand the propagation of diseases [1, 2]. The transmission of diseases can be influenced by many factors, such as the age and social structure of the population, the contact network among individuals, and the metapopulation characteristics. So it is difficult to establish an accurate epidemic model which is completely consistent with the real world. In recent years, a lot of compartmental epidemic models have been studied by many researchers [3–5], and complex networks also have been used to study the spread of diseases [6–17].In this paper we consider an SI model with the birth and death in complex networks. As mentioned in the paper [6, 13], the birth and death do not affect the degree of nodes. Suppose Sk(t),Ik(t) are the number of the healthy and infected nodes with the degree k at time t; the mean-field equations can be written as
(1)dSkdt=bk-λkSkθ-dSk,dIkdt=λkSkθ-(d+ϵ)Ik,
where θ=(1/〈k〉)∑k=1nλkP(k). For system (1), it can be written as the following form:
(2)dSk(t)=(bk-1〈k〉∑j=1n‍λkjP(j)Sk(t)Ij(t)-dSk(t))dt,dIk(t)=(1〈k〉∑j=1n‍λkjP(j)Sk(t)Ij(t)-(d+ϵ)Ik(t))dt.
We denote βkj=(1/〈k〉)λkjP(j), so we obtain
(3)dSk(t)=(bk-∑j=1n‍βkjSk(t)Ij(t)-dSk(t))dt,dIk(t)=(∑j=1n‍βkjSk(t)Ij(t)-(d+ϵ)Ik(t))dt.
It always has the disease-free equilibrium E0=(S10,0,…,Sn0,0), where Sk0=bk/d, k=1,2,…,n. If A=(βkj)n×n is irreducible and R0≤1, then E0 is globally stable in D, while if R0>1, E0 is unstable and there is an endemic equilibrium E*=(S1*,I1*,…,Sn*,In*) belonging to D which is globally asymptotically stable in D; here
(4)D={(S1,I1,…,Sn,In)∈ℝ+2n:Sk,Ik≤bkd,Sk+Ik≤bkd,k=1,2,…,nbkd},M0=M(S10,…,Sn0)=(βkjSk0d+ϵ)n×n,R0=ρ(M0),
and ρ(M0) denotes the spectral radius of M0.The deterministic models have some limitations in describing the spread of disease. The accident in the process of disease transmission can not be reflected by the deterministic models. This is because of the fact that the deterministic models ignore the effect of the environmental noise. In an ecosystem, the environmental noise is inevitably in the real world; thus stochastic models are more realistic. In the research of stochastic epidemic models, many researchers make a lot of contributions [17–26].In this paper, we consider the following stochastic system:(5)dSk(t)=(bk-∑j=1n‍βkjSk(t)Ij(t)-dSk(t))dt+σk1Sk(t)dBk1(t),dIk(t)=(∑j=1n‍βkjSk(t)Ij(t)-(d+ϵ)Ik(t))dt+σk2Ik(t)dBk2(t),
where Bki(t), k=1,2,…,n, i=1,2, are independent standard Brownian motions with Bki(0)=0, and σki2≥0, k=1,2,…,n, i=1,2, represent the intensities of Bki(t).The remaining parts of this paper are as follows. In the next section we show the existence and uniqueness of a global positive solution of model (5). In Section 3, we analyze the asymptotic behavior around the disease-free equilibrium. In Section 4, we study the dynamic of system (5) around the endemic of the deterministic model. In Section 5, numerical simulations and conclusions are carried out.


