Research ArticleGlobal Stability of Multigroup SIRS EpidemicModel with Varying Population Sizes and StochasticPerturbation around Equilibrium

Xiaoming Fan

School of Mathematical Sciences Harbin Normal University Harbin 150500 China

Correspondence should be addressed to Xiaoming Fan fanxm093163com

Received 8 October 2013 Accepted 17 December 2013 Published 20 January 2014

Academic Editor Francisco Solıs Lozano

Copyright copy 2014 Xiaoming FanThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We discuss multigroup SIRS (susceptible infectious and recovered) epidemic models with random perturbations We carry out adetailed analysis on the asymptotic behavior of the stochastic model when reproduction numberR

0gt 1 we deduce the globally

asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium ofthe deterministic model in time average Numerical methods are employed to illustrate the dynamic behavior of the model andsimulate the system of equations developed The effect of the rate of immunity loss on susceptible and recovered individuals is alsoanalyzed in the deterministic model

1 Introduction

To curb the spread and impact of viruses it is importantto study their feature propagating methods means andlimitation Many studies about the outbreak and spread ofdisease have been done by means of establishing epidemicmodels These researches provided some useful and validreference for the characteristic of disease transmission Basedon these results of theoretical analysis one can predict thefuture course of an outbreak and evaluate strategies to controlan epidemic In 1927 Kermack and McKendrick created anSIR model in which they considered a fixed population withonly three compartments of three classes susceptible 119878(119905)infected 119868(119905) and removed 119877(119905) 119878(119905) represents the numberof individuals not yet infected with the disease at time 119905 orthose susceptible to the disease 119868(119905) denotes the number ofindividuals who have been infected with the disease and arecapable of spreading the disease to those in the susceptibleclass 119877(119905) is the compartment used for those individuals whohave been infected and then removed from the disease eitherdue to immunization or due to death Those in this classare not able to be infected again or to transmit the infection

to others A single group SIRS model is an extension of theSIR model The SIRS model for infections that do not conferpermanent immunity that is an infection that does not leavea long lasting immunity thus individuals that have recoveredreturn to being susceptible again moving back into the 119878(119905)compartment The only difference between SIR and SIRS isthat SIRS model allows members of the recovered class to befree of infection and rejoin the susceptible class

Considering different contact patterns a distinct numberof sexual partners or different geography and so forth itis more appropriate to divide individual hosts into groupsin modeling epidemic disease [1] Therefore it is reasonableto propose multigroup models to describe the transmissiondynamics of viruses in heterogeneous host populations onepidemic models In fact there are already many schol-ars focusing their study on various forms of multigroupepidemic models (see [2ndash7]) They have also proved theglobal stability of the unique endemic equilibrium throughLyapunov function which is one of the main mathematicalchallenges in analyzing multigroup models Recently Guo etal have paid more attention to multigroup models [8ndash10]They first proposed a graph-theoretic approach to themethod

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 154725 14 pageshttpdxdoiorg1011552014154725

2 Abstract and Applied Analysis

of global Lyapunov functions and used it to establish theglobal stability of the interior equilibrium for more generalmodels [8]Muroya et al considered a class of 119899-group (119899 ⩾ 2)SIRS epidemic models described by the following system ofequations [11]

119896= Λ119896minus 119889119878

119896119878119896minus

119899

sum

119895=1

120573119896119895119878119896 (119905) 119868119895 (119905) + 120575119896119877119896

119896=

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) minus (119889

119868

119896+ 120574119896) 119868119896

119896= 120574119896119868119896minus (119889119877

119896+ 120575119896) 119877119896 119896 = 1 119899

(1)

Nakata et al [12] and Enatsu et al [13] proposed an idea toextend Lyapunov functional techniques inMcCluskey [14] forSIR epidemic models to SIRS epidemic models By extendingwell-known Lyapunov function techniques Muroya et al[11] succeeded to prove the global stability of system (1)without use of the grouping technique by graph theory4 in Guo et al [10] Note that because of environmentalnoises the deterministic approach has some limitations in themathematicalmodeling transmission of an infectious diseaseseveral authors began to consider the effect of white noise inepidemic models [15ndash18] Beretta et al proved the stabilityof epidemic model with stochastic time delays influencedby probability under certain conditions [19] Such type ofstochastic perturbations firstly was proposed in [19 20]and later was successfully used in many other papers formany other different systems (see eg [21ndash27]) Yuan etal in [28] and Yu et al in [29] all investigated epidemicmodels with fluctuations around the positive equilibrium andthey proved locally stochastically asymptotic stability of thepositive equilibrium Ji et al also discuss a multigroup SIRmodel with stochastic perturbation and deduce the globallyasymptotic stability of the disease-free equilibrium when1198770le 1 which means the disease will die out When

1198770gt 1 they derive the disease will prevail which is

measured through the difference between the solution andthe endemic equilibrium of the deterministic model in timeaverage [1] Imhof and Walcher [30] considered a stochasticchemostat model and they proved that the stochastic modelled to extinction even though the deterministic counterpartpredicts persistence In our previous work we consideredan SEIR epidemic model with constant immigration andrandom fluctuation around the endemic equilibrium wecarried out a detailed analysis on the asymptotic behaviorof the stochastic model [31] and we also investigated a two-group epidemic model with distributed delays and randomperturbation [32] In addition we investigated multigroupSEIQR models with and without random perturbation incomputer network [33] In the present paper based on system(1) to examine the influence of white noise on system (1)we also consider a stochastic version of the SIRS modelby perturbing the deterministic system (1) by a white noiseand assume that the perturbations are around the positiveendemic equilibrium of epidemic models

This paper is organized as follows We begin in Section 2with necessary background with respect to the deterministic

multigroup SIRS model We establish the global dynamicsdetermined by the basic reproduction number R

0and

introduce some results of graph theory used by Guo etal We also review the important results in Theorem 2 ondeterministic model (1) by Muroya et al In Section 3 wederive the stochastic version from the deterministic model(1) In Section 4 we analyse the asymptotic behavior of thestochastic model by means of the method of Lyapunovfunctions and the theories of stochastic differential equationin Theorem 5 Numerical methods are employed to simulatethe dynamic behavior of the model and the effect of therate of immunity loss on the recovered is also analyzed inthe deterministic models and the corresponding stochasticmodels in Section 5 Finally we give the conclusion of ourpaper in Section 6

2 Global Stability of DeterministicMultigroup SIRS Models

First let us review some theories and results on deterministicmultigroup SIRS models we summarize the parameters inthe model (1) by the following list

Λ119896 the recruitment rate of the population

120573119896119895 transmission coefficient between compartments

119878119896and 119868119895

119889119878

119896 119889119868

119896 119889119877

119896 the natural death rates of susceptible

infected and recovered individuals in city 119896 respec-tively

120575119896 the rate of immunity loss in the 119896th group

120574119896 the natural recovery rate of the infected individuals

in city 119896

We assume 119889119878119896 119889119868

119896 119889119877

119896 Λ119896gt 0 and the rest of the parameters

are nonnegative for all 119896 In particular 120573119896119895= 0 if there is no

transmission of the disease between compartments 119878119896and 119868119895

By the biological meanings we may assume that

119889119878

119896⩽ min 119889119868

119896 119889119877

119896 119896 = 1 2 119899 (2)

For each 119896 adding the three equations in (1) gives(119878119896+ 119868119896+ 119877119896)1015840

⩽ Λ119896minus 119889119878

119896(119878119896+ 119868119896+ 119877119896) Hence

lim sup119905rarrinfin

(119878119896+119868119896+119877119896) ⩽ Λ

119896119889119878

119896Therefore omega limit sets

of system (1) are contained in the following bounded regionin the non-negative cone of 1198773119899

Γ = (1198781 1198681 1198771 119878

119899 119868119899 119877119899) isin 1198773119899

+| 119878119896⩽ 1198780

119896

119878119896+ 119868119896+ 119877119896⩽

Λ119896

119889119878

119896

1 ⩽ 119896 ⩽ 119899

(3)

Abstract and Applied Analysis 3

It can be verified that region Γ is positively invariant andsystem (1) always has the disease-free equilibrium 119875

0=

(1198780

1 0 0 0 0 119878

0

119899 0 0 0 0) where 1198780

119896= Λ

119896119889119878

119896is the

equilibrium of the 119878119896population in the absence of disease

(1198681= 1198682= sdot sdot sdot = 119868

119899= 0) Let

∘

Γ denote the interior of Γ Anendemic equilibrium119875lowast = (119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) belongs

to∘

Γ satisfying the equilibrium equations

Λ119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895minus 119889119878

119896119878lowast

119896+ 120575119896119877lowast

119896 (4)

(119889119868

119896+ 120574119896) 119868lowast

119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895 (5)

(119889119868

119896+ 120575119896) 119877lowast

119896= 120574119896119868lowast

119896 (6)

LetR0= 120588 (119872

0) (7)

denote the spectral radius of the matrix

1198720= (

1205731198961198951198780

119896

119889119868

119896+ 120574119896

)

1⩽119896119895⩽119899

(8)

The parameter R0is referred to as the basic reproduction

number Its biological significance is that if R0lt 1 the

disease dies outwhile ifR0gt 1 the disease becomes endemic

[9] In the following theorem we show that the multi-groupmodel (1) has at least one endemic equilibrium119875lowastwhenR

