Applied Mathematics and Computation 243 (2014) 546–558

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Dynamics of the deterministic and stochastic SIQS epidemicmodel with non-linear incidence q

http://dx.doi.org/10.1016/j.amc.2014.05.1360096-3003/� 2014 Elsevier Inc. All rights reserved.

q This work was partially supported by Hongliu Young Teachers of Lanzhou University of Technology (No. Q201313), the NNSF of China (10961NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of Uof Gansu Province of China and the Development Program for HongLiu Distinguished Young Scholars in Lanzhou University of Technology.⇑ Corresponding author at: College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, Gansu 730050,

Republic of China.E-mail address: hfhuo@lut.cn (H.-F. Huo).

Xiao-Bing Zhang a,b, Hai-Feng Huo a,b,⇑, Hong Xiang b, Xin-You Meng b

a College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of Chinab Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Keywords:Lyapunov functionRandom perturbationsNon-linear incidenceItô’s formulaStationary distribution

The deterministic and stochastic SIQS models with non-linear incidence are introduced. Fordeterministic model, the basic reproductive rate R0 is derived. Moreover, if R0 6 1, the dis-ease-free equilibrium is globally asymptotically stable and if R0 > 1, there exists a uniqueendemic equilibrium which is globally asymptotically stable. For stochastic model, suffi-cient condition for extinction of the disease that is regardless of the value of R0 is pre-sented. In addition, if the intensities of the white noises are sufficiently small and R0 > 1,then there exists a unique stationary distribution to stochastic model. Numerical simula-tions are also carried out to confirm the analytical results.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Herbert et al. [1] introduced the following SIQS model

S0ðtÞ ¼ A� bIS� lSþ cI þ eQ ;

I0ðtÞ ¼ bIS� ðlþ aþ dþ cÞI;Q 0ðtÞ ¼ dI � ðlþ aþ eÞQ ;

8><>: ð1:1Þ

where S is the number of individuals in the susceptible class, I is the number of individuals who are infectious but not iso-lated, and Q is the number of individuals who are isolated. Constant A is the recruitment rate of susceptibles correspondingto births and immigration. Constant l is the per capita natural mortality rate. Constant d is the rate constant for individualsleaving the infective compartment I for the quarantine compartment Q. Constant a is the disease-related death rate constantin I and Q. Constants c and e are the rates at which individuals recover and return to S from I and Q, respectively. They pre-sented the basic reproduction number R1

0 ¼bðA=lÞ

cþdþlþa. Moreover, if R10 6 1, model (1.1) has only the disease-free equilibrium

E1ðAl ;0;0Þ which is globally asymptotical stable, and if R10 P 1; E1 becomes unstable and there exists a global asymptotically

stable endemic equilibrium E2ðSD; ID;QDÞ, where SD ¼ A=lR1

0; ID ¼ Að1�1=R1

0ÞðlþaÞð1þd=ðeþlþaÞÞ ;Q

D ¼ dID

eþlþa.

018), theniversity

People’s

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 547

It is well known that an incidence rate played an important role in guaranteeing that the model can give a reasonableapproximative description for the disease dynamics. In most classical epidemic models, the bilinear incidence rate andthe standard incidence rate are frequently used. However, there are several reasons for using non-linear incidence rates suchas saturating and nearly bilinear. For instance, Capasso and Serio [2] introduced a saturated incidence rate gðIÞS into epi-demic models, where gðIÞ tends to a saturation level when I gets large, i.e., gðIÞ ¼ bI

1þqI. Ruan and Wang [3] studied an epidemic

model with a specific nonlinear incident rate gðIÞS ¼ bI2S1þqI2. Liu et al. [4] proposed the general incidence rate gðIÞS ¼ bIpS

1þqIq. If the

function gðIÞ is nonmonotone, that is, gðIÞ is increasing when I is small and decreasing when I is large (e.g. gðIÞ ¼ bI1þqI2). It can

be used to interpret the ‘‘psychological’’ effect: for a very large number of infective individuals the infection force maydecrease as the number of infective individuals increases, because in the presence of large number of infectives the popu-lation may tend to reduce the number of contacts per unit time [5]. Lahrouz et al. [6] introduced the more generalized inci-dence rate gðIÞS ¼ I

f ðIÞ S.

To make model (1.1) more realistic, we suppose that the disease-related death rate in I and Q is different and adopt thenonlinear incidence rate IS

f ðIÞ which has been used in [6], where f ðIÞ is continuously differentiable with f ð0Þ ¼ 1, and f 0ðIÞP 0.Firstly, in this paper, we will consider the following model:

S0ðtÞ ¼ A� bISf ðIÞ � lSþ cI þ eQ ;

I0ðtÞ ¼ bISf ðIÞ � ðlþ a2 þ dþ cÞI;

Q 0ðtÞ ¼ dI � ðlþ a3 þ eÞQ :

8>><>>: ð1:2Þ

If a2 ¼ a3 ¼ a and f ðIÞ ¼ 1, then the model (1.2) will become the model (1.1). By using Lyapunov functions, we will show thatthe disease-free equilibrium of (1.2) is globally asymptotically stable when the basic reproduction number is less than orequal to one, and that there is an endemic equilibrium of (1.2) which is globally asymptotically stable when it is greater thanone.