## 2. Global Positive Solution

When we study a dynamical behavior, a global solution is important for the system. In this section we show that the solution of system (5) is global and nonnegative. As we know, for a stochastic differential equation, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition. It is a sufficient condition for a stochastic differential equation has a unique global (i.e., no explosion in a finite time) solution for any given initial value [27, 28]. Although the coefficients of system (5) satisfy locally Lipschitz continuous, they are not satisfied with the linear growth condition, so the solution of system (5) may explode at a finite time. In this section, Lyapunov analysis method (mentioned in [29]) is used to show that the solution of system (5) is positive and global.Theorem 1.
For any given initial value(S1(0),I1(0),…,Sn(0),In(0))∈ℝ+2n, there is a unique positive solution (S1(t),I1(t),…,Sn(t),In(t)) of model (5) on t≥0 and the solution will remain in ℝ+2n with probability 1, namely, (S1(t),I1(t),…,Sn(t),In(t))∈ℝ+2n for all t≥0 a.s.Proof.
Due to the fact that the coefficients of the system (5) are locally Lipschitz continuous, for any given initial value (S1(0),I1(0),…,Sn(0),In(0))∈ℝ+2n, it has a unique local solution (S1(t),I1(t),…,Sn(t),In(t)) on t∈[0,τe), where τe is the explosion time [30]. If we show that τe=∞ a.s., it suggests that this solution is global. Let l0>0 be sufficiently large so that Sk(0), Ik(0)   (k=1,2,…,n) all lie within the interval [1/l0,l0]. For each integer l≥l0, defining the stopping time
(6)τl=inf⁡{t∈[0,τe):min⁡{Sk(t),Ik(t),k=1,…,n}≤1lormax⁡{Sk(t),Ik(t),k=1,…,n}≥l1l},
we set inf⁡∅=∞ (as usual ∅ denotes the empty set). Obviously, τl is increasing as l→∞. Set τ∞=lim⁡l→∞τl; therefore τ∞≤τe a.s. If τ∞=∞ a.s. is true, then τe=∞ a.s. and (S1(t),I1(t),…,Sn(t),In(t))∈ℝ+2n a.s. for t≥0. In other words, to complete the proof it is required to show that τ∞=∞ a.s. If this statement is false, then there is a pair of constants T>0 and ɛ∈(0,1) such that P{τ∞≤T}>ɛ. Thus there is an integer l1≥l0, such that
(7)P{τl≤T}≥ɛ,∀l≥l1.
Define a C2-function V:ℝ+2n→ℝ+ as follows:
(8)V(S1,I1,…,Sn,In)=∑k=1n‍[(Sk-1-ln⁡Sk)+(Ik-1-ln⁡Ik)].
Applying Itô’s formula, we obtain
(9)d(S1,I1,…,Sn,In)=∑k=1n‍[(1-1Sk)dSk+121Sk2(dSk)2+(1-1Ik)dIk+121Ik2(dIk)2(1-1Sk)]=∑k=1n‍[(1-1Sk)[(bk-∑j=1n‍βkjSkIj-dSk)dt(∑j=1n‍βkjSk-dSk)+σk1Sk(t)dBk1(t)∑j=1n‍]+12σk12dt+(1-1Ik)×[(∑j=1n‍βkjSkIj-(d+ϵ)Ik)dt+σk2IkdBk2(t)]+12σk22dt[(bk-∑j=1n‍βkjSkIj-dSk)dt]=∑k=1n‍[{∑j=1n‍βkjSkIj-(d+ϵ)Ik-∑j=1nβkjSkIjIk}bk-∑j=1n‍βkjSkIj-dSk-bkSk+∑j=1n‍βkjSkIj+d+12σk12+∑j=1n‍βkjSkIj-(d+ϵ)Ik-∑j=1nβkjSkIjIk+(d+ϵ)+12σk22{{∑j=1n‍βkjSkIj-(d+ϵ)Ik-∑j=1nβkjSkIjIk}}]dt+∑k=1n‍[σk1(Sk-1)dBk1(t)+σk2IkdBk2(t)]≤∑k=1n‍(bk+2d+ϵ+∑j=1n‍βkjIj+12(σk12+σk22))dt+∑k=1n‍[σk1(Sk-1)dBk1(t)+σk2(Ik-1)dBk2(t)]=:Kdt+[σk1(Sk-1)dBk1(t)+σk2IkdBk2(t)].
We can now integrate both sides of (9) from 0 to τl∧T and then take the expectations
(10)E[V(S1(τl∧T),I1(τl∧T),…,Sn(τl∧T),In(τl∧T))]≤V(S1(0),I1(0),…,Sn(0),In(0))+E[∫0τl∧TKdt‍]≤V(S1(0),I1(0),…,Sn(0),In(0))+KT.
Let Ωl={τl≤T} for l≥l1 and, by (7), P(Ωl)≥ɛ. Note that, for every ω∈Ωl, there is at least one of Sk(τl,ω) and Ik(τl,ω), k=1,2,…,n, that equals either l or 1/l, and therefore V(S1(τl,ω),I1(τl,ω),…,Sn(τl,ω),In(τl,ω)) is not less than either
(11)l-1-ln⁡lor1l-1-ln⁡1l=1l-1+ln⁡l.
Hence,
(12)V(S1(τl,ω),I1(τl,ω),…,Sn(τl,ω),In(τl,ω))≥(l-1-ln⁡l)∧(1l-1+ln⁡l).
It then follows from (7) and (10) that
(13)V(S1(0),I1(0),…,Sn(0),In(0))+KT≥E[1Ωl(ω)V(S1(τl,ω),I1(τl,ω),…,Sn(τl,ω),In(τl,ω))]≥ɛ[(l-1-ln⁡l)∧(1l-1+ln⁡l)],
where 1Ωl(ω) is the indicator function of Ωl. Letting l→∞, we have that
(14)∞>V(S1(0),I1(0),…,Sn(0),In(0))+KT≥∞
is a contradiction. So we must have τ∞=∞. Therefore, it implies Sk(t), Ik(t), k=1,2,…,n, will not explode in a finite time with probability one.