0gt

1 and 119875lowast is globally stable [9]The matrix 119861 = (120573

119896119895) denotes the contact matrix

Associated with 119861 one can construct a directed graph L =

119866(119861) whose vertex 119896 represents the 119896th group 119896 = 1 119899A directed edge exists from vertex 119896 to vertex 119895 if andonly if 120573

119896119895gt 0 Throughout the paper we assume that

119861 is irreducible This is equivalent to 119866(119861) being stronglyconnected Biologically this is the same as assuming thatany two groups 119896 and 119895 have a direct or indirect route oftransmission More specifically individuals in 119868

119895can infect

ones in 119878119896directly or indirectly Now consider the linear

system

Bℓ = 0 (9)where

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(10)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899 LetL = 119866(119861) denote

the directed graph associated with matrix 119861 (and (120573119896119895)) and

let 119862119895119896denote the cofactor of the (119895 119896) entry ofB

We have the following fundamental lemma[8]

Lemma 1 (Kirchhoff rsquos Matrix-Tree Theorem) Assume that(120573119896119895)119899times119899

is irreducible and 119899 ⩾ 2 Then the following resultshold

(1) The solution space of system (9) has dimension 1 witha basis (ℓ

1 ℓ2 ℓ

119899) = (119862

11 11986222 119862

119899119899)

(2) For 1 ⩽ 119896 ⩽ 119899

119862119896119896= sum

119879isinT119896

119882(119879) = sum

119879isinT119896

prod

(119903119898)isin119864(119879)

120573119903119898gt 0 (11)

where T119896is the set of all directed spanning subtrees ofL that are

rooted at vertex 119896119882(119879) is the weight of a directed tree T and119864(119879) denotes the set of directed arcs in a directed tree T

Y Muroya et al proved the following theorem

Theorem 2 Assume that 119861 = (120573119896119895) is irreducible and

inequality (2) holds If R0gt 1 and 119889119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 then

system (1) has at least one endemic equilibrium 119875lowast in

∘

Γ and119875lowast is globally asymptotically stable

3 Multigroup Stochastic SIRS Model

Under assumptions that R0gt 1 119889

119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 119889119878

119896⩽

min119889119868119896 119889119877

119896 and B = (120573

119896119895) is irreducible in Theorem 2

we know from Section 2 that there exists a positive endemicequilibrium 119875

lowast in∘

Γ Furthermore we assume stochasticperturbations are of white noise type which are directlyproportional to distances 119878

119896(119905) 119868119896(119905) and119877

119896(119905) from values of

119878lowast

119896 119868lowast

119896 119877lowast

119896 influence on the 119878

119896(119905) 119868119896(119905) and119877

119896(119905) respectively

So system (1) results in

119896= Λ119896minus 119889119878

119896119878119896minus

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) + 120575

119896119877119896

+ 1205901119896(119878119896minus 119878lowast

119896) 1119896

119896=

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) minus (119889

119868

119896+ 120574119896) 119868119896+ 1205902119896(119868119896minus 119868lowast

119896) 2119896

119896= 120574119896119868119896minus (119889119877

119896+ 120575119896) 119877119896+ 1205903119896(119877119896minus 119877lowast

119896) 3119896

119896 = 1 119899

(12)

where 1198611119896(119905) 1198612119896(119905) and 119861

3119896(119905) are independent standard

Brownian motions and 1205902119894119896gt 0 represent the intensities

of 119861119894119896(119905) (119894 = 1 2 3) respectively Obviously stochastic

system (12) has the same equilibrium points as system (1)Hence when the stochasticmodel (12) satisfies the conditionsof Theorem 2 we can ensure the existence of the positiveequilibrium point of the stochastic model (12) Note thatbecause of the influence of the white noise the componentsof solutions to (12) can also be negative at some time Butbecause it is impossible for populations to be less than zerophysically once the negative sections appear on componentsof solutions of (12) we only need to consider the nonnegative

4 Abstract and Applied Analysis

section of the components of solutions In the next sectionwe will investigate asymptotic stability of the equilibrium119875lowast of system (12) To this end we will construct a class

of different Lyapunov functions to achieve our proof undercertain conditions

4 Stochastic Stability ofthe Endemic Equilibrium

Let us now proceed to discuss asymptotic stability of thesystem (12) In this paper unless otherwise specified let(ΩF F

119905119905⩾1199050

119875) be a complete probability space with afiltration F

119905119905⩾1199050

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all 119875-

null sets) Let 119861119894119896(119905) be the Brownian motions defined on

this probability space If R0gt 1 then the stochastic

system (12) can be centered at its endemic equilibrium 119875lowast=

(119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) by the change of variables

u119896= 119878119896minus 119878lowast

119896 k

119896= 119868119896minus 119868lowast

119896 w

119896= 119877119896minus 119877lowast

119896

(13)

we obtain the following system

u119896= minus 119889

119878

119896u119896minus

119899

sum

119895=1

120573119896119895119878lowast

119896k119895minus

119899

sum

119895=1

120573119896119895u119896k119895

minus

119899

sum

119895=1

120573119896119895u119896119868lowast

119895+ 120575119896w119896+ 1205901119896u1198961119896

k119896=

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus (119889119868

119896+ 120574119896) k119896+ 1205902119896k1198962119896

w119896= 120574119896k119896minus (119889119877

119896+ 120575119896)w119896+ 1205903119896w1198963119896

119896 = 1 119899

(14)

It is clear that the stability of equilibrium of system(12) is equivalent to the stability of zero solution of system(14) Considering the 119889-dimensional stochastic differentialequation

119889x (119905) = 119891 (x (119905) 119905) 119889119905 + 119892 (x (119905) 119905) 119889119861 (119905) 119905 ⩾ 1199050

(15)

If the assumptions of the existence and uniqueness theoremare satisfied then for any given initial value x(119905

0) = 1199090isin R119889

(15) has a unique global solution denoted by x(119905 1199050 1199090) For

the purpose of stability we assume in this section 119891(0 119905) = 0and 119892(0 119905) = 0 for all 119905 ⩾ 119905

0 So (15) admits a solution

x(119905) equiv 0 which is called the trivial solution or equilibriumposition We let 11986221(R119889 times [119905

0infin)R

+) denote the family of

all nonnegative functions 119881(x 119905) on R119889 times [1199050infin) which are

continuously twice differentiable in x and once in 119905 Definethe differential operator 119871 associated with (15) by

119871 =

120597

120597119905

+

119889

sum

119894=1

119891119894(x 119905) 120597

120597119909119894

+

1

2

119889

sum

119894119895=1

[119892119879(x 119905) 119892 (x 119905)]

119894119895

1205972

120597119909119894119909119895

(16)

If 119871 acts on a function 119881 isin 11986221(R119889 times [1199050infin)R

+) then

119871119881 (x 119905) = 119881119905 (x 119905) + 119881119905 (x 119905) 119891 (x 119905)

+

1

2

trace [119892119879 (x 119905) 119881119909119909 (x 119905) 119892 (x 119905)] (17)

Definition 3 (1) The trivial solution of (15) is said to bestochastically stable or stable in probability if for every pairof 120576 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120576 119903 119905

0) gt 0 such

that

119875 1003816100381610038161003816x (119905 1199050 1199090)1003816100381610038161003816lt 119903 forall119905 ⩾ 119905

0 ⩾ 1 minus 120576 (18)

whenever |1199090| lt 120575 Otherwise it is said to be stochastically

unstable(2) The trivial solution is said to be stochastically asymp-

totically stable if it is stochastically stable and for every 120576 isin(0 1) there exists a 120575

0= 1205750(120576 1199050) gt 0 such that

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 ⩾ 1 minus 120576 (19)

whenever |1199090| lt 1205750

(3) The trivial solution is said to be stochastically asymp-totically stable in the large if it is stochastically stable and forall 1199090isin R119889

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 = 1 (20)

Before proving themain theoremweput forward a lemmain [34]

Lemma 4 (see [34]) If there exists a positive-definite decres-cent radially unbounded function 119881(x 119905) isin 119862

21(R119889 times

[1199050infin)R

+) such that 119871119881(x 119905) is negative-definite then the

trivial solution of (15) is stochastically asymptotically stable inthe large

From the above lemma we can obtain the stochasticallyasymptotical stability of equilibrium as following

Theorem 5 Assume that B = (120573119896119895) is irreducible and

inequality (2) holds and R0gt 1 119889119878

119896119878lowast

119896minus 120575119896119877lowast

119896⩾ 0 then if

the following condition is satisfied

Abstract and Applied Analysis 5

0 5 10 150

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

5

10

15

t

S1(t)

I1(t)

R1(t)

t

S2(t)

I2(t)

R2(t)

Figure 1 Deterministic trajectories of SIRSmodel (1) for initial condition 1198781(0) = 2 119868

1(0) = 4119877

1(0) = 6 119878

2(0) = 9 119868

2(0) = 4 and119877

2(0) = 05

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 2 Stochastic trajectories of SIRS model (12) for 12059011= 04 120590

21= 073 120590

31= 045 120590

12= 05 120590

22= 067 120590

32= 05 and Δ119905 = 10minus3

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

of global Lyapunov functions and used it to establish theglobal stability of the interior equilibrium for more generalmodels [8]Muroya et al considered a class of 119899-group (119899 ⩾ 2)SIRS epidemic models described by the following system ofequations [11]

119896= Λ119896minus 119889119878

119896119878119896minus

119899

sum

119895=1

120573119896119895119878119896 (119905) 119868119895 (119905) + 120575119896119877119896

119896=

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) minus (119889

119868

119896+ 120574119896) 119868119896

119896= 120574119896119868119896minus (119889119877

119896+ 120575119896) 119877119896 119896 = 1 119899

(1)