On the other hand, any system is always subject to environmental noise. Environmental noise can provide an additionaldegree of realism in compared to their deterministic counterparts, so it is an important component in realism. Consequently,there are many authors studied stochastic biological systems and stochastic epidemic models. They introduce random effectsinto models by three main ways. The first is directly to perturb parameters of deterministic model by Gaussian white noise(see [7–21]). The second is assumption that stochastic perturbations are around the positive endemic equilibrium of determin-istic model (see [22–25]). The last is assumption that the system will switch from one regime to the other according to the prob-ability law of the Markov chain (see [26–29]). Here, as an extension of model (1.2), we introduce randomness into the (1.2) byreplacing the parameters b; a2 and a3 with bdt þ r1dW1; a2dt þ r2dW2, and a3dt þ r3dW3 as [7], where Wi ði ¼ 1;2;3Þ areindependent standard Brownian motions and r2

i > 0 ði ¼ 1;2;3Þ represent the intensities of Wi ði ¼ 1;2;3Þ, respectively.Hence, we also consider a reasonable stochastic similar of model (1.2) in the following form in this paper:

dSðtÞ ¼ A� bISf ðIÞ � lSþ cI þ eQ

� �dt � r1

ISf ðIÞdW1;

dIðtÞ ¼ bISf ðIÞ � ðlþ a2 þ dþ cÞI� �

dt � r2IdW2 þ r1IS

f ðIÞdW1;

dQðtÞ ¼ ðdI � ðlþ a3 þ eÞQÞdt � r3QdW3:

8>>><>>>:

ð1:3Þ

The organization is as follows. In Section 2, we analysis existence and uniqueness of disease-free equilibrium and ende-mic equilibrium of deterministic model (1.2). Then, using Lyapunov functions, we investigate stability of all equilibria. InSection 3, Firstly, we show the existence and uniqueness of a global positive solution of the model (1.3). Next, we studythe extinction of the model (1.3). At last, we prove that the model (1.3) admits a unique stationary distribution under certainparametric conditions. In Section 4, we make some numerical simulations to illustrate our results. In Section 5, we give somediscussions.

2. The deterministic SIQS model (1.2)

Denote the total population NðtÞ ¼ SðtÞ þ IðtÞ þ QðtÞ, then we have N0ðtÞ ¼ A� lN � a2I � a3Q . Obviously,

D ¼ ðS; I;QÞjS P 0; I P 0;Q P 0; Sþ I þ Q 6 Al

n ois positive invariant set of model (1.2). So, it suffices that we only consider

solutions in the region D. The model (1.2) always has the disease-free equilibrium E0Al ; 0;0� �

. Define the reproduction num-

ber as R0 ¼ Abðcþdþlþa2Þl

.

To get the endemic of equilibrium the model (1.2), we let

A� bISf ðIÞ � lSþ cI þ eQ ¼ 0;

bISf ðIÞ � ðlþ a2 þ dþ cÞI ¼ 0;

dI � ðlþ a3 þ eÞQ ¼ 0:

8>><>>: ð2:1Þ

548 X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558

By f ð0Þ ¼ 1; f 0ðIÞP 0, we have the following equivalent equations to (2.1)

A� bISf ðIÞ � lSþ cI þ eQ ¼ 0;

S ¼ ðlþa2þdþcÞf ðIÞb ;

Q ¼ dIðlþa3þeÞ ;

8>><>>: ð2:2Þ

Substituting S and Q into the first equation of (2.2), we have

KðIÞ¼M A� ðlþ a2 þ dÞðlþ a3 þ eÞ � edlþ a3 þ e

I � lðlþ a2 þ dþ cÞf ðIÞb

¼ 0:

Because of f 0ðIÞP 0, then K0ðIÞ < � ðlþa2þdÞðlþa3þeÞ�edlþa3þe . This results in limI!1KðIÞ ¼ �1. Moreover Kð0Þ ¼ A� lðlþa2þdþcÞ

b . Hence,

If and only if KðIÞ > 0, i.e., R0 > 1; KðIÞ has a unique positive zero. That is, there exists unique endemic equilibrium of the

model (1.2), denoting E� ðS�; I�;Q �Þ.Next, we study the global stability of the disease-free equilibrium E0

Al ;0;0� �

and endemic equilibrium E� ðS�; I�;Q �Þ ofmodel (1.2).

Theorem 2.1. If R0 6 1, then the disease-free equilibrium E0 is globally asymptotically stable.The proof is omitted, because it is similar to the proof of Theorem 2.1 in [1], just notice that S 6 A

l and f ðIÞP 1.

Theorem 2.2. If R0 > 1, then the endemic equilibrium E� is globally asymptotically stable.

Proof. Firstly, we note that the model (1.2) is equivalent to the following system involving N; I and Q:

N0ðtÞ ¼ A� lN � a2I � a3Q ;

I0ðtÞ ¼ bIf ðIÞ ðN � I � QÞ � ðlþ a2 þ dþ cÞI;

Q 0ðtÞ ¼ dI � ðlþ a3 þ eÞQ :

8><>: ð2:3Þ

For convenience, model (2.3) can be rewritten in the following form:

N0ðtÞ ¼ �lðN � N�Þ � a2ðI � I�Þ � a3ðQ � Q �Þ;I0ðtÞ ¼ bI

f ðIÞ ðN � N�Þ � ðI � I�Þ � ðQ � Q �Þ þ S� f ðI�Þ�f ðIÞf ðI�Þ

h i;

Q 0ðtÞ ¼ dðI � I�Þ � ðlþ a3 þ eÞðQ � Q �Þ:

8>><>>: ð2:4Þ

Now, we consider the Lyapunov function

V ¼ 2a2a3 þ da3 þ 2la2

bf ðI�Þ I � I� � I� ln

II�

� �þZ I

I�

ðf ðrÞ � f ðI�ÞÞðr � I�Þr

dr� �

þ a3

2ðN � N�Þ � ðQ � Q �Þ½ �2

þ ð2lþ a3Þ2

ðN � N�Þ2 þ ð2lþ a3Þa2

2dðQ � Q �Þ2:

Obviously, V is positive definite function. The derivative of V along (2.4) is

V 0 ¼ �ð2a2a3 þ da3 þ 2la2ÞðI � I�Þ2 � ð2a2a3 þ da3 þ 2la2ÞS�ðf ðIÞ � f ðI�ÞÞðI � I�Þ

f ðI�Þ � lð2lþ 2a3ÞðN � N�Þ2

� a3ðlþ a3Þdþ a2ð2lþ a3Þðlþ a3 þ eÞd

ðQ � Q �Þ2:

It is easy to know that V 0 is negative definite, using ðf ðIÞ � f ðI�ÞÞðI � I�ÞP 0. Thus, by Lyapunov theorem [30], the endemicequilibrium E� is globally asymptotically stable. h

3. Stochastic SIQS model

3.1. Preliminaries

First, we introduce some notations and results. Let ðX;F; Ftf gtP0; PÞ be a complete probability space with a filtrationFtf gtP0 satisfying the usual conditions (i.e. it is increasing and right continuous while F0 contains all P-null sets). Set

Rdþ ¼ x 2 Rd; xi > 0; 1 6 1 6 d

� .