## 3. Asymptotic Behavior around the Disease-Free Equilibrium

As mentioned in the Introduction,E0=(b1/d,0,…,bn/d,0) is the disease-free equilibrium of system (3), and when R0≤1, E0 is globally stable, which means that the disease will be extinct in the limited time. In this section, we will study the asymptotic behavior around E0 of system (5).Lemma 2.
IfA is nonnegative and irreducible, then the spectral radius ρ(A) of A is a simple eigenvalue, and A has a positive eigenvector ω=(ω1,ω2,…,ωn) corresponding to ρ(A). Besides, if 0≤A≤B, then ρ(A)≤ρ(B). (This lemma can be found in [20].)Theorem 3.
AssumeA=(βkj)n×n is irreducible. If R0≤1 and the following condition is satisfied:
(15)σk12≤43d,σk22≤2(d+ϵ),
then for any given initial value (S1(0),I1(0),…,Sn(0),In(0))∈ℝ+2n, the solution of system (5) has the property
(16)limsup⁡t→∞1tE∫0t[(Sk(r)-bkd)2+Ik2(r)]‍dr≤32∑k=1n‍akσk12bk2d2K1,
where
(17)K1=min⁡{ωkβkk(2d+ϵ)(d+ϵ)(d-34σk12),d+ϵ-12σk22}.Proof.
First change the variablessk=Sk-bk/d, ik=Ik; then -bk/d≤sk≤0, ik≥0 and system (5) can be written as
(18)dsk=(-∑j=1n‍βkj(sk+bkd)ij-dsk)dt+σk1(sk+bkd)dBk1(t),dik=(∑j=1n‍βkj(sk+bkd)ij-(d+ϵ)ik)dt+σk2ikdBk2(t).
Let S0=(S10,S20,…,Sn0), where Sk0=bk/d, k=1,2,…,n. Define
(19)M(s)=[β11S10d+ϵβ12(S10+s1)d+ϵ⋯β1n(S10+s1)d+ϵβ21(S20+s2)d+ϵβ22S20d+ϵ⋯β2n(S20+s2)d+ϵ⋮⋮⋱⋮βn1(Sn0+sn)d+ϵβn2(Sn0+sn)d+ϵ⋯βnnSn0d+ϵ];
then it is nonnegative and irreducible. By Lemma 2, there is a positive eigenvector ω=(ω1,ω2,…,ωn) of M(s) corresponding to ρ(M(s)), such that
(20)(ω1,ω2,…,ωn)M(s)=(ω1,ω2,…,ωn)ρ(M(s)).
Define a C2-function V:ℝ+2n→ℝ¯+ by
(21)V(s1,i1,…,sn,in)=12∑k=1n‍ak(sk+ik)2+∑k=1n‍ωkd+ϵik,
where ak, k=1,2,…,n, are positive constants. Then the function V is positive definite, and
(22)dV=LVdt+∑k=1n‍ak(sk+ik)∑k=1n‍ak(zxz)×(σk1(sk+bkd)dBk1(t)+σk2ikdBk2(t))+∑k=1n‍ωkd+ϵσk2ikdBk2(t),
where
(23)LV=∑k=1nak(sk+ik)[-dsk-(d+ϵ)ik]‍+12∑k=1nak[σk12(sk+bkd)2+σk22ik2]‍+∑k=1nωkd+ϵ‍[∑j=1nβkj(sk+bkd)ij‍-(d+ϵ)ik]=-∑k=1nak[(d-12σk12)sk2+(d+ϵ-12σk22)ik2]‍+∑k=1n∑j≠kωkd+ϵβkjskik-∑k=1n[ak(2d+ϵ)-ωkd+ϵβkk]skik‍+∑k=1n∑j=1nωkd+ϵβkjbkdik‍-∑k=1nωkik‍‍+12∑k=1nakσk12(2skbkd+bk2d2)‍.
Choose  ak=ωkβkk/(2d+ϵ)(d+ϵ),  k=1,2,…,n; then ak(2d+ϵ)-(ωk/(d+ϵ))βkk=0. And we note that
(24)∑k=1n∑j≠kωkd+ϵβkjskik‍+∑k=1n∑j=1nωkd+ϵβkjbkdik‍‍=(ω1,ω2,…,ωn)×[β11S10d+ϵβ12(S10+s1)d+ϵ⋯β1n(S10+s1)d+ϵβ21(S20+s2)d+ϵβ22S20d+ϵ⋯β2n(S20+s2)d+ϵ⋮⋮⋱⋮βn1(Sn0+sn)d+ϵβn2(Sn0+sn)d+ϵ⋯βnnSn0d+ϵ]×[i1i2⋮in]=ωM(s)i.
Then
(25)LV=-∑k=1nak[(d-12σk12)sk2+(d+ϵ-12σk22)ik2]‍-ωi+ωM(s)i+12∑k=1nakσk12(2skbkd+bk2d2)‍≤-∑k=1nak[(d-34σk12)sk2+(d+ϵ-12σk22)ik2]‍+ω(M(s)-1)i+32∑k=1nakσk12bk2d2‍=-∑k=1nak[(d-34σk12)sk2+(d+ϵ-12σk22)ik2]‍+ω(ρ(M(s))-1)i+32∑k=1nakσk12bk2d2‍,
where the last equality is derived from (20). Since -bk/d≤sk≤0, then 0≤M(s)≤M(S0)=(Sk0βkj/(d+ϵ))n×n=M0, and so ρ(M(s))≤ρ(M0) according to Lemma 2. Besides, R0≤1, and then ρ(M(s))≤1. Therefore
(26)dV≤[-∑k=1nak[(d-34σk12)sk2+(d+ϵ-12σk22)ik2]+32∑k=1nakσk12bk2d2]dt+∑k=1nak(sk+ik)σk1(sk+bkd)dBk1(t)‍+∑k=1n(ak(sk+ik)σk2+ωkd+ϵσk2)ikdBk2(t)‍.
Integrating both sides of (26) from 0 to t, and taking expectation, yields
(27)0≤E[V(s1(t),i1(t),…,sn(t),in(t))]≤E[V(s1(0),i1(0),…,sn(0),in(0))]+E∫0t[-∑k=1nak[(d-34σk12)sk2(r)‍+(d+ϵ-12σk22)ik2(r)(d-34σk12)]+32∑k=1nakσk12bk2d2]dr,
which implies
(28)E∫0t[∑k=1nak[(d-34σk12)sk2(r)+(d+ϵ-12σk22)ik2(r)]‍]‍dr≤E[V(s1(0),i1(0),…,sn(0),in(0))]+32∑k=1nakσk12bk2d2t‍.
Therefore
(29)limsup⁡t→∞1tE∫0t‍[∑k=1n‍ak[(d-34σk12)sk2(r)lim⁡sup⁡t→∞1t∫0t‍+(d+ϵ-12σk22)ik2(r)]∑k=1n‍]dr≤32∑k=1n‍akσk12bk2d2;
that is,(30)limsup⁡t→∞1tE∫0t‍[∑k=1n‍ak[(d-34σk12)(Sk(r)-bkd)2lim⁡sup⁡t→∞1tE∫0t‍+(d+ϵ-12σk22)Ik2(r)(Sk(r)-bkd)2]]dr≤32∑k=1n‍akσk12bk2d2.
If we let
(31)K1=min⁡{ak(d-34σk12),d+ϵ-12σk22},
then
(32)limsup⁡t→∞1tE∫0t‍[(Sk(r)-bkd)2+Ik2(r)]dr≤32∑k=1n‍akσk12bk2d2K1,
as the theorem is proved.Remark 4.
From Theorem3, we can get the conclusion that the solution of the stochastic system will oscillate around the disease-free equilibrium of the deterministic model; the values of σk1 and σk2 have bearing on the intensity of turbulence. If the stochastic perturbations become small, the solution of system (5) will be close to the disease-free equilibrium of system (3).
Besides, ifσk1=0, then E0 is also the disease-free equilibrium of system (5). From the proof of Theorem 3, we can obtain
(33)LV≤-∑k=1n‍ak[(d-34σk12)sk2+(d+ϵ-12σk22)ik2]≤0.
Therefore, E0 is globally asymptotically stable.