Nakata et al [12] and Enatsu et al [13] proposed an idea toextend Lyapunov functional techniques inMcCluskey [14] forSIR epidemic models to SIRS epidemic models By extendingwell-known Lyapunov function techniques Muroya et al[11] succeeded to prove the global stability of system (1)without use of the grouping technique by graph theory4 in Guo et al [10] Note that because of environmentalnoises the deterministic approach has some limitations in themathematicalmodeling transmission of an infectious diseaseseveral authors began to consider the effect of white noise inepidemic models [15ndash18] Beretta et al proved the stabilityof epidemic model with stochastic time delays influencedby probability under certain conditions [19] Such type ofstochastic perturbations firstly was proposed in [19 20]and later was successfully used in many other papers formany other different systems (see eg [21ndash27]) Yuan etal in [28] and Yu et al in [29] all investigated epidemicmodels with fluctuations around the positive equilibrium andthey proved locally stochastically asymptotic stability of thepositive equilibrium Ji et al also discuss a multigroup SIRmodel with stochastic perturbation and deduce the globallyasymptotic stability of the disease-free equilibrium when1198770le 1 which means the disease will die out When

1198770gt 1 they derive the disease will prevail which is

measured through the difference between the solution andthe endemic equilibrium of the deterministic model in timeaverage [1] Imhof and Walcher [30] considered a stochasticchemostat model and they proved that the stochastic modelled to extinction even though the deterministic counterpartpredicts persistence In our previous work we consideredan SEIR epidemic model with constant immigration andrandom fluctuation around the endemic equilibrium wecarried out a detailed analysis on the asymptotic behaviorof the stochastic model [31] and we also investigated a two-group epidemic model with distributed delays and randomperturbation [32] In addition we investigated multigroupSEIQR models with and without random perturbation incomputer network [33] In the present paper based on system(1) to examine the influence of white noise on system (1)we also consider a stochastic version of the SIRS modelby perturbing the deterministic system (1) by a white noiseand assume that the perturbations are around the positiveendemic equilibrium of epidemic models

This paper is organized as follows We begin in Section 2with necessary background with respect to the deterministic

multigroup SIRS model We establish the global dynamicsdetermined by the basic reproduction number R

0and

introduce some results of graph theory used by Guo etal We also review the important results in Theorem 2 ondeterministic model (1) by Muroya et al In Section 3 wederive the stochastic version from the deterministic model(1) In Section 4 we analyse the asymptotic behavior of thestochastic model by means of the method of Lyapunovfunctions and the theories of stochastic differential equationin Theorem 5 Numerical methods are employed to simulatethe dynamic behavior of the model and the effect of therate of immunity loss on the recovered is also analyzed inthe deterministic models and the corresponding stochasticmodels in Section 5 Finally we give the conclusion of ourpaper in Section 6

2 Global Stability of DeterministicMultigroup SIRS Models

First let us review some theories and results on deterministicmultigroup SIRS models we summarize the parameters inthe model (1) by the following list

Λ119896 the recruitment rate of the population

120573119896119895 transmission coefficient between compartments

119878119896and 119868119895

119889119878

119896 119889119868

119896 119889119877

119896 the natural death rates of susceptible

infected and recovered individuals in city 119896 respec-tively

120575119896 the rate of immunity loss in the 119896th group

120574119896 the natural recovery rate of the infected individuals

in city 119896

We assume 119889119878119896 119889119868

119896 119889119877

119896 Λ119896gt 0 and the rest of the parameters

are nonnegative for all 119896 In particular 120573119896119895= 0 if there is no

transmission of the disease between compartments 119878119896and 119868119895

By the biological meanings we may assume that

119889119878

119896⩽ min 119889119868

119896 119889119877

119896 119896 = 1 2 119899 (2)

For each 119896 adding the three equations in (1) gives(119878119896+ 119868119896+ 119877119896)1015840

⩽ Λ119896minus 119889119878

119896(119878119896+ 119868119896+ 119877119896) Hence

lim sup119905rarrinfin

(119878119896+119868119896+119877119896) ⩽ Λ

119896119889119878

119896Therefore omega limit sets

of system (1) are contained in the following bounded regionin the non-negative cone of 1198773119899

Γ = (1198781 1198681 1198771 119878

119899 119868119899 119877119899) isin 1198773119899

+| 119878119896⩽ 1198780

119896

119878119896+ 119868119896+ 119877119896⩽

Λ119896

119889119878

119896

1 ⩽ 119896 ⩽ 119899

(3)

Abstract and Applied Analysis 3

It can be verified that region Γ is positively invariant andsystem (1) always has the disease-free equilibrium 119875

0=

(1198780

1 0 0 0 0 119878

0

119899 0 0 0 0) where 1198780

119896= Λ

119896119889119878

119896is the

equilibrium of the 119878119896population in the absence of disease

(1198681= 1198682= sdot sdot sdot = 119868

119899= 0) Let

∘

Γ denote the interior of Γ Anendemic equilibrium119875lowast = (119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) belongs

to∘

Γ satisfying the equilibrium equations

Λ119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895minus 119889119878

119896119878lowast

119896+ 120575119896119877lowast

119896 (4)

(119889119868

119896+ 120574119896) 119868lowast

119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895 (5)

(119889119868

119896+ 120575119896) 119877lowast

119896= 120574119896119868lowast

119896 (6)

LetR0= 120588 (119872

0) (7)

denote the spectral radius of the matrix

1198720= (

1205731198961198951198780

119896

119889119868

119896+ 120574119896

)

1⩽119896119895⩽119899

(8)

The parameter R0is referred to as the basic reproduction

number Its biological significance is that if R0lt 1 the

disease dies outwhile ifR0gt 1 the disease becomes endemic

[9] In the following theorem we show that the multi-groupmodel (1) has at least one endemic equilibrium119875lowastwhenR

0gt

1 and 119875lowast is globally stable [9]The matrix 119861 = (120573

119896119895) denotes the contact matrix

Associated with 119861 one can construct a directed graph L =

119866(119861) whose vertex 119896 represents the 119896th group 119896 = 1 119899A directed edge exists from vertex 119896 to vertex 119895 if andonly if 120573

119896119895gt 0 Throughout the paper we assume that

119861 is irreducible This is equivalent to 119866(119861) being stronglyconnected Biologically this is the same as assuming thatany two groups 119896 and 119895 have a direct or indirect route oftransmission More specifically individuals in 119868

119895can infect

ones in 119878119896directly or indirectly Now consider the linear

system

Bℓ = 0 (9)where

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(10)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899 LetL = 119866(119861) denote

the directed graph associated with matrix 119861 (and (120573119896119895)) and

let 119862119895119896denote the cofactor of the (119895 119896) entry ofB

We have the following fundamental lemma[8]

Lemma 1 (Kirchhoff rsquos Matrix-Tree Theorem) Assume that(120573119896119895)119899times119899

is irreducible and 119899 ⩾ 2 Then the following resultshold

(1) The solution space of system (9) has dimension 1 witha basis (ℓ

1 ℓ2 ℓ

119899) = (119862

11 11986222 119862

119899119899)

(2) For 1 ⩽ 119896 ⩽ 119899

119862119896119896= sum

119879isinT119896

119882(119879) = sum

119879isinT119896

prod

(119903119898)isin119864(119879)

120573119903119898gt 0 (11)

where T119896is the set of all directed spanning subtrees ofL that are

rooted at vertex 119896119882(119879) is the weight of a directed tree T and119864(119879) denotes the set of directed arcs in a directed tree T

Y Muroya et al proved the following theorem

Theorem 2 Assume that 119861 = (120573119896119895) is irreducible and

inequality (2) holds If R0gt 1 and 119889119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 then

system (1) has at least one endemic equilibrium 119875lowast in

∘

Γ and119875lowast is globally asymptotically stable

3 Multigroup Stochastic SIRS Model

Under assumptions that R0gt 1 119889

119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 119889119878

119896⩽

min119889119868119896 119889119877

119896 and B = (120573

119896119895) is irreducible in Theorem 2

we know from Section 2 that there exists a positive endemicequilibrium 119875

lowast in∘

Γ Furthermore we assume stochasticperturbations are of white noise type which are directlyproportional to distances 119878

119896(119905) 119868119896(119905) and119877

119896(119905) from values of

119878lowast

119896 119868lowast

119896 119877lowast

119896 influence on the 119878

119896(119905) 119868119896(119905) and119877

119896(119905) respectively

So system (1) results in

119896= Λ119896minus 119889119878

119896119878119896minus

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) + 120575

119896119877119896

+ 1205901119896(119878119896minus 119878lowast

119896) 1119896

119896=

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) minus (119889

119868

119896+ 120574119896) 119868119896+ 1205902119896(119868119896minus 119868lowast

119896) 2119896

119896= 120574119896119868119896minus (119889119877

119896+ 120575119896) 119877119896+ 1205903119896(119877119896minus 119877lowast

119896) 3119896

119896 = 1 119899

(12)

where 1198611119896(119905) 1198612119896(119905) and 119861

3119896(119905) are independent standard

Brownian motions and 1205902119894119896gt 0 represent the intensities

of 119861119894119896(119905) (119894 = 1 2 3) respectively Obviously stochastic

system (12) has the same equilibrium points as system (1)Hence when the stochasticmodel (12) satisfies the conditionsof Theorem 2 we can ensure the existence of the positiveequilibrium point of the stochastic model (12) Note thatbecause of the influence of the white noise the componentsof solutions to (12) can also be negative at some time Butbecause it is impossible for populations to be less than zerophysically once the negative sections appear on componentsof solutions of (12) we only need to consider the nonnegative