In general, the d-dimensional stochastic system:

dXðtÞ ¼ f ðt;XðtÞÞdt þ gðt;XðtÞÞdWt; ð3:1Þ

where f ðt; xÞ is a function in Rd defined in t0;1½ � � Rd, and gðt; xÞ is a d�m matrix, f ; g are locally Lipschitz functions in x. Wt

is an m-dimensional standard Wiener process defined on the above probability space. Denote C2;1ðRd � t0;1½ �; RþÞ as the

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 549

family of all nonnegative functions Vðx; tÞ defined on Rd � t0;1½ � such that they are continuously twice differentiable in x andonce in t. The differential operator L of Eq. (3.1) is defined [31] by

L ¼ @

@tþXd

i¼1

fiðtÞ@

@xiþ 1

2

Xd

i;j¼1

gTðx; tÞgðx; tÞ �

ij

@2

@xi@xj: ð3:2Þ

If L acts on a function V 2 C2;1ðRd � t0:;1½ �; RþÞ, then

LVðx; tÞ ¼ Vtðx; tÞ þ Vxðx; tÞf ðx; tÞ þ12

trace gTðx; tÞVxxgðx; tÞ �

:

where Vtðx; tÞ ¼ @V@t ; Vxðx; tÞ ¼ @V

@x1; � � � ; @V

@xd

� �; Vxx ¼ @2V

@xixj

� �d�d

. By Itô’s formula, if xðtÞ 2 Rd, then dVðx; tÞ ¼ LVðx; tÞdtþVxðx; tÞgðx; tÞdWt .

The following lemma is known as the exponential martingale inequality.

Lemma 3.1.1 [31]. Let g ¼ ðg1; . . . ; gmÞ 2 Ł2ðRþ;R1�mÞ, and let T; c; h be any positive numbers. Then

P sup06t6T

Z t

0gðsÞdWðsÞ � c

2

Z t

0gðsÞj j2ds

� �> h

� �6 expð�chÞ:

To show the existence of a stationary distribution, let us cite a known result from Hasminskii [32]. Let XðtÞ be a regulartime-homogeneous Markov process in Rn

þ described by

dXðtÞ ¼ bðXÞdt þXk

r¼1

rrðXÞdBrðtÞ:

The diffusion matrix is defined as follows:

AðXÞ ¼ ðaijðxÞÞ; aijðxÞ ¼Xk

r¼1

rirðxÞrj

rðxÞ:

Lemma 3.1.2 [32]. The markov process XðtÞ has a unique stationary distribution m if there exists a bounded domain U 2 Rd withregular boundary such that its closure �U � Rd, having the following properties:

(i) In the open domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix AðtÞ is bounded awayfrom zero.

(ii) If x 2 Rd n U, the mean time s at which a path issuing from x reaches the set U is finite, and supx2K Exs <1 for every compactsubset K � Rn. Moreover, if f ð�Þ is a function integrable with respect to the measure p, then

P limT!1

1T

Z T

0f ðXxðtÞÞ ¼

ZRd

f ðxÞpðdxÞ�

¼ 1

for all x 2 Rd.

Remark 1. To verify condition (i) it is sufficient to prove that F is uniformly elliptical in U, where

Fu ¼ bðxÞux þ 12 traceðAðxÞuxxÞ, that is, there is a positive number M such that

Pdi;j¼1aijðxÞfifj > M fj j2; x 2 U; f 2 Rd (see [33]

and Rayleigh’s principle in [34]). To validate condition (ii), it is sufficient to show that there is a nonnegative C2-functionW and a neighborhood U such that for some j > 0; LWðxÞ < �j; x 2 Rd n U (see [35]).

3.2. Existence and uniqueness of the positive solution

In this section, we show there is a unique global and positive solution of model (1.3).

Theorem 3.2.1. For any given initial value ðSð0Þ; Ið0Þ;Qð0ÞÞ 2 R3þ, there is a unique positive solution ðSðtÞ; IðtÞ;QðtÞÞ of model

(1.3) on t P 0 and the solution will remain in 2 R3þ with probability 1, namely ðSðtÞ; IðtÞ;QðtÞÞ 2 R3

þ for t P 0 almost surely.

Proof. Since the coefficients of the equation are locally Lipschitz continuous, for any given initial valueðSð0Þ; Ið0Þ;Qð0ÞÞ 2 R3

þ, there is a unique local solution ðSðtÞ; IðtÞ;QðtÞÞ 2 R3þ on t 2 ½0; seÞ, where se is the explosion time

(see [31]). To show that this solution is global, we need to show that se ¼ 1 almost surely (briefly a.s.). Define the stoppingtime s� by

s� ¼ inf t 2 ½0; seÞ : SðtÞ 6 0; or IðtÞ 6 0 or QðtÞ 6 0f g;

550 X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558

where throughout this paper we set inf / ¼ 1 (as usual / denotes the empty set). We have s� 6 se. So, if we can show thats� ¼ 1 a.s., then se ¼ 1 and ðSðtÞ; IðtÞ;QðtÞÞ 2 R3

þ a.s. for all t P 0. Assume that s� <1, then there exists a T > 0 such thatP s� < Tf g > 0. Define a C2-function V : R3

þ ! R by VðS; I;QÞ ¼ ln SIQ . Using Itô’s formula, we get, for x 2 ðs� < TÞ, and for allt 2 ½0; s�Þ,

dVðS; I;QÞ ¼ AS� bI

f ðIÞ � lþ cISþ e

QS� 1

2r1If ðIÞ

� 2 !