## 4. The Dynamic of System (5) around the Endemic of System (3)

In the deterministic model, ifR0>1, there exists the endemic equilibrium E*. But E* is not the endemic equilibrium of stochastic system (5), because there is no endemic equilibrium for the stochastic system (5). In fact, we still want to find the relation between the solution of stochastic system and E*.Given a weighted digraph(𝒢,A) with n vertices, where A=(akj)nn is the weight matrix, whose entry akj equals the weight of arc (j,k) if it exists, and 0 otherwise, the Laplacian matrix of A is defined as
(34)LA=[∑k≠1‍a1k-a12⋯-a1n-a21∑k≠2‍a2k⋯-a2n⋮⋮⋱⋮-an1-an2⋯∑k≠n‍ank].
Let ck denote the cofactor of the kth diagonal element of LA, and we have the following results.Theorem 5.
AssumeA=(βkj)n×n is irreducible and R0>1. For any given initial value (S1(0),I1(0),…,Sn(0),In(0))∈ℝ+2n, the solution of system (5) has the property
(35)limsup⁡t→∞1t∑k=1n∫0t‍[pkd(Sk-Sk*)2+mk(d+ϵ)(Ik-Ik*)2]ds≤∑k=1n[(ac¯kbkd+(mk+pk)bk2d2)σk12‍+((a+1)c¯kbkd+mkbk2d2)σk22],a.s.,
where E*=(S1*,I1*,…,Sn*,In*) is the endemic equilibrium of system (3) and c¯k, k=1,2,…,n, denote the cofactor of the kth diagonal element of LA¯(A¯=(β¯kj)n×n=(βkjSk*Ij*)n×n), and a, mk, pk, k=1,2,…,n, are positive constants defined as in the proof.Proof.
SinceE* is the endemic equilibrium of system (3), we have
(36)∑j=1n‍βkjSk*Ij*+dSk*=bk,∑j=1n‍βkjSk*Ij*=(d+ϵ)Ik*.
Define
(37)V(S1,I1,…,Sn,In)=a∑k=1nc¯k(Sk-Sk*-Sk*ln⁡SkSk*+Ik-Ik*-Ik*ln⁡IkIk*)‍+∑k=1nc¯k(Ik-Ik*-Ik*ln⁡IkIk*)‍+12∑k=1nmk(Sk-Sk*+Ik-Ik*)2‍+12∑k=1npk(Sk-Sk*)2‍≔aV1+V2+V3+V4,
where a, mk, pk, k=1,2,…,n, are positive constants to be determined later. From the property (1) of Lemma A.2 (see [20]), we know c¯k>0, k=1,2,…,n. Hence V is positive definite. Let L be the generating operator of system (5). Then we get
(38)LV1=∑k=1n‍c¯k(1-Sk*Sk)(bk-∑j=1nβkjSkIj‍-dSk)+∑k=1nc¯kSk*σk122‍+∑k=1nc¯k(1-Ik*Ik)‍(∑j=1nβkjSkIj-(d+ϵ)Ik)+∑k=1nc¯kIk*σk222‍=∑k=1nc¯k[bk-dSk-bkSk*Sk+∑j=1nβkjSk*Ij‍+dSk*‍-(d+ϵ)Ik+(d+ϵ)Ik*-∑j=1nβkjSkIjIk*Ik‍+12(Sk*σk12+Ik*σk22)∑j=1nβkj]=∑k=1nc¯k[∑j=1nβ¯kj‍+dSk*-dSk-∑j=1nβ¯kjSk*Sk‍-dSk*Sk*Sk‍+∑j=1nβ¯kjIjIj*+dSk*-∑j=1nβ¯kjIkIk*‍-∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍+∑j=1nβ¯kj‍+12(Sk*σk12+Ik*σk22)∑j=1nβkj]=∑k=1nc¯k[-dSk*(Sk*Sk+SkSk*-2)+∑j=1nβ¯kj(IjIj*-IkIk*)‍+∑j=1nβ¯kj(2-Sk*Sk-SkIjIk*IkSk*Ij*)‍+12(Sk*σk12+Ik*σk22)∑j=1nβ¯kj(IjIj*-IkIk*)].(39)LV2=∑k=1nc¯k(1-Ik*Ik)‍(∑j=1nβkjSkIj‍-(d+ϵ)Ik)+∑k=1nc¯kIk*σk222‍=∑k=1nc¯k[∑j=1nβkjSkIj‍-(d+ϵ)Ik-Ik*Ik∑j=1nβkjSkIj+(d+ϵ)Ik*+Ik*σk222∑j=1nβkjSkIj]=∑k=1nc¯k[∑j=1nβkjSkIj‍-∑j=1nβ¯kjIkIk*-∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍‍+∑j=1nβ¯kj‍+Ik*σk222].(40)LV3=∑k=1nmk(Sk-Sk*+Ik-Ik*)(bk-dSk-(d+ϵ)Ik)‍+∑k=1nmkσk12Sk2+σk22Ik22‍=∑k=1nmk(Sk-Sk*+Ik-Ik*)‍×(-d(Sk-Sk*)-(d+ϵ)(Ik-Ik*))+∑k=1nmkσk12Sk2+σk22Ik22‍=∑k=1nmk[σk12Sk2+σk22Ik22-d(Sk-Sk*)2-(d+ϵ)(Ik-Ik*)2‍-(2d+ϵ)(Sk-Sk*)(Ik-Ik*)+σk12Sk2+σk22Ik22]≤∑k=1nmk[-(d-(2d+ϵ)22(d+ϵ))(Sk-Sk*)2‍-(d+ϵ)2(Ik-Ik*)2+σk12Sk2+σk22Ik22(d-(2d+ϵ)22(d+ϵ))]=∑k=1nmk[(d+ϵ)2+d22(d+ϵ)(Sk-Sk*)2-(d+ϵ)2(Ik-Ik*)2+σk12Sk2+σk22Ik22].LV4=∑k=1npk(Sk-Sk*)(bk-∑j=1nβkjSkIj‍-dSk)‍+∑k=1npkσk12Sk22‍=∑k=1npk(Sk-Sk*)×(∑j=1n‍βkj(Sk*Ij*-SkIj)-d(Sk-Sk*))‍+∑k=1npkσk12Sk22‍=-∑k=1npk[∑j=1nβkjSk*(Sk-Sk*)(Ij-Ij*)+∑j=1nβkjIj(Sk-Sk*)2+d(Sk-Sk*)2‍-σk12Sk22∑j=1nβkjIj(Sk-Sk*)2].
By property (2) of Lemma A.2 (see [20]), we know
(41)∑k=1n‍c¯k(∑j=1n‍β¯kjIjIj*-∑j=1n‍β¯kjIkIk*)=0,∑k=1n‍c¯k(∑j=1n‍β¯kjln⁡IjIj*-∑j=1n‍β¯kjln⁡IkIk*)=0.
Besides, note that x-1-ln⁡x≥0 for x>0; then
(42)Sk*Sk≥1+ln⁡Sk*Sk,SkIjIk*IkSk*Ij*≥1+ln⁡SkIjIk*IkSk*Ij*.
According to (41) and (42), we get
(43)∑k=1nc¯k∑j=1nβ¯kj(2-Sk*Sk-SkIjIk*IkSk*Ij*)‍‍≤∑k=1nc¯k∑j=1nβ¯kj[2-(1+ln⁡Sk*Sk)-(1+ln⁡SkIjIk*IkSk*Ij*)]‍=∑k=1nc¯k∑j=1nβ¯kj(ln⁡IkIk*-ln⁡IjIj*)‍‍=0,(44)∑k=1nc¯k∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍‍≥∑k=1nc¯k∑j=1nβ¯kj(1+ln⁡SkIjIk*IkSk*Ij*)‍‍∑k=1nc¯k∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍=∑k=1nc¯k∑j=1nβ¯kj(1-ln⁡Sk*Sk)‍‍∑k=1nc¯k∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍+∑k=1nc¯k∑j=1nβ¯kj(ln⁡IjIj*-ln⁡IkIk*)‍‍∑k=1nc¯k∑j=1nβ¯kjSkIjIk*IkSk*Ij*‍≥∑k=1nc¯k∑j=1nβ¯kj(2-Sk*Sk).
Substituting (41) and (43) into (38), we get
(45)LV1≤∑k=1nc¯k[-dSk*(Sk*Sk+SkSk*-2)+12(Sk*σk12+Ik*σk22)]‍=-∑k=1nc¯kd(Sk-Sk*)2Sk+∑k=1nc¯k2(Sk*σk12+Ik*σk22)‍.
Substituting (44) into (39), we get
(46)LV2≤∑k=1nc¯k[∑j=1nβkjSkIj-∑j=1nβ¯kjIkIk*‍-∑j=1nβ¯kj(2-Sk*Sk)‍+∑j=1nβ¯kj‍+Ik*σk222]=∑k=1nc¯k[∑j=1nβkj(Sk-Sk*)(Ij-Ij*)‍‍-∑j=1nβ¯kj(IkIk*-IjIj*)-∑j=1nβ¯kj(2-Sk*Sk-SkSk*)‍+Ik*σk222]=∑k=1n‍c¯k[∑j=1nβkj(Sk-Sk*)(Ij-Ij*)‍+∑j=1nβkjIj*(Sk-Sk*)2Sk+Ik*σk222].
Therefore,
(47)LV=aLV1+LV2+LV3+LV4≤-∑k=1n[pkd-mk(d+ϵ)2+d22(d+ϵ)](Sk-Sk*)2‍-12∑k=1nmk(d+ϵ)(Ik-Ik*)2‍-∑k=1nc¯k(ad-∑j=1nβkjIj*)(Sk-Sk*)2Sk‍+∑k=1n∑j=1nβkj(c¯k-pkSk*)(Sk-Sk*)(Ij-Ij*)‍‍-∑k=1n∑j=1npkβkjIj(Sk-Sk*)2‍‍+∑k=1nac¯kSk*+(mk+pk)Sk22σk12‍+∑k=1n(a+1)c¯kIk*+mkIk22σk22‍≤-∑k=1n[pkd-mk(d+ϵ)2+d22(d+ϵ)](Sk-Sk*)2‍-12∑k=1nmk(d+ϵ)(Ik-Ik*)2‍-∑k=1nc¯k(ad-∑j=1nβkjIj*)(Sk-Sk*)2Sk‍+∑k=1n∑j=1nβkj(c¯k-pkSk*)(Sk-Sk*)(Ij-Ij*)‍+∑k=1nac¯kSk*+(mk+pk)Sk22σk12‍+∑k=1n(a+1)c¯kIk*+mkIk22σk22‍.
Choose  a=max⁡{(∑j=1nβkjIj*)/d,k=1,2,…,n}, mk=(d+ϵ)pkd/((d+ϵ)2+d2), pk=c¯k/Sk*, k=1,2,…,n; then
(48)LV≤-∑k=1npkd2‍(Sk-Sk*)2-12∑k=1nmk(d+ϵ)(Ik-Ik*)2‍+12∑k=1n[(ac¯kbd+(mk+pk)b2d2)σk12‍+((a+1)c¯kbd+mkb2d2)σk22]≔F(t).
Therefore,
(49)dV≤F(t)dt+∑k=1n[[ac¯kσk1(Sk-Sk*)+(mk+pk)σk1(Sk-Sk*)Sk]‍×dBk1(t)+[c¯k(a+1)σk2(Ik-Ik*)+mkσk2(Ik-Ik*)Ik]×dBk2(t)].
Integrating both sides of (49) from 0 to t yields
(50)V(t)-V(0)≤∫0tF(s)ds‍+∫0t∑k=1n[ac¯kσk1(Sk-Sk*)+(mk+pk)σk1(Sk-Sk*)Sk]‍×dBk1(s)+∫0t∑k=1n‍[c¯k(a+1)σk2(Ik-Ik*)+mkσk2(Ik-Ik*)Ik]×dBk2(s).
Let M1(t):=∫0t‍∑‍k=1n[ac¯kσk1(Sk-Sk*)+(mk+pk)σk1(Sk-Sk*)Sk]dBk1(s), M2(t):=∫0t‍∑‍k=1n[c¯k(a+1)σk2(Ik-Ik*)+mkσk2(Ik-Ik*)Ik]dBk2(s), which are local continuous martingale, and M1(0)=M2(0)=0. Moreover
(51)limsup⁡t→∞〈M1,M1〉tt≤8∑k=1n‍σk12[a2c¯k2+(mk+pk)2bk2d2]bk2d2<∞,limsup⁡t→∞〈M2,M2〉tt≤8∑k=1n‍σk22[c¯k2(a+1)2+mk2bk2d2]bk2d2<∞.
By Lemma A.4 (see [20]), we obtain
(52)lim⁡t→∞M1(t)t=0,lim⁡t→∞M2(t)t=0a.s.,
which together with (50) implies
(53)liminf⁡t→∞∫0t‍F(s)dst≥0a.s.
Consequently,
(54)limsup⁡t→∞1t∑k=1n∫0t[pkd(Sk-Sk*)2+mk(d+ϵ)(Ik-Ik*)2]ds‍≤∑k=1n[(ac¯kbkd+(mk+pk)bk2d2)σk12‍+((a+1)c¯kbkd+mkbk2d2)σk22],a.s.
Thus Theorem 5 is proved.Remark 6.
Theorem5 shows that the solution of system (5) fluctuates around the certain level which is relevant to E* of system (3) and σk12, σk22, k=1,2,…,n. The distance between the solution X(t)=(S1(t),I1(t),…,Sn(t),In(t)) and E* of system (3) has the following form:
(55)limsup⁡t→∞1t∫0t‍∥X(s)-E*∥2ds≤C∥σ∥2,
where C is a positive constant and ∥σ∥2=∑‍k=1n(σk12+σk22). Although the solution of system (5) does not have stability as the deterministic system, we can draw a conclusion that system (5) is persistent on the basis of the result of Theorem 5, which also accounts for the fact that the disease is prevalent.


## 5. Simulations and Conclusions

### 5.1. Numerical Simulations

In order to confirm the results above, we numerically simulate the solution of system (5) with n=50, bk=0.25, d=0.3, ϵ=0.01, and initial value Sk(0)=0.5, Ik(0)=0.1, k=1,2,…,50. Using Milstein’s Higer Order Method [31], we get the discretization equation:
(56)Sk,i+1=Sk,i+Δt(bk-∑j=12‍βkjSk,iIj,i-dSk,i)+σk1Sk,iΔtξk1,i+σk122Sk,iΔt(ξk1,i2-1),Ik,i+1=Ik,i+Δt(∑j=12‍βkjSk,iIj,i-(d+ϵ)Ik,i)+σk2Ik,iΔtξk2,i+σk222Ik,iΔt(ξk2,i2-1),
where k=1,2,…,n and ξk1,i, ξk2,i, i=1,2,…,N, are the independent Gaussian random variables N(0,1).From Theorem3 and Remark 4, it is shown that the expectations of Sk(t), Ik(t), k=1,2,…,n, are converging under some conditions, and the solution of system (5) will oscillate around the disease-free equilibrium of system (3). In Figure 1, we choose parameters λ=0.025, σk1=0.005, and σk2=0.03, such that R0≤1, and in Figures 1(a) and 1(b) we choose P(k)=mkexp⁡(-m)/k!, and in Figures 1(c) and 1(d) we choose P(k)=2m2k-3. From Figure 1, we can see that the disease-free equilibrium E0 of system (3) (imaginary lines) is globally asymptotically stable and the curves of system (5) (real lines) always fluctuate around the curves of system (3) (imaginary lines). From Figure 2, we can see that, due to the difference of the degree distribution, the critical value of spread is different.λ
=
0.025, σk1=0.005, σk2=0.03, R0≤1. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=0.4704≤1; (c), (d) P(k)=2m2k-3, m=3, R0=0.9071≤1.
(a)
(b)
(c)
(d)λ
=
0.05, σk1=0.01, σk2=0.02. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=0.9409≤1; (c), (d) P(k)=2m2k-3, m=3, R0=1.8142≥1.
(a)
(b)
(c)
(d)In Figure3, parameters λ=0.08, σk1=0.01, and σk2=0.01 and others are the same as the previous. From Figure 3, we can see that the position of the equilibrium state is different due to the difference of the degree distribution. From Figures 1(d), 2(b), 4(b), and 4(d), we found that the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system with the increase of noise intensity.λ
=
0.08, σk1=0.01, σk2=0.01, R0≥1. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=1.5054≥1; (c), (d) P(k)=2m2k-3, m=3, R0=2.9027≥1.
(a)
(b)
(c)
(d)(a), (b)λ=0.05, σk1=0.015, σk2=0.05, P(k)=mkexp⁡(-m)/k!, m=6, R0=0.94.9≤1. (c), (d)    λ=0.025, σk1=0.01, σk2=0.1, P(k)=2m2k-3, m=3, R0=0.9071≤1.
(a)
(b)
(c)
(d)


### 5.2.  Conclusions

The numerical simulations illustrate the mathematical theorems well. Due to the existence of the noise, the solution of the stochastic system goes around the solution of the deterministic system. With intensities decreasing, the turbulence intensity is weaker. From numerical simulations, we have a new discovery. WhenR0≤1, with the increase of noise intensity, the solution of stochastic system converging to the disease-free equilibrium is faster than the deterministic system. This is because of the fact that, when R0≤1, the disease will die out after some time. However in the real world many stochastic factors contributed to the extinction of the disease.


## 5.1. Numerical Simulations

In order to confirm the results above, we numerically simulate the solution of system (5) with n=50, bk=0.25, d=0.3, ϵ=0.01, and initial value Sk(0)=0.5, Ik(0)=0.1, k=1,2,…,50. Using Milstein’s Higer Order Method [31], we get the discretization equation:
(56)Sk,i+1=Sk,i+Δt(bk-∑j=12‍βkjSk,iIj,i-dSk,i)+σk1Sk,iΔtξk1,i+σk122Sk,iΔt(ξk1,i2-1),Ik,i+1=Ik,i+Δt(∑j=12‍βkjSk,iIj,i-(d+ϵ)Ik,i)+σk2Ik,iΔtξk2,i+σk222Ik,iΔt(ξk2,i2-1),
where k=1,2,…,n and ξk1,i, ξk2,i, i=1,2,…,N, are the independent Gaussian random variables N(0,1).From Theorem3 and Remark 4, it is shown that the expectations of Sk(t), Ik(t), k=1,2,…,n, are converging under some conditions, and the solution of system (5) will oscillate around the disease-free equilibrium of system (3). In Figure 1, we choose parameters λ=0.025, σk1=0.005, and σk2=0.03, such that R0≤1, and in Figures 1(a) and 1(b) we choose P(k)=mkexp⁡(-m)/k!, and in Figures 1(c) and 1(d) we choose P(k)=2m2k-3. From Figure 1, we can see that the disease-free equilibrium E0 of system (3) (imaginary lines) is globally asymptotically stable and the curves of system (5) (real lines) always fluctuate around the curves of system (3) (imaginary lines). From Figure 2, we can see that, due to the difference of the degree distribution, the critical value of spread is different.λ
=
0.025, σk1=0.005, σk2=0.03, R0≤1. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=0.4704≤1; (c), (d) P(k)=2m2k-3, m=3, R0=0.9071≤1.
(a)
(b)
(c)
(d)λ
=
0.05, σk1=0.01, σk2=0.02. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=0.9409≤1; (c), (d) P(k)=2m2k-3, m=3, R0=1.8142≥1.
(a)
(b)
(c)
(d)In Figure3, parameters λ=0.08, σk1=0.01, and σk2=0.01 and others are the same as the previous. From Figure 3, we can see that the position of the equilibrium state is different due to the difference of the degree distribution. From Figures 1(d), 2(b), 4(b), and 4(d), we found that the solution of stochastic system converging to the disease-free equilibrium is faster than that of the deterministic system with the increase of noise intensity.λ
=
0.08, σk1=0.01, σk2=0.01, R0≥1. (a), (b) P(k)=mkexp⁡(-m)/k!, m=6, R0=1.5054≥1; (c), (d) P(k)=2m2k-3, m=3, R0=2.9027≥1.
(a)
(b)
(c)
(d)(a), (b)λ=0.05, σk1=0.015, σk2=0.05, P(k)=mkexp⁡(-m)/k!, m=6, R0=0.94.9≤1. (c), (d)    λ=0.025, σk1=0.01, σk2=0.1, P(k)=2m2k-3, m=3, R0=0.9071≤1.
(a)
(b)
(c)
(d)


## 5.2.  Conclusions

The numerical simulations illustrate the mathematical theorems well. Due to the existence of the noise, the solution of the stochastic system goes around the solution of the deterministic system. With intensities decreasing, the turbulence intensity is weaker. From numerical simulations, we have a new discovery. WhenR0≤1, with the increase of noise intensity, the solution of stochastic system converging to the disease-free equilibrium is faster than the deterministic system. This is because of the fact that, when R0≤1, the disease will die out after some time. However in the real world many stochastic factors contributed to the extinction of the disease.



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*Source: 610959-2014-03-13.xml*