4 Abstract and Applied Analysis

section of the components of solutions In the next sectionwe will investigate asymptotic stability of the equilibrium119875lowast of system (12) To this end we will construct a class

of different Lyapunov functions to achieve our proof undercertain conditions

4 Stochastic Stability ofthe Endemic Equilibrium

Let us now proceed to discuss asymptotic stability of thesystem (12) In this paper unless otherwise specified let(ΩF F

119905119905⩾1199050

119875) be a complete probability space with afiltration F

119905119905⩾1199050

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all 119875-

null sets) Let 119861119894119896(119905) be the Brownian motions defined on

this probability space If R0gt 1 then the stochastic

system (12) can be centered at its endemic equilibrium 119875lowast=

(119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) by the change of variables

u119896= 119878119896minus 119878lowast

119896 k

119896= 119868119896minus 119868lowast

119896 w

119896= 119877119896minus 119877lowast

119896

(13)

we obtain the following system

u119896= minus 119889

119878

119896u119896minus

119899

sum

119895=1

120573119896119895119878lowast

119896k119895minus

119899

sum

119895=1

120573119896119895u119896k119895

minus

119899

sum

119895=1

120573119896119895u119896119868lowast

119895+ 120575119896w119896+ 1205901119896u1198961119896

k119896=

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus (119889119868

119896+ 120574119896) k119896+ 1205902119896k1198962119896

w119896= 120574119896k119896minus (119889119877

119896+ 120575119896)w119896+ 1205903119896w1198963119896

119896 = 1 119899

(14)

It is clear that the stability of equilibrium of system(12) is equivalent to the stability of zero solution of system(14) Considering the 119889-dimensional stochastic differentialequation

119889x (119905) = 119891 (x (119905) 119905) 119889119905 + 119892 (x (119905) 119905) 119889119861 (119905) 119905 ⩾ 1199050

(15)

If the assumptions of the existence and uniqueness theoremare satisfied then for any given initial value x(119905

0) = 1199090isin R119889

(15) has a unique global solution denoted by x(119905 1199050 1199090) For

the purpose of stability we assume in this section 119891(0 119905) = 0and 119892(0 119905) = 0 for all 119905 ⩾ 119905

0 So (15) admits a solution

x(119905) equiv 0 which is called the trivial solution or equilibriumposition We let 11986221(R119889 times [119905

0infin)R

+) denote the family of

all nonnegative functions 119881(x 119905) on R119889 times [1199050infin) which are

continuously twice differentiable in x and once in 119905 Definethe differential operator 119871 associated with (15) by

119871 =

120597

120597119905

+

119889

sum

119894=1

119891119894(x 119905) 120597

120597119909119894

+

1

2

119889

sum

119894119895=1

[119892119879(x 119905) 119892 (x 119905)]

119894119895

1205972

120597119909119894119909119895

(16)

If 119871 acts on a function 119881 isin 11986221(R119889 times [1199050infin)R

+) then

119871119881 (x 119905) = 119881119905 (x 119905) + 119881119905 (x 119905) 119891 (x 119905)

+

1

2

trace [119892119879 (x 119905) 119881119909119909 (x 119905) 119892 (x 119905)] (17)

Definition 3 (1) The trivial solution of (15) is said to bestochastically stable or stable in probability if for every pairof 120576 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120576 119903 119905

0) gt 0 such

that

119875 1003816100381610038161003816x (119905 1199050 1199090)1003816100381610038161003816lt 119903 forall119905 ⩾ 119905

0 ⩾ 1 minus 120576 (18)

whenever |1199090| lt 120575 Otherwise it is said to be stochastically

unstable(2) The trivial solution is said to be stochastically asymp-

totically stable if it is stochastically stable and for every 120576 isin(0 1) there exists a 120575

0= 1205750(120576 1199050) gt 0 such that

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 ⩾ 1 minus 120576 (19)

whenever |1199090| lt 1205750

(3) The trivial solution is said to be stochastically asymp-totically stable in the large if it is stochastically stable and forall 1199090isin R119889

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 = 1 (20)

Before proving themain theoremweput forward a lemmain [34]

Lemma 4 (see [34]) If there exists a positive-definite decres-cent radially unbounded function 119881(x 119905) isin 119862

21(R119889 times

[1199050infin)R

+) such that 119871119881(x 119905) is negative-definite then the

trivial solution of (15) is stochastically asymptotically stable inthe large

From the above lemma we can obtain the stochasticallyasymptotical stability of equilibrium as following

Theorem 5 Assume that B = (120573119896119895) is irreducible and

inequality (2) holds and R0gt 1 119889119878

119896119878lowast

119896minus 120575119896119877lowast

119896⩾ 0 then if

the following condition is satisfied

Abstract and Applied Analysis 5

0 5 10 150

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

5

10

15

t

S1(t)

I1(t)

R1(t)

t

S2(t)

I2(t)

R2(t)

Figure 1 Deterministic trajectories of SIRSmodel (1) for initial condition 1198781(0) = 2 119868

1(0) = 4119877

1(0) = 6 119878

2(0) = 9 119868

2(0) = 4 and119877

2(0) = 05

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 2 Stochastic trajectories of SIRS model (12) for 12059011= 04 120590

21= 073 120590

31= 045 120590

12= 05 120590

22= 067 120590

32= 05 and Δ119905 = 10minus3

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

It can be verified that region Γ is positively invariant andsystem (1) always has the disease-free equilibrium 119875

0=

(1198780

1 0 0 0 0 119878

0

119899 0 0 0 0) where 1198780

119896= Λ

119896119889119878

119896is the

equilibrium of the 119878119896population in the absence of disease

(1198681= 1198682= sdot sdot sdot = 119868

119899= 0) Let

∘

Γ denote the interior of Γ Anendemic equilibrium119875lowast = (119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) belongs

to∘

Γ satisfying the equilibrium equations

Λ119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895minus 119889119878

119896119878lowast

119896+ 120575119896119877lowast

119896 (4)

(119889119868

119896+ 120574119896) 119868lowast

119896=

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119895 (5)

(119889119868

119896+ 120575119896) 119877lowast

119896= 120574119896119868lowast

119896 (6)

LetR0= 120588 (119872

0) (7)

denote the spectral radius of the matrix

1198720= (

1205731198961198951198780

119896

119889119868

119896+ 120574119896

)

1⩽119896119895⩽119899

(8)

The parameter R0is referred to as the basic reproduction

number Its biological significance is that if R0lt 1 the

disease dies outwhile ifR0gt 1 the disease becomes endemic

[9] In the following theorem we show that the multi-groupmodel (1) has at least one endemic equilibrium119875lowastwhenR

0gt

1 and 119875lowast is globally stable [9]The matrix 119861 = (120573

119896119895) denotes the contact matrix

Associated with 119861 one can construct a directed graph L =

119866(119861) whose vertex 119896 represents the 119896th group 119896 = 1 119899A directed edge exists from vertex 119896 to vertex 119895 if andonly if 120573

119896119895gt 0 Throughout the paper we assume that

119861 is irreducible This is equivalent to 119866(119861) being stronglyconnected Biologically this is the same as assuming thatany two groups 119896 and 119895 have a direct or indirect route oftransmission More specifically individuals in 119868

119895can infect

ones in 119878119896directly or indirectly Now consider the linear

system

Bℓ = 0 (9)where

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(10)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899 LetL = 119866(119861) denote

the directed graph associated with matrix 119861 (and (120573119896119895)) and

let 119862119895119896denote the cofactor of the (119895 119896) entry ofB

We have the following fundamental lemma[8]

Lemma 1 (Kirchhoff rsquos Matrix-Tree Theorem) Assume that(120573119896119895)119899times119899

is irreducible and 119899 ⩾ 2 Then the following resultshold

(1) The solution space of system (9) has dimension 1 witha basis (ℓ

1 ℓ2 ℓ

119899) = (119862

11 11986222 119862

119899119899)

(2) For 1 ⩽ 119896 ⩽ 119899

119862119896119896= sum

119879isinT119896

119882(119879) = sum

119879isinT119896

prod

(119903119898)isin119864(119879)

120573119903119898gt 0 (11)

where T119896is the set of all directed spanning subtrees ofL that are

rooted at vertex 119896119882(119879) is the weight of a directed tree T and119864(119879) denotes the set of directed arcs in a directed tree T

Y Muroya et al proved the following theorem

Theorem 2 Assume that 119861 = (120573119896119895) is irreducible and

inequality (2) holds If R0gt 1 and 119889119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 then

system (1) has at least one endemic equilibrium 119875lowast in

∘

Γ and119875lowast is globally asymptotically stable

3 Multigroup Stochastic SIRS Model

Under assumptions that R0gt 1 119889

119878

119896119878lowastminus 120575119896119877lowast

119896⩾ 0 119889119878

119896⩽

min119889119868119896 119889119877

119896 and B = (120573

119896119895) is irreducible in Theorem 2

we know from Section 2 that there exists a positive endemicequilibrium 119875

lowast in∘

Γ Furthermore we assume stochasticperturbations are of white noise type which are directlyproportional to distances 119878