dt � r1I

f ðIÞdW1

þ bSf ðIÞ � ðlþ aþ dþ cÞ � 1

2r2

2 þr1Sf ðIÞ

� 2" # !

dt � r2dW2 þ r1S

f ðIÞdW1

þ dIQ� ðlþ aþ eÞ � 1

2ðr3Þ2

� dt � r3dW3 ¼

AS �

bIf ðIÞ � lþ c I

Sþ e QS þ

bSf ðIÞ � ðlþ aþ dþ cÞ þ dI

Q

�ðlþ aþ eÞ � 12

r21I2

f 2ðIÞ �12 r

22 � 1

2r2

1S2

f 2ðIÞ �12 r

23

0@

1Adt

� r1I

f ðIÞdW1 � r2dW2 þ r1S

f ðIÞ dW1 � r3dW3 P�bI � l� ðlþ aþ dþ cÞ � ðlþ aþ eÞ

� 12 r

21I2 � 1

2 r22 � 1

2 r21S2 � 1

2 r23

!dt

� r1I

f ðIÞdW1 � r2dW2 þ r1S

f ðIÞ dW1 � r3dW3;

where f ðIÞP 1 is used. Consequently,

dVðS; I;QÞP�bI � 3l� 2a� d� c� e� 1

2 r21I2 � 1

2 r22 � 1

2 r21S2 � 1

2 r23

!dt � r1

If ðIÞdW1 � r2dW2 þ r1

Sf ðIÞdW1 � r3dW3:

Letting FðS; I;QÞ ¼ �bI � 3l� 2a� d� c� e� 12 r

21I2 � 1

2 r22 � 1

2 r21S2 � 1

2 r23, then

dVðS; I;QÞP FðS; I;QÞdt � r1IdW1 � r2dW2 þ r1SdW1 � r3dW3:

So, we obtain

VðSðtÞ; IðtÞ;QðtÞÞP VðSð0Þ; Ið0Þ;Qð0ÞÞ þZ t

0FðSðuÞ; IðuÞ;QðuÞÞdu� r2W2ðtÞ � r3W3ðtÞ þ r1

Z t

0ðSðuÞ � IðuÞÞdW1ðuÞ:

ð3:3Þ

Note that some components of ðSðs�Þ; Iðs�Þ;Qðs�ÞÞ equal 0. Thereby,

limt!s�

VðSðtÞ; IðtÞ;QðtÞÞ ¼ �1:

Letting t ! s� in (3.3), we have

�1P VðSð0Þ; Ið0Þ;Qð0ÞÞ þZ s�

0FðSðuÞ; IðuÞ;QðuÞÞdu� r2dW2ðs�Þ � r3dW3ðs�Þ þ r1

Z s�

0ðSðuÞ � IðuÞÞdW1ðuÞ > �1;

which leads to contradicts. So we must have s� ¼ 1 a.s., which completes the proof of Theorem 3.2.1. h

3.3. Extinction

In this section, we investigate the condition for the extinction of the disease.

Theorem 3.3.1. For any given initial value ðSð0Þ; Ið0Þ;Qð0ÞÞ 2 R3þ; limt!1 sup ln IðtÞ

t 6 Hðr21;r2

2Þ holds a.s. Further, Hðr21;r2

2Þ < 0,

then IðtÞ tends to zero exponentially almost surely, where Hðr21;r2

2Þ ¼b2

2r21� ðlþ a2 þ dþ cþ 1

2 r22Þ.

Proof. By (1.3) and using Itô’s formula, we have

d ln IðtÞ ¼ bSf ðIÞ � ðlþ a2 þ dþ cÞ � 1

2r2

2 þ r21

S2

f 2ðIÞ

! !dt � r2dW2 þ r1

Sf ðIÞdW1:

Thus, we get

ln IðtÞ ¼ ln Ið0Þ þZ t

0b

SðuÞf ðIðuÞÞ � ðlþ a2 þ dþ cÞ � 1

2r2

2 þ r21

S2ðuÞf 2ðIðuÞÞ

!" #du� r2W2ðtÞ þ

Z t

0r1

SðuÞf ðIðuÞÞdW1ðuÞ:

Denote UðtÞ ¼R t

0 r1SðuÞ

f ðIðuÞÞdW1ðuÞ, which is a continuous local martingale and whose quadratic variation is

UðtÞh i ¼ r21

R t0

S2ðuÞf 2ðIðuÞÞdu.

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 551

By Lemma 3.1.1. we have

P sup06t6k

UðtÞ � c2

UðtÞh ih i

>2c

ln k

( )6 k�

2c ;

where 0 < c < 1; k is a random integer. Making use of Borel–Cantelli lemma yields that for almost all x 2 X, there is a ran-dom integer k0ðxÞ such that for k > k0; sup06t6k UðtÞ � c

2 UðtÞh i �

62c ln k. That is to say

Z t

0r1

SðuÞf ðIðuÞÞ dW1ðuÞ 6

12

cr21

Z t

0

S2ðuÞf 2ðIðuÞÞ duþ 2

cln k; t 2 0; k½ �:

So, we obtain that

ln IðtÞ 6 ln Ið0Þ þZ t

0

b SðuÞf ðIðuÞÞ � ðlþ a2 þ dþ cÞ� 1

2 ðr22 þ r2

1S2ðuÞ

f 2ðIðuÞÞÞ

" #du� r2W2ðtÞ þ

12

cr21

Z t

0

S2ðuÞf 2ðIðuÞÞ duþ 2

cln k

¼ ln Ið0Þ þZ t

0

b SðuÞf ðIðuÞÞ � ðlþ a2 þ dþ cþ 1

2 r22Þ

� 12 ð1� cÞr2

1S2ðuÞ

f 2ðIðuÞÞ

" #du� r2W2ðtÞ þ

2c

ln k:

Since

bSðuÞ

f ðIðuÞÞ �12ð1� cÞr2

1S2ðuÞ

f 2ðIðuÞÞ ¼ �12ð1� cÞr2

1

S2ðuÞf 2ðIðuÞÞ �

2bð1�cÞr2

1

SðuÞf ðIðuÞÞ

þ bð1�cÞr2

1

� �2� b

ð1�cÞr21

� �2

264

375 6 b2

2ð1� cÞr21

;

we have

ln IðtÞ 6 ln Ið0Þ þZ t

0

b2

2ð1� cÞr21

� lþ a2 þ dþ cþ 12r2

2

� " #du� r2W2ðtÞ þ

2c

ln k

¼ ln Ið0Þ þ b2

2ð1� cÞr21

� lþ a2 þ dþ cþ 12r2

2

� " #t � r2W2ðtÞ þ

2c

ln k:

Hence, for k� 1 6 t 6 k, we obtain

ln IðtÞt6

ln Ið0Þtþ b2

2ð1� cÞr21

� lþ a2 þ dþ cþ 12r2

2

� � r2

W2ðtÞtþ 2

cln k

k� 1:

Moreover, let k!1, so t !1 and by the strong law of large numbers to the Brownian motion, we have limt!1 sup W2ðtÞt ¼ 0.

Thus,

limt!1

supln IðtÞ

t6

b2

2ð1� cÞr21

� lþ a2 þ dþ cþ 12r2

2

� :

Finally, let c ! 0, we have

limt!1

supln IðtÞ

t6

b2

2r21

� lþ a2 þ dþ cþ 12r2

2

� :

This completes the proof of Theorem 3.3.1. h

Remark 3.3.2. Hðr21;r2

2Þ ¼b2

2r21� lþ a2 þ dþ cþ 1

2 r22

� �is decreasing in r2

1 and r22. So, as long as r2

1 and r22 are large enough,

then we always can let Hðr21;r2

2Þ < 0, that is, the disease will extinct eventually.

Next, let us study a special case in which assumption that r22 ¼ 0 and r2

3 ¼ 0, thus, the model (1.3) become the followingform:

dSðtÞ ¼ A� bISf ðIÞ � lSþ cI þ eQ

� �dt � r1

ISf ðIÞdW1;

dIðtÞ ¼ bISf ðIÞ � ðlþ a2 þ dþ cÞI� �

dt þ r1IS

f ðIÞ dW1;

dQðtÞ ¼ ðdI � ðlþ a3 þ eÞQÞdt:

8>>><>>>:

ð3:4Þ

� �

To be based on the Theorem 3.2.1, the region D ¼

ðS; I;QÞjS > 0; I > 0;Q > 0; Sþ I þ Q 6 A

lis positive invariant set of model (3.4). So, it

suffices that we only consider solutions in the region D.

For model (3.4), we have the following theorem.

552 X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558

Theorem 3.3.3. For any solution ðSðtÞ; IðtÞ;QðtÞÞ of model (3.4) with initial value ðSð0Þ; Ið0Þ;Qð0ÞÞ 2 D,

limt!1 sup ln IðtÞt 6 ðR0 � 1Þðlþ a2 þ dþ cÞ and limt!1 sup ln IðtÞ

t 6 ðlþ a2 þ dþ cÞ blR0

2r21� 1

� �hold. Further, if R0 < 1 or

r21 >

blR02 , then limt!1 sup ln IðtÞ

t < 0, namely, IðtÞ tends to zero exponentially almost surely, where R0 is the reproductive numberof the deterministic model (1.2).

Proof. By (3.4) and using Itô’s formula, we obtain

d ln IðtÞ ¼ bSf ðIÞ � ðlþ a2 þ dþ cÞ � 1

2r2

1S2

f 2ðIÞ

!dt þ r1

Sf ðIÞdW1:

Integrating and dividing by t, we get

ln IðtÞt¼ ln Ið0Þ

tþ 1

t

Z t

0b

SðuÞf ðIðuÞÞ � ðlþ a2 þ dþ cÞ � 1

2r2

1S2ðuÞ

f 2ðIðuÞÞ

" #duþ 1

t

Z t

0r1

SðuÞf ðIðuÞÞ dW1ðuÞ: ð3:5Þ

Case (i):By SðtÞ 6 A

l and f ðIÞP 1, lead to

ln IðtÞt6

ln Ið0Þtþ 1

t

Z t

0b

Al� ðlþ a2 þ dþ cÞ

� �duþ 1

t

Z t

0r1

SðuÞf ðIðuÞÞ dW1ðuÞ

¼ ln Ið0Þtþ ðR0 � 1Þðlþ a2 þ dþ cÞ þ 1

t

Z t

0r1

SðuÞf ðIðuÞÞ dW1ðuÞ:

Denote UðtÞ ¼R t

0 r1SðuÞ

f ðIðuÞÞdW1ðuÞ, which is a continuous local martingale and whose quadratic variation is

UðtÞh i ¼ r21

R t0

S2ðuÞf 2ðIðuÞÞdu. limt!1 sup UðtÞh i

t 6r2

1A2

l2 <1, a.s. By the large number theorem for martingales (see [36]), we have

limt!1 sup UðtÞh it ¼ 0, a.s. Consequently, we obtain

limt!1

supln IðtÞ

t6 ðR0 � 1Þðlþ a2 þ dþ cÞ:

Moreover, if R0 < 1, then limt!1 sup ln IðtÞt < 0, a.s.

Case (ii):In order to prove case (ii), as long as let r2

2 ¼ 0 in the proof of the Theorem 3.3.1, we can obtain

limt!1

supln IðtÞ

t6

b2

2r21

� ðlþ a2 þ dþ cÞ ¼ ðlþ a2 þ dþ cÞ b2

2r21ðlþ a2 þ dþ cÞ � 1

!¼ ðlþ a2 þ dþ cÞ blR0

2r21

� 1�

:

So, if r21 >

blR02 , then limt!1 sup ln IðtÞ

t < 0 a.s.This completes the proof of Theorem 3.3.3. h

Remark 3.3.4. Theorem 3.3.3 imply that as long as R0 < 1 or r21 >

blR02 holds, then the disease will extinct with probability 1.

3.4. Stationary distribution

In this section, we study the existence of a unique stationary distribution of the model (1.3).

Theorem 3.4.1. If R0 > 1 and the following conditions are satisfied

0 < F < minðg1S�2;g2I�2;g1Q �2Þ; ð3:6Þ

where

F ¼ ebar23Q �2 þ edð2lþ aÞð2lþ aþ dþ eÞI�r2

1S�2

þ dbð2lþ aÞr22I�2 þ edb r2

2I�2 þ r23Q �2

h i;

g1 ¼ bdlð2lþ aþ eÞ � df ðI�Þð2lþ aÞð2lþ aþ dþ eÞI�r21;

g2 ¼ bdð2lþ aÞðlþ aþ d� r22Þ þ bedðlþ a� r2

2Þ;g3 ¼ beaðlþ aþ e� r2

3Þ þ bedðlþ a� r23Þ;

then for any given initial value ðSð0Þ; Ið0Þ;Qð0ÞÞ 2 R3þ, there exists stationary distribution p, and the solution of model (1.3) is

ergodic.

0 50 100 150 2000

2

4

6

8

10

t

S(t)

,I(t),

Q(t) S(t)

I(t)Q(t)

(a)

0 50 100 150 2000

1

2

3

4

5

6

t

S(t)

,I(t),

Q(t)

S(t) I(t)Q(t)

(b)

Fig. 1. (a) A ¼ 1; b ¼ 0:05; l ¼ 0:1; c ¼ 0:2; e ¼ 0:1; a2 ¼ 0:1, a3 ¼ 0:1; d ¼ 0:2; R0 < 1; (b) A ¼ 1; b ¼ 0:112; l ¼ 0:1; c ¼ 0:2; e ¼ 0:1,a2 ¼ 0:1; a3 ¼ 0:1; d ¼ 0:2; R0 > 1.

0 50 100 150 2000

2

4

6

8

10

t

S (t

)

(a)

2 4 6 8 10 120

2000

4000

6000

8000

10000

S (t)

Fre

quen

cy o

f S (t

)

(b)

0 50 100 150 2000

0.5

1

1.5

2

2.5

t

I (t)

(c)

−1 0 1 2 30

2000

4000

6000

8000

10000

I (t)

Fre

quen

cy o

f I (t

)

(d)

0 50 100 150 2000

0.5

1

1.5

2

t

Q (t

)

(e)

−0.5 0 0.5 1 1.50

2000

4000

6000

8000

10000

Q (t)

Fre

quen

cy o

f Q (t

)

(f)

Fig. 2. r1 ¼ 0:11; r2 ¼ 0:001; r3 ¼ 0:001. (a) time-series of SðtÞ; (b) frequency histograms of SðtÞ; (c) time-series of IðtÞ; (d) frequency histograms of IðtÞ;(e) time-series of QðtÞ; (f) frequency histograms of QðtÞ.

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 553

554 X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558

Especially, we have

limt!1

1t

EZ t

0g1ðSðuÞ � S�Þ2 þ g2ðIðuÞ � I�Þ2 þ g3ðQðuÞ � Q�Þ2h i

du 6 F;

where ðS�; I�;Q �Þ is the unique endemic equilibrium of (1.2).

Proof. Define a C2-function V: by

WðS; I;QÞ ¼ beaW1ðQÞ þ df ðI�Þð2lþ aÞð2lþ aþ dþ eÞW2ðIÞ þ bdð2lþ aÞW3ðS; IÞ þ bedW4ðS; I;QÞ; ð3:7Þ

where W1ðQÞ ¼ 12 ðQ � Q �Þ2; W2ðIÞ ¼ I � I� � I� ln I

I� ; W3ðS; IÞ ¼ 12 ðSþ I � S� � I�Þ2; W4ðS; I;QÞ ¼ 1

2 ðSþ I þ Q � S� � I� � Q �Þ2.First, we compute

LW1 ¼ ðQ � Q �ÞðdI � ðlþ aþ eÞQÞ þ 12r2

3Q 2 ¼ ðQ � Q �ÞðdðI � I�Þ � ðlþ aþ eÞðQ � Q �ÞÞ þ 12r2

3ðQ � Q � þ Q �Þ2

6 dðI � I�ÞðQ � Q �Þ � ðlþ aþ eÞðQ � Q �Þ2 þ r23ðQ � Q �Þ2 þ r2

3Q �2

¼ dðI � I�ÞðQ � Q �Þ � ðlþ aþ e� r23ÞðQ � Q �Þ2 þ r2

3Q �2: ð3:8Þ

Next, we have

LW2 ¼ 1� I�

I

� bISf ðIÞ � ðlþ aþ dþ cÞI�

þ 12

I�

I2 ðr22I2 þ r2

1S2I2

f 2ðIÞÞ ¼ ðI � I�Þ bSf ðIÞ � ðlþ aþ dþ cÞ�

þ 12

I� r22 þ

r21S2

f 2ðIÞ

!

¼ ðI � I�Þ bSf ðIÞ �

bS�

f ðI�Þ

� þ 1

2I� r2

2 þr2

1S2

f 2ðIÞ

!¼ ðI � I�Þ � bSðf ðIÞ � f ðI�ÞÞ

f ðIÞf ðI�Þ þ bðS� S�Þf ðI�Þ

� þ 1

2I� r2

2 þr2

1S2

f 2ðIÞ

!:

Noting that bSðf ðIÞ�f ðI�ÞÞðI�I�Þf ðIÞf ðI�Þ P 0 and f ðIÞ > 1, so

LW2 6bðI � I�ÞðS� S�Þ

f ðI�Þ þ 12

I�r22 þ I�r2

1ðS� S� þ S�Þ2

26

bðI � I�ÞðS� S�Þf ðI�Þ þ 1

2I�r2

2 þ I�r21ðS� S�Þ2 þ I�r2

1S�2: ð3:9Þ

Now, we calculate

LW3 ¼ ðSþ I � S� � I�ÞðA� lSþ eQ � ðlþ aþ dÞIÞ þ 12r2

2I2

¼ ðSþ I � S� � I�Þð�lðS� S�Þ þ eðQ � Q �Þ � ðlþ aþ dÞðI � I�ÞÞ þ r22ðI � I� þ I�Þ2

2

6 �lðS� S�Þ2 � ðlþ aþ dÞðI � I�Þ2 þ eðS� S�ÞðQ � Q �Þ � ð2lþ aþ dÞðS� S�ÞðI � I�Þ

þ eðI � I�ÞðQ � Q �Þ þ r22ðI � I�Þ2 þ r2

2I�2 ¼ �lðS� S�Þ2 � ðlþ aþ d� r22ÞðI � I�Þ2

þ eðS� S�ÞðQ � Q �Þ � ð2lþ aþ dÞðS� S�ÞðI � I�Þ þ eðI � I�ÞðQ � Q �Þ þ r22I�2: ð3:10Þ

At last, for W4ðS; I;QÞ, we have

LW4 ¼ ðSþ I þ Q � S� � I� � Q �ÞðA� lS� ðlþ aÞI � ðlþ aÞQÞ þ 12r2

2I2 þ 12r2

3Q 2

¼ ðSþ I þ Q � S� � I� � Q �Þ�lðS� S�Þ � ðlþ aÞðI � I�Þ

�ðlþ aÞðQ � Q �Þ

!þ 1

2r2

2ðI � I� þ I�Þ2

þ 12r2

3ðQ � Q � þ Q �Þ2 6 �lðS� S�Þ2 � ðlþ aÞðI � I�Þ2 � ðlþ aÞðQ � Q �Þ2

� ð2lþ aÞðS� S�ÞðI � I�Þ � ð2lþ aÞðS� S�ÞðQ � Q �Þ � ð2lþ 2aÞðI � I�ÞðQ � Q �Þ

þ r22ðI � I�Þ2 þ r2

2I�2 þ r23ðQ � Q �Þ2 þ r2

3Q �2 ¼ �lðS� S�Þ2 � ðlþ a� r22ÞðI � I�Þ2

� ðlþ a� r23ÞðQ � Q �Þ2 � ð2lþ aÞðS� S�ÞðI � I�Þ � ð2lþ aÞðS� S�ÞðQ � Q �Þ

� ð2lþ 2aÞðI � I�ÞðQ � Q �Þ þ r22I�2 þ r2

3Q �2: ð3:11Þ

0 50 100 150 2003

3.5

4

4.5

5

5.5

6

t

S (t

)

(a)

5.35 5.4 5.45 5.5 5.550

500

1000

1500

2000

S (t)

Fre

quen

cy o

f S (t

)

(b)

0 50 100 150 2000.8

1

1.2

1.4

1.6

1.8

2

t

I (t)

(c)

1.3 1.35 1.4 1.45 1.50

500

1000

1500

2000

I (t)

Fre

quen

cy o

f I (t

)

(d)

0 50 100 150 2000.5

1

1.5

2

t

Q (t

)

(e)

0.9 0.92 0.94 0.960

500

1000

1500

2000

Q (t)

Fre

quen

cy o

f Q (t

)

(f)

Fig. 3. r1 ¼ 0:001; r2 ¼ 0:0001; r3 ¼ 0:001. (a) time-series of SðtÞ; (b) frequency histograms of SðtÞ; (c) time-series of IðtÞ; (d) frequency histograms of IðtÞ;(e) time-series of QðtÞ; (f) frequency histograms of QðtÞ.

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 555

Substituting (3.8)–(3.11) into (3.7), simultaneously considering (3.6), we get

WðS; I;QÞ ¼ beaW1ðRÞ þ df ðI�Þð2lþ aÞð2lþ aþ dþ eÞW2ðIÞ þ bdð2lþ aÞW3ðS; IÞ þ bedW4ðS; I;QÞ

6 bea �ðlþ aþ e� r23ÞðQ � Q �Þ2 þ r2

3Q �2h i

þ df ðI�Þð2lþ aÞð2lþ aþ dþ eÞ

12

I�r22 þ I�r2

1ðS� S�Þ2 þ I�r21S�2

� �þ bdð2lþ aÞ �lðS� S�Þ2 � ðlþ aþ d� r2

2ÞðI � I�Þ2 þ r22I�2

h i

þ bed�lðS� S�Þ2 � ðlþ a� r2

2ÞðI � I�Þ2

�ðlþ a� r23ÞðQ � Q �Þ2 þ r2

2I�2 þ r23Q �2

" #¼

� bdlð2lþ aþ eÞ � df ðI�Þð2lþ aÞð2lþ aþ dþ eÞI�r21

�ðS� S�Þ2

� bdð2lþ aÞðlþ aþ d� r22Þ þ bedðlþ a� r2

2Þ �

ðI � I�Þ2

� beaðlþ aþ e� r23Þ þ bedðlþ a� r2

3Þ �

ðQ � Q �Þ2

þ bear23Q �2 þ df ðI�Þð2lþ aÞð2lþ aþ dþ eÞ 1

2I�r2

2 þ I�r21S�2

� �þ bdð2lþ aÞr2

2I�2 þ bedðr22I�2 þ r2

3Q �2Þ

¼ �g1ðS� S�Þ2 � g2ðI � I�Þ2 � g3ðQ � Q �Þ2 þ F:

556 X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558

By (3.6), we know that ellipsoid

Fig. 4.

�g1ðS� S�Þ2 � g2ðI � I�Þ2 � g3ðQ � Q �Þ2 þ F ¼ 0

lies entirely in R3þ. we can take D as any neighborhood of the ellipsoid such that �D � R3

þ, where �D is the closure of D. So, weobtain LWðS; I;QÞ < 0 for ðS; I;QÞ 2 R3

þ n D which implies condition (ii) in Lemma 3.1.2 holds. Besides, it is easy to know thatthe diffusion matrix associated to (1.3) is uniformly elliptic D (see [33]). Hence, by Rayleigh’ principle [34], condition (i) inLemma 3.1.2 is verified. Consequently, the stochastic system (1.3) has a stable stationary distribution. h

Remark 3.4.2. From Theorem 3.4.1, we get

limr2

1 ;r22 ;r

23!1

F ¼ 0;

limr2

1!1g1 ¼ bdlð2lþ aþ eÞ > 0;

limr2

2!1g2 ¼ bdð2lþ aÞðlþ aþ dÞ þ bedðlþ aÞ > 0;

limr2

3!1g3 ¼ beaðlþ aþ eÞ þ bedðlþ aÞ > 0;

that is, the solution of model (1.3) fluctuates around the endemic equilibrium ðS�; I�;Q �Þ of (1.1). Moreover, with the values ofr2

1; r22; r2

3 decreasing, the difference between them also decreases.

4. Numerical simulations

In order to confirm the results above, we numerically simulate the solution of system (1.2) and (1.3). We always choosethe same initial value ðSð0Þ; Ið0Þ; Qð0ÞÞ ¼ ð3;2;1Þ and f ðIÞ ¼ 1þ 3I þ 2I2.

In Fig. 1(a), let A ¼ 1; b ¼ 0:05; l ¼ 0:1; c ¼ 0:2; e ¼ 0:1; a2 ¼ 0:1; a3 ¼ 0:1; d ¼ 0:2 such that R0 ¼ 0:8333 < 1. So, The-orem 2.1 holds, namely, the disease-free equilibrium of deterministic model (1.2) is globally asymptotically stable; InFig. 1(b), we choose the same parameters in Fig. 1(a) except b ¼ 0:112 such that R0 ¼ 1:8867 > 1, that is, Theorem 2.2 holds.

In Figs. 2 and 3, we choose the same parameters in Fig. 1(a) except r1; r2; r3. In Fig. 2, we chooser1 ¼ 0:11; r2 ¼ 0:001; r3 ¼ 0:001 which are satisfied the Theorem 3.3.1, that is, the disease is disappear eventually in sto-chastic model (1.3). Fig. 2 confirms this, where (b), (d) and (f) are frequency histograms of SðtÞ; IðtÞ and QðtÞ respectively,based on 10,000 stochastic simulations at time t ¼ 200 using the above parameter values. Moreover, In Fig. 3, we chooser1 ¼ 0:001; r2 ¼ 0:01; r3 ¼ 0:001 such that the conditions of Theorem 3.4.1 hold. Namely, there is a stationary distribution

0 50 100 150 2000

0.5

1

1.5

2

t

I (t)

(a)

0 50 100 150 2002

4

6

8

10

t

S (t

)

(b)

0 50 100 150 2000

0.5

1

1.5

2

t

Q (t

)

(c)

A ¼ 1; b ¼ 0:0672; l ¼ 0:1; c ¼ 0:2; e ¼ 0:1; a2 ¼ 0:1; a3 ¼ 0:1; d ¼ 0:2;r1 ¼ 0:05; r2 ¼ 0, r3 ¼ 0; R0 ¼ 1:1200 > 0, and blR02 � r2

1 ¼ 0:0013 > 0.

0 1

1

2

3

disease willdie out

disease willdie out disease will

die outR0 = 1

σ21 = βμR0

2

R0

σ21

R0 = ϕ σ21

Fig. 5. For model (3.4), when R0 < 1 or r21 >

blR02 is satisfied, the disease will extinct with probability.

X.-B. Zhang et al. / Applied Mathematics and Computation 243 (2014) 546–558 557

for model (1.3). Fig. 3 supports this result, where (b), (d) and (f) are frequency histograms of SðtÞ; IðtÞ and QðtÞ, respectively,based on 40,000 stochastic simulations at time t ¼ 200 using the given parameter values.

5. Discussion

In this paper, we explore a deterministic and stochastic SIQS model with non-linear incidence. For deterministic model(1.2), we present the basic reproduction number R0. If R0 6 1, the disease-free equilibrium is globally asymptotically stable,and if R0 > 1, the endemic equilibrium is globally asymptotically stable. Moreover, for stochastic model (1.3), we have obtainsufficient conditions for the extinction of a disease and existence of a unique stationary distribution. Further more, weobserve that R0 < 1 not is necessary condition that Theorem 3.3.1 holds. This implies that R0 will not act as the thresholdto determine the extinction or the persistence of epidemics as that of the deterministic model to stochastic model (1.3). Thatis, the introduction of noise modifies the threshold R0 of deterministic model (1.2).

Especially, the Theorem 3.3.3 tell us that in I = ðr21;R0Þ R0 < 1;r2

1 >blR0

2

��n o(see Fig. 5), the disease will extinct with prob-

ability 1. In Fig. 4, we choose A ¼ 1; b ¼ 0:0672; l ¼ 0:1; c ¼ 0:2; e ¼ 0:1; a2 ¼ 0:1; a3 ¼ 0:1; d ¼ 0:2; r1 ¼ 0:05; r2 ¼ 0and r3 ¼ 0. By calculating, we have R0 ¼ 1:1200 > 0; blR0

2 � r21 ¼ 0:0013 > 0, that is, conditions of Theorem 3.3.3 can not

be satisfied. However, in Fig. 4, we can find that the disease still extinct, that means that conditions of the Theorem 3.3.3are just sufficient conditions of disappearance of disease. So, we have a conjecture of model (2.1) that there is a functioncurve R0 ¼ uðr2

1Þ passing through point ðr21;R0Þ ¼ ð0;1Þ, and when R0 < uðr2

1Þ (see Fig. 5), the disease extinct. Namely, there

may exist a function R0 ¼ uðr21Þ such that the disease is still disappear in II ¼ ðr2

1;R0Þ R0 < uðr21Þ; R0 > 1 and r2

1 <blR0

2

��n o(see Fig. 5). We leave this conjecture for open problem and look forward to solve it in the future.

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