119896(119905) 119868119896(119905) and119877

119896(119905) from values of

119878lowast

119896 119868lowast

119896 119877lowast

119896 influence on the 119878

119896(119905) 119868119896(119905) and119877

119896(119905) respectively

So system (1) results in

119896= Λ119896minus 119889119878

119896119878119896minus

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) + 120575

119896119877119896

+ 1205901119896(119878119896minus 119878lowast

119896) 1119896

119896=

119899

sum

119895=1

120573119896119895119878119896(119905) 119868119895(119905) minus (119889

119868

119896+ 120574119896) 119868119896+ 1205902119896(119868119896minus 119868lowast

119896) 2119896

119896= 120574119896119868119896minus (119889119877

119896+ 120575119896) 119877119896+ 1205903119896(119877119896minus 119877lowast

119896) 3119896

119896 = 1 119899

(12)

where 1198611119896(119905) 1198612119896(119905) and 119861

3119896(119905) are independent standard

Brownian motions and 1205902119894119896gt 0 represent the intensities

of 119861119894119896(119905) (119894 = 1 2 3) respectively Obviously stochastic

system (12) has the same equilibrium points as system (1)Hence when the stochasticmodel (12) satisfies the conditionsof Theorem 2 we can ensure the existence of the positiveequilibrium point of the stochastic model (12) Note thatbecause of the influence of the white noise the componentsof solutions to (12) can also be negative at some time Butbecause it is impossible for populations to be less than zerophysically once the negative sections appear on componentsof solutions of (12) we only need to consider the nonnegative

4 Abstract and Applied Analysis

section of the components of solutions In the next sectionwe will investigate asymptotic stability of the equilibrium119875lowast of system (12) To this end we will construct a class

of different Lyapunov functions to achieve our proof undercertain conditions

4 Stochastic Stability ofthe Endemic Equilibrium

Let us now proceed to discuss asymptotic stability of thesystem (12) In this paper unless otherwise specified let(ΩF F

119905119905⩾1199050

119875) be a complete probability space with afiltration F

119905119905⩾1199050

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all 119875-

null sets) Let 119861119894119896(119905) be the Brownian motions defined on

this probability space If R0gt 1 then the stochastic

system (12) can be centered at its endemic equilibrium 119875lowast=

(119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) by the change of variables

u119896= 119878119896minus 119878lowast

119896 k

119896= 119868119896minus 119868lowast

119896 w

119896= 119877119896minus 119877lowast

119896

(13)

we obtain the following system

u119896= minus 119889

119878

119896u119896minus

119899

sum

119895=1

120573119896119895119878lowast

119896k119895minus

119899

sum

119895=1

120573119896119895u119896k119895

minus

119899

sum

119895=1

120573119896119895u119896119868lowast

119895+ 120575119896w119896+ 1205901119896u1198961119896

k119896=

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus (119889119868

119896+ 120574119896) k119896+ 1205902119896k1198962119896

w119896= 120574119896k119896minus (119889119877

119896+ 120575119896)w119896+ 1205903119896w1198963119896

119896 = 1 119899

(14)

It is clear that the stability of equilibrium of system(12) is equivalent to the stability of zero solution of system(14) Considering the 119889-dimensional stochastic differentialequation

119889x (119905) = 119891 (x (119905) 119905) 119889119905 + 119892 (x (119905) 119905) 119889119861 (119905) 119905 ⩾ 1199050

(15)

If the assumptions of the existence and uniqueness theoremare satisfied then for any given initial value x(119905

0) = 1199090isin R119889

(15) has a unique global solution denoted by x(119905 1199050 1199090) For

the purpose of stability we assume in this section 119891(0 119905) = 0and 119892(0 119905) = 0 for all 119905 ⩾ 119905

0 So (15) admits a solution

x(119905) equiv 0 which is called the trivial solution or equilibriumposition We let 11986221(R119889 times [119905

0infin)R

+) denote the family of

all nonnegative functions 119881(x 119905) on R119889 times [1199050infin) which are

continuously twice differentiable in x and once in 119905 Definethe differential operator 119871 associated with (15) by

119871 =

120597

120597119905

+

119889

sum

119894=1

119891119894(x 119905) 120597

120597119909119894

+

1

2

119889

sum

119894119895=1

[119892119879(x 119905) 119892 (x 119905)]

119894119895

1205972

120597119909119894119909119895

(16)

If 119871 acts on a function 119881 isin 11986221(R119889 times [1199050infin)R

+) then

119871119881 (x 119905) = 119881119905 (x 119905) + 119881119905 (x 119905) 119891 (x 119905)

+

1

2

trace [119892119879 (x 119905) 119881119909119909 (x 119905) 119892 (x 119905)] (17)

Definition 3 (1) The trivial solution of (15) is said to bestochastically stable or stable in probability if for every pairof 120576 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120576 119903 119905

0) gt 0 such

that

119875 1003816100381610038161003816x (119905 1199050 1199090)1003816100381610038161003816lt 119903 forall119905 ⩾ 119905

0 ⩾ 1 minus 120576 (18)

whenever |1199090| lt 120575 Otherwise it is said to be stochastically

unstable(2) The trivial solution is said to be stochastically asymp-

totically stable if it is stochastically stable and for every 120576 isin(0 1) there exists a 120575

0= 1205750(120576 1199050) gt 0 such that

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 ⩾ 1 minus 120576 (19)

whenever |1199090| lt 1205750

(3) The trivial solution is said to be stochastically asymp-totically stable in the large if it is stochastically stable and forall 1199090isin R119889

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 = 1 (20)

Before proving themain theoremweput forward a lemmain [34]

Lemma 4 (see [34]) If there exists a positive-definite decres-cent radially unbounded function 119881(x 119905) isin 119862

21(R119889 times

[1199050infin)R

+) such that 119871119881(x 119905) is negative-definite then the

trivial solution of (15) is stochastically asymptotically stable inthe large

From the above lemma we can obtain the stochasticallyasymptotical stability of equilibrium as following

Theorem 5 Assume that B = (120573119896119895) is irreducible and

inequality (2) holds and R0gt 1 119889119878

119896119878lowast

119896minus 120575119896119877lowast

119896⩾ 0 then if

the following condition is satisfied

Abstract and Applied Analysis 5

0 5 10 150

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

5

10

15

t

S1(t)

I1(t)

R1(t)

t

S2(t)

I2(t)

R2(t)

Figure 1 Deterministic trajectories of SIRSmodel (1) for initial condition 1198781(0) = 2 119868

1(0) = 4119877

1(0) = 6 119878

2(0) = 9 119868

2(0) = 4 and119877

2(0) = 05

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 2 Stochastic trajectories of SIRS model (12) for 12059011= 04 120590

21= 073 120590

31= 045 120590

12= 05 120590

22= 067 120590

32= 05 and Δ119905 = 10minus3

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

section of the components of solutions In the next sectionwe will investigate asymptotic stability of the equilibrium119875lowast of system (12) To this end we will construct a class

of different Lyapunov functions to achieve our proof undercertain conditions

4 Stochastic Stability ofthe Endemic Equilibrium

Let us now proceed to discuss asymptotic stability of thesystem (12) In this paper unless otherwise specified let(ΩF F

119905119905⩾1199050

119875) be a complete probability space with afiltration F

119905119905⩾1199050

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all 119875-

null sets) Let 119861119894119896(119905) be the Brownian motions defined on

this probability space If R0gt 1 then the stochastic

system (12) can be centered at its endemic equilibrium 119875lowast=

(119878lowast

1 119868lowast

1 119877lowast

1 119878

lowast

119899 119868lowast

119899 119877lowast

119899) by the change of variables

u119896= 119878119896minus 119878lowast

119896 k

119896= 119868119896minus 119868lowast

119896 w

119896= 119877119896minus 119877lowast

119896

(13)

we obtain the following system

u119896= minus 119889

119878

119896u119896minus

119899

sum

119895=1

120573119896119895119878lowast

119896k119895minus

119899

sum

119895=1

120573119896119895u119896k119895

minus

119899

sum

119895=1

120573119896119895u119896119868lowast

119895+ 120575119896w119896+ 1205901119896u1198961119896

k119896=

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus (119889119868

119896+ 120574119896) k119896+ 1205902119896k1198962119896

w119896= 120574119896k119896minus (119889119877

119896+ 120575119896)w119896+ 1205903119896w1198963119896

119896 = 1 119899

(14)

It is clear that the stability of equilibrium of system(12) is equivalent to the stability of zero solution of system(14) Considering the 119889-dimensional stochastic differentialequation

119889x (119905) = 119891 (x (119905) 119905) 119889119905 + 119892 (x (119905) 119905) 119889119861 (119905) 119905 ⩾ 1199050

(15)

If the assumptions of the existence and uniqueness theoremare satisfied then for any given initial value x(119905

0) = 1199090isin R119889

(15) has a unique global solution denoted by x(119905 1199050 1199090) For

the purpose of stability we assume in this section 119891(0 119905) = 0and 119892(0 119905) = 0 for all 119905 ⩾ 119905

0 So (15) admits a solution

x(119905) equiv 0 which is called the trivial solution or equilibriumposition We let 11986221(R119889 times [119905

0infin)R

+) denote the family of

all nonnegative functions 119881(x 119905) on R119889 times [1199050infin) which are

continuously twice differentiable in x and once in 119905 Definethe differential operator 119871 associated with (15) by

119871 =

120597

120597119905

+

119889

sum

119894=1

119891119894(x 119905) 120597

120597119909119894

+

1

2

119889

sum

119894119895=1

[119892119879(x 119905) 119892 (x 119905)]

119894119895

1205972

120597119909119894119909119895

(16)

If 119871 acts on a function 119881 isin 11986221(R119889 times [1199050infin)R

+) then

119871119881 (x 119905) = 119881119905 (x 119905) + 119881119905 (x 119905) 119891 (x 119905)

+

1

2

trace [119892119879 (x 119905) 119881119909119909 (x 119905) 119892 (x 119905)] (17)

Definition 3 (1) The trivial solution of (15) is said to bestochastically stable or stable in probability if for every pairof 120576 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120576 119903 119905

0) gt 0 such

that

119875 1003816100381610038161003816x (119905 1199050 1199090)1003816100381610038161003816lt 119903 forall119905 ⩾ 119905

0 ⩾ 1 minus 120576 (18)

whenever |1199090| lt 120575 Otherwise it is said to be stochastically

unstable(2) The trivial solution is said to be stochastically asymp-

totically stable if it is stochastically stable and for every 120576 isin(0 1) there exists a 120575

0= 1205750(120576 1199050) gt 0 such that

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 ⩾ 1 minus 120576 (19)

whenever |1199090| lt 1205750

(3) The trivial solution is said to be stochastically asymp-totically stable in the large if it is stochastically stable and forall 1199090isin R119889

119875 lim119905rarrinfin

x (119905 1199050 1199090) = 0 = 1 (20)

Before proving themain theoremweput forward a lemmain [34]

Lemma 4 (see [34]) If there exists a positive-definite decres-cent radially unbounded function 119881(x 119905) isin 119862

21(R119889 times

[1199050infin)R

+) such that 119871119881(x 119905) is negative-definite then the

trivial solution of (15) is stochastically asymptotically stable inthe large

From the above lemma we can obtain the stochasticallyasymptotical stability of equilibrium as following

Theorem 5 Assume that B = (120573119896119895) is irreducible and

inequality (2) holds and R0gt 1 119889119878

119896119878lowast

119896minus 120575119896119877lowast

119896⩾ 0 then if

the following condition is satisfied

Abstract and Applied Analysis 5

0 5 10 150

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

5

10

15

t

S1(t)

I1(t)

R1(t)

t

S2(t)

I2(t)

R2(t)

Figure 1 Deterministic trajectories of SIRSmodel (1) for initial condition 1198781(0) = 2 119868

1(0) = 4119877

1(0) = 6 119878

2(0) = 9 119868

2(0) = 4 and119877

2(0) = 05

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 2 Stochastic trajectories of SIRS model (12) for 12059011= 04 120590

21= 073 120590

31= 045 120590

12= 05 120590

22= 067 120590

32= 05 and Δ119905 = 10minus3

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

0 5 10 150

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

5

10

15

t

S1(t)

I1(t)

R1(t)

t

S2(t)

I2(t)

R2(t)

Figure 1 Deterministic trajectories of SIRSmodel (1) for initial condition 1198781(0) = 2 119868

1(0) = 4119877

1(0) = 6 119878

2(0) = 9 119868

2(0) = 4 and119877

2(0) = 05

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 2 Stochastic trajectories of SIRS model (12) for 12059011= 04 120590

21= 073 120590

31= 045 120590

12= 05 120590

22= 067 120590

32= 05 and Δ119905 = 10minus3

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

0 5 10 15 20 250

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

20

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 3 Stochastic trajectories of SIRS model (12) for 12059011= 07 120590

21= 12 120590

31= 13 120590

12= 08 120590

22= 08 120590

32= 14 and Δ119905 = 10minus3

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

0 1 2 3 40

10

20

30

40

50

60

70

80

90

100

t t

S1(t)

I1(t)

R1(t)

S2(t)

I2(t)

R2(t)

Figure 4 Stochastic trajectories of SIRS model (12) for 12059011= 405 120590

21= 3273 120590

31= 3692 120590

12= 334 120590

22= 3967 and 120590

32= 305

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

5 55 6 65 7 750

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 5 The relation between 119878119896and 119877

119896for the deterministic SIRS model

5 55 6 65 7 750

05

1

15

2

25

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

Figure 6 The relation between 119878119896and 119877

119896for the stochastic SIRS model

1205902

1119896lt 2119889119878

119896 120590

2

2119896lt

2 (119889119868

119896+ 120574119896)sum119899

119895=1120573119896119895119868lowast

119895

sum119899

119895=1120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896

1205902

3119896lt 2 (119889

119877

119896+ 120575119896)

1205752

119896

(119889119878

119896minus (12) 120590

2

1119896) (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

+

21205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

lt

(119889119877

119896+ 120575119896minus (12) 120590

2

3119896)119872119896

21205742

119896sum119899

119895=1120573119896119895119868lowast

119895

(21)

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

55 6 65 70

05

1

15

2

25

0 1 2 3 40

1

2

3

4

5

6

7

8

9

10

S1(t)

R1(t)

S2(t)

R2(t)

1205751 = 001

1205751 = 002

1205751 = 0025

1205751 = 01

1205752 = 0012

1205752 = 015

1205752 = 025

1205752 = 035

Figure 7 Comparison of relationships of 119878119896and 119877

119896for the different values 120575

119896in the deterministic SIRS model

the endemic equilibrium 119875lowast of system (12) is stochastically

asymptotically stable in the large where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(22)

Proof Set

B =

[[[[[[[[[[[[[

[

sum

119894 = 1

1205731119894minus12057321

sdot sdot sdot minus1205731198991

minus12057312

sum

119894 = 2

1205732119894sdot sdot sdot minus120573

1198992

d

minus1205731119899

1205732119899

sdot sdot sdot sum

119894 = 119899

120573119899119894

]]]]]]]]]]]]]

]

(23)

and 120573119896119895= 120573119896119895119878lowast

119896119868lowast

119895 120573119896119895gt 0 1 ⩽ 119896 119895 ⩽ 119899

Note thatB is the Laplacian matrix of the matrix (120573119896119895)119899times119899

(see Lemma 1) Since 120573119896119895is irreducible the matrices (120573

119896119895)119899times119899

andB are also irreducible Let 119862119896119895denote the cofactor of the

(119896 119895) entry of B We know that the system Bℓ = 0 has apositive solution ℓ = (ℓ

1 ℓ2 ℓ

119899) where ℓ

119896= 119862119896119896gt 0

for 119896 = 1 2 119899 by Lemma 1It is easy to see thatwe only need to prove the zero solution

of (14) is stochastically asymptotically stable in the large Letx119896(119905) = (u

119896(119905) k119896(119905)w119896(119905))119879isin 1198773

+ 119896 = 1 119899 and x(119905) =

(x1(119905) x

119899(119905))119879isin 1198773119899

+ We define the Lyapunov function

119881(x(119905)) as follows

119881 (x) = 12

119899

sum

119896=1

(119886119896k2119896+ 119887119896(u119896+ k119896)2+ 119888119896w2119896) (24)

where 119886119896gt 0 119887

119896gt 0 119888119896gt 0 are real positive constants to be

chosen later Then it can be described as the quadratic form

119881 (x) = 12

119899

sum

119896=1

x119879119896119876x119896 (25)

where

119876 = (

119887119896

119887119896

0

119887119896119886119896+ 1198871198960

0 0 119888119896

) (26)

is a symmetric positive-definite matrix So it is obviousthat 119881(x) is positive-definite decrescent radially unboundedFor the sake of simplicity (24) may be divided into threefunctions 119881(x) = 119881

1(x) + 119881

2(x) + 119881

3(x) where

1198811 (x) =

1

2

119886119896k2119896 119881

2 (x) =1

2

119887119896(u119896+ k119896)2

1198813(x) = 1

2

119888119896w2119896

(27)

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Using Itorsquos formula and (5) we compute

1198711198811=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus(119889119868

119896+ 120574119896) k119896)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896k119896(

119899

sum

119895=1

120573119896119895119878lowast

119896k119895+

119899

sum

119895=1

120573119896119895u119896k119895

+

119899

sum

119895=1

120573119896119895u119896119868lowast

119895minus

sum119899

119895=1120573119896119895119878lowast

119896119868lowast

119895

119868lowast

119896

k119896)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119878lowast

119896119868lowast

119896119868lowast

119895(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895119868lowast

119896

k119896

119868lowast

119896

k119895

119868lowast

119895

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

⩽

119899

sum

119896=1

119886119896(

1

2

119899

sum

119895=1

120573119896119895119868lowast

119896((

k119896

119868lowast

119896

)

2

+ (

k119895

119868lowast

119895

)

2

)

minus

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

=

1

2

(

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119895

119868lowast

119895

)

2

minus

119899

sum

119896=1

119899

sum

119895=1

120573119896119895119868lowast

119896(

k119896

119868lowast

119896

)

2

)

+

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(28)

Let 119886119896= 119862119896119896119868lowast

119896 so ℓ119896= 119886119896119868lowast

119896 It follows from Bℓ = 0 and

120573119895119896= 120573119895119896119878lowast

119895119868lowast

119896that

119899

sum

119895=1

120573119895119896ℓ119895=

119899

sum

119896=1

120573119896119895ℓ119896 (29)

which implies

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119895

119868lowast

119895

)

2

=

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895119868lowast

119896(

w119896

119868lowast

119896

)

2

(30)

Hence inequality (28) becomes

1198711198811⩽

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

119899

sum

119896=1

1198861198961205902

2119896k2119896

(31)

Similarly from Itorsquos formula we obtain

1198711198812=

119899

sum

119896=1

119887119896(u119896+ k119896) (minus119889119878

119896u119896minus (119889119868

119896+ 120574119896) k119896+ 120575119896w119896)

+

1

2

119899

sum

119896=1

119887119896(1205902

1119896u2119896+ 1205902

2119896k2119896)

= minus

119899

sum

119896=1

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119899

sum

119896=1

119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus

119899

sum

119896=1

119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+

119899

sum

119896=1

119887119896120575119896(u119896+ k119896)w119896

1198711198813=

119899

sum

119896=1

119888119896w119896(120574119896k119896minus (119889119877

119896+ 120575119896)w119896)

+

1

2

119899

sum

119896=1

1198881198961205902

3119896w2119896

= minus

119899

sum

119896=1

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896+

119899

sum

119896=1

119888119896120574119896k119896w119896

(32)

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Moreover using Cauchy inequality we can obtain

119888119896120574119896k119896w119896⩽

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

119887119896120575119896u119896w119896⩽

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

119887119896120575119896k119896w119896⩽ +

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

(33)

where

119872119896= 2 (119889

119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus (

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896

(34)

Hence we can calculate

119871119881 = 1198711198811+ 1198711198812+ 1198711198813

=

119899

sum

119896=1

119886119896(

119899

sum

119895=1

120573119896119895u119896k119896k119895+

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895)

+

1

2

1198861198961205902

2119896k2119896minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896minus119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896

+ 119887119896120575119896(u119896+ k119896)w119896minus 119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+119888119896120574119896k119896w119896

⩽

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896119868lowast

119895+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896

minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896) u119896k119896minus119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

+

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

+ (119886119896

119899

sum

119895=1

120573119896119895119868lowast

119895minus 119887119896(119889119878

119896+ 119889119868

119896+ 120574119896)) u119896k119896

+

1

2

1198861198961205902

2119896k2119896

minus 119887119896(119889119868

119896+ 120574119896minus

1

2

1205902

2119896) k2119896+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(35)

Set

119886119896=

(119889119878

119896+ 119889119868

119896+ 120574119896)

sum119899

119895=1120573119896119895119868lowast

119895

119887119896 (36)

then we have

119871119881 ⩽

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus

119887119896

2sum119899

119895=1120573119896119895119868lowast

119895

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

times (2 (119889119868

119896+ 120574119896)

119899

sum

119895=1

120573119896119895119868lowast

119895

minus(

119899

sum

119895=1

120573119896119895119868lowast

119895+ 119889119878

119896+ 119889119868

119896+ 120574119896)1205902

2119896) k2119896

+

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896

+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896minus

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896k2119896

+

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

k2119896

minus

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896)w2119896

+

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

w2119896+

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

w2119896

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896 +

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

(37)

where

1198711198810=

119899

sum

119896=1

minus

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896) u2119896

minus (

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896

minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

) k2119896

minus (

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

)w2119896

(38)

We let

A119896=

1

2

119887119896(119889119878

119896minus

1

2

1205902

1119896)

B119896=

119887119896

4sum119899

119895=1120573119896119895119868lowast

119895

119872119896minus

1198881198961205742

119896

2 (119889119877

119896+ 120575119896minus (12) 120590

2

3119896)

C119896=

1

2

119888119896(119889119877

119896+ 120575119896minus

1

2

1205902

3119896) minus

1198871198961205752

119896

2 (119889119878

119896minus (12) 120590

2

1119896)

minus

1198871198961205752

119896sum119899

119895=1120573119896119895119868lowast

119895

119872119896

(39)

The proofs above show that if the condition (21) is sat-isfied A

119896 B119896 and C

119896are positive constants Let 120582 =

min119896isin1119899

A119896B119896C119896 then 120582 gt 0 From (37) and (38) one

sees that

119871119881 ⩽ 1198711198810+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus

119899

sum

119896=1

(A119896u2119896+B119896k2119896+C119896w2119896)

+

119899

sum

119896=1

119886119896

119899

sum

119895=1

120573119896119895u119896k119896k119895

= minus 120582

119899

sum

119896=1

(1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2+ 119900 (

1003816100381610038161003816x119896 (119905)

1003816100381610038161003816

2))

= minus 120582|x (119905)|2 + 119900 (|x (119905)|2)

(40)

where |x119896(119905)| = radicu2

119896(119905) + k2

119896(119905) + w2

119896(119905) |x(119905)| =

(sum119899

119896=1|x119896(119905)|)12 and 119900(|x(119905)|2) is an infinitesimal of higher

order of |x(119905)|2 for 119905 rarr infin Hence 119871119881(x 119905) is negative-definite for 119905 ⩾ 0 According to Lemma 4 we thereforeconclude that the zero solution of (12) is stochasticallyasymptotically stable in the largeThe proof is complete

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

5 Numerical Simulation

Numerical methods are employed to solve the system (1) and(12) and to depict the behavior of the susceptible infectiousand recovered nodes with respect to time We numericallysimulate the solution of system (1) and (12) with 119899 = 2 Inthis case we have

1198720=

[[[[[[

[

12057311(Λ1119889119878

1)

119889119868

1+ 1205741

12057312(Λ1119889119878

1)

119889119868

1+ 1205741

12057321(Λ2119889119878

2)

119889119868

2+ 1205742

12057322(Λ2119889119878

2)

119889119868

2+ 1205742

]]]]]]

]

= [120573111198701120573121198701

120573211198702120573221198702

]

R0

= 120588 (1198720)

=

120573111198701+ 120573221198702+ radic(120573

111198701minus 120573221198702)2+ 4120573121205732111987011198702

2

(41)

Let

Λ1= 25 120573

11= 055 120573

12= 035 119889

119878

1= 015

119889119868

1= 02 119889

119877

1= 025 120575

1= 001 120574

1= 04

Λ2= 10 120573

21= 05 120573

22= 02 119889

119878

2= 01 119889

119868

2= 015

119889119877

2= 02 120575

2= 0012 120574

2= 015

(42)

Hence we obtain 119878lowast1= 07205 119868lowast

1= 40914 119877lowast

1= 62945

119878lowast

2= 03663 119868lowast

2= 33048 119877lowast

2= 23383 We can calculate

R0=

250

9

gt 1 119889119878

1119878lowast

1minus 1205751119877lowast

1= 004513 gt 0

119889119878

2119878lowast

2minus 1205752119877lowast

2= 00085704 gt 0

(43)

In the absence of noise we simulate the global stability of theendemic equilibrium of deterministic system (1) in Figure 1and we always choose initial value 119878

1(0) = 20 119868

1(0) = 40

1198771(0) = 60 119878

2(0) = 90 119868

2(0) = 40 and 119877

2(0) = 05

Next we consider the effect of stochastic fluctuations ofenvironment to the endemic equilibrium of the correspond-ing deterministic system Given the discretization of system(12) for 119905 = 0 Δ119905 2Δ119905 119899Δ119905 and 119896 = 1 2

119878119896119894+1

= 119878119896119894+ (Λ119896minus 119889119878

119896119878119896119894minus 12057311989611198781198961198941198681119894

minus12057311989621198781198961198941198682119894+ 120575119896119877119896119894) Δ119905

+ 1205901119896(119878119896119894minus 119878lowast

119896)radicΔ119905120576

1119896119894

119868119896119894+1

= 119868119896119894+ (12057311989611198781198961198941198681119894+ 12057311989621198781198961198941198682119894minus (119889119868

119896+ 120574119896) 119868119896119894) Δ119905

+ 1205902119896(119868119896119894minus 119868lowast

119896)radicΔ119905120576

2119896119894

119877119896119894+1

= 119877119896119894+ (120574119896119868119896119894minus (119889119868

119896+ 120575119896) 119877119896119894) Δ119905

+ 1205903119896(119877119896119894minus 119877lowast

119896)radicΔ119905120576

3119896119894

(44)

where time increment Δ119905 gt 0 and 1205761119896119894

1205762119896119894

1205763119896119894

are119873(0 1)-distributed independent random variables whichcan be generated numerically by pseudorandom numbergenerators

As mentioned above there is an endemic equilibrium119875lowast= (119878lowast

1 119868lowast

1 119877lowast

1 119878lowast

2 119868lowast

2 119877lowast

2)of system (1)whenR

0gt 1 where

119878lowast

1= 07205 119868

lowast

1= 40914 119877

lowast

1= 62945

119878lowast

2= 03663 119868

lowast

2= 33048 119877

lowast

2= 23383

(45)

Figure 2 corresponds to 12059011= 04 120590

21= 073 120590

31= 045

12059012= 05 120590

22= 067 and 120590

32= 05 the numerical simulation

shows that the endemic equilibrium of stochastic system (12)is global asymptotically stable under the condition (21) whileFigure 3 corresponds to 120590

11= 07 120590

21= 12 120590

31= 13

12059012= 08 120590

22= 08 and 120590

32= 14 Moreover comparison

of Figures 2 and 3 suggests that fluctuations enhance as thenoise level increases Note that the condition (21) is just asufficient condition When this condition is not satisfiedthe stochastic system (12) may be or may not be stable Forinstance the intensities of Brownian motions 120590

11= 07

12059021= 12 120590

31= 13 120590

12= 08 120590

22= 08 and 120590

32=

14 do not meet the condition (21) but we can see fromFigure 3 that the stochastic system (12) is still asymptoticallystable If we choose 120590

11= 405 120590

21= 3273 120590

31= 3692

12059012= 334 120590

22= 3967 and 120590

32= 305 then the solution

of the stochastic system (12) is not asymptotically stable butexplodes to infinity at the finite time (see Figure 4)

The relationships between 119878119896and 119877

119896are also observed

and are depicted in Figures 5 and 6Note that these two curvesof Figure 5 show that the number of susceptible individualssharply declines to near 119878lowast

119896as the number of the recovery

individuals increases slightly and then increases with thenumber of the increasing recovered individuals gently andgoes on increasing with the number of the decreasing recov-ered individuals andfinally both reach the steady state valuesFor the stochastic version Figure 6 corresponds to 120590

11= 04

12059021= 073 120590

31= 045 120590

12= 05 120590

22= 067 and 120590

32= 05

oscillation appears under environmental driving forceswhich

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 13

Table 1 Values of (119878lowast1 119877lowast

1) when fixing 120575

2= 0012 and changing 120575

1

1205751= 01 120575

1= 0025 120575

1= 002 120575

1= 001

119878lowast

1= 07627 119878

lowast

1= 07290 119878

lowast

1= 07263 119878

lowast

1= 07205

119877lowast

1= 56130 119877

lowast

1= 61694 119877

lowast

1= 62105 119877

lowast

1= 62945

Table 2 Values of (119878lowast2 119877lowast

2) when fixing 120575

1= 001 and changing 120575

2

1205752= 035 120575

2= 025 120575

2= 015 120575

2= 0012

119878lowast

2= 04678 119878

lowast

2= 04502 119878

lowast

2= 04254 119878

lowast

2= 03663

119877lowast

2= 12710 119877

lowast

2= 14692 119877

lowast

2= 17408 119877

lowast

2= 23383

actually affect the deterministic curves shown in Figure 5 ButFigure 6 still maintains the same total trend as Figure 5 andreaches the same equilibrium point as deterministic versionSimulation result agrees with the real life situation

Figure 7 and Tables 1 and 2 show that when the rate ofimmunity loss 120575

119896gradually reduces 119878lowast

119896decreases but 119877lowast

119896

increases the lower the rate of immunity loss 120575119896is the lower

the steady state value of the susceptible is the higher thesteady state value of the recovered is Thus it will be of greatimportance for health management to control the rate ofimmunity loss to keep it in a lower level for example whenantibody concentrations of recovered individuals are at a lowlevel or are zero they can be required to undergo vaccinationto achieve the protective antibody levels

6 Conclusion

This paper presented a mathematical study describing thedynamical behavior of an SIRS epidemicmodel with stochas-tic perturbations Our purpose was based on analyzing thisbehavior using a stochastic model When the reproductionnumber R

0is greater than one we obtained sufficient con-

ditions for stochastic stability of the endemic equilibrium 119875lowastby using a suitable Lyapunov function andother techniques ofstochastic analysis The investigation of this stochastic modelrevealed that the stochastic stability of 119875lowast depends on themagnitude of the intensity of noise as well as the param-eters involved within the model system Finally numericalsimulations are given to validate the theoretical results Theproposed model a more accurate epidemic model can helpus to understand the dynamic behavior of virus Moreoverthe theoretical results may provide some useful guidance formaking effective countermeasures on virus propagation

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author was partially supported by Project of Science andTechnology of Heilongjiang Province of China no 12531187

References

[1] C Ji D Jiang andN Shi ldquoMultigroup SIR epidemicmodel withstochastic perturbationrdquo Physica A vol 390 no 10 pp 1747ndash1762 2011

[2] A Lajmanovich and J A Yorke ldquoA deterministic model forgonorrhea in a nonhomogeneous populationrdquo MathematicalBiosciences vol 28 no 3-4 pp 221ndash236 1976

[3] E Beretta and V Capasso ldquoGlobal stability results for amultigroup SIR epidemic modelrdquo in Mathematical Ecology TG Hallam L J Gross and S A Levin Eds pp 317ndash342 WorldScientific Teaneck NJ USA 1988

[4] H W Hethcote ldquoAn immunization model for a heterogeneouspopulationrdquo Theoretical Population Biology vol 14 no 3 pp338ndash349 1978

[5] H W Hethcote and H R Thieme ldquoStability of the endemicequilibrium in epidemic models with subpopulationsrdquo Math-ematical Biosciences vol 75 no 2 pp 205ndash227 1985

[6] H R Thieme ldquoLocal stability in epidemic models for hetero-geneous populationsrdquo inMathematics in Biology and MedicineV Capasso E Grosso and S L Paveri-Fontana Eds vol 57 ofLecture Notes in Biomathematics pp 185ndash189 Springer BerlinGermany 1985

[7] H R Thieme Mathematics in Population Biology PrincetonUniversity Press Princeton NJ USA 2003

[8] H Guo M Y Li and Z Shuai ldquoA graph-theoretic approach tothe method of global Lyapunov functionsrdquo Proceedings of theAmerican Mathematical Society vol 136 no 8 pp 2793ndash28022008

[9] M Y Li Z Shuai and C Wang ldquoGlobal stability of multi-group epidemic models with distributed delaysrdquo Journal ofMathematical Analysis and Applications vol 361 no 1 pp 38ndash47 2010

[10] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianAppliedMathematics Quarterly vol 14 no 3 pp 259ndash284 2006

[11] Y Muroya Y Enatsu and T Kuniya ldquoGlobal stability for amulti-group SIRS epidemic model with varying populationsizesrdquo Nonlinear Analysis Real World Applications vol 14 no3 pp 1693ndash1704 2013

[12] Y Nakata Y Enatsu and Y Muroya ldquoOn the global stability ofan SIRS epidemic model with distributed delaysrdquo Discrete andContinuous Dynamical Systems A vol 2011 pp 1119ndash1128 2011

[13] Y Enatsu Y Nakata and Y Muroya ldquoLyapunov functionaltechniques for the global stability analysis of a delayed SIRSepidemic modelrdquo Nonlinear Analysis Real World Applicationsvol 13 no 5 pp 2120ndash2133 2012

[14] C CMcCluskey ldquoComplete global stability for an SIR epidemicmodel with delaymdashdistributed or discreterdquo Nonlinear AnalysisReal World Applications vol 11 no 1 pp 55ndash59 2010

[15] N Dalal D Greenhalgh and X Mao ldquoA stochastic model ofAIDS and condom userdquo Journal of Mathematical Analysis andApplications vol 325 no 1 pp 36ndash53 2007

[16] C Ji D Jiang and N Shi ldquoAnalysis of a predator-prey modelwith modified Leslie-Gower and Holling-type II schemes withstochastic perturbationrdquo Journal of Mathematical Analysis andApplications vol 359 no 2 pp 482ndash498 2009

[17] C Ji D Jiang and X Li ldquoQualitative analysis of a stochasticratio-dependent predator-prey systemrdquo Journal of Computa-tional and Applied Mathematics vol 235 no 5 pp 1326ndash13412011

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Abstract and Applied Analysis

[18] E Tornatore S M Buccellato and P Vetro ldquoStability of astochastic SIR systemrdquo Physica A vol 354 no 1ndash4 pp 111ndash1262005

[19] E Beretta V Kolmanovskii and L Shaikhet ldquoStability ofepidemic model with time delays influenced by stochasticperturbationsrdquoMathematics and Computers in Simulation vol45 no 3-4 pp 269ndash277 1998

[20] L Shaikhet ldquoStability of predator-preymodel with aftereffect bystochastic perturbationrdquo Stability and Control vol 1 no 1 pp3ndash13 1998

[21] M Carletti ldquoOn the stability properties of a stochastic modelfor phage-bacteria interaction in open marine environmentrdquoMathematical Biosciences vol 175 no 2 pp 117ndash131 2002

[22] M Bandyopadhyay and J Chattopadhyay ldquoRatio-dependentpredator-prey model effect of environmental fluctuation andstabilityrdquo Nonlinearity vol 18 no 2 pp 913ndash936 2005

[23] R R Sarkar and S Banerjee ldquoCancer self remission and tumorstabilitymdasha stochastic approachrdquoMathematical Biosciences vol196 no 1 pp 65ndash81 2005

[24] M Bandyopadhyay T Saha and R Pal ldquoDeterministic andstochastic analysis of a delayed allelopathic phytoplanktonmodel within fluctuating environmentrdquo Nonlinear AnalysisHybrid Systems vol 2 no 3 pp 958ndash970 2008

[25] L Shaikhet ldquoStability of a positive point of equilibrium of onenonlinear system with aftereffect and stochastic perturbationsrdquoDynamic Systems and Applications vol 17 no 1 pp 235ndash2532008

[26] N Bradul and L Shaikhet ldquoStability of the positive point ofequilibrium of Nicholsonrsquos blowflies equation with stochasticperturbations numerical analysisrdquoDiscrete Dynamics in Natureand Society vol 2007 Article ID 92959 25 pages 2007

[27] B Mukhopadhyay and R Bhattacharyya ldquoA nonlinear math-ematical model of virus-tumor-immune system interactiondeterministic and stochastic analysisrdquo Stochastic Analysis andApplications vol 27 no 2 pp 409ndash429 2009

[28] C YuanD Jiang DOrsquoRegan andR P Agarwal ldquoStochasticallyasymptotically stability of the multi-group SEIR and SIR mod-els with random perturbationrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 6 pp 2501ndash25162012

[29] J Yu D Jiang and N Shi ldquoGlobal stability of two-group SIRmodel with random perturbationrdquo Journal of MathematicalAnalysis and Applications vol 360 no 1 pp 235ndash244 2009

[30] L Imhof and S Walcher ldquoExclusion and persistence in deter-ministic and stochastic chemostat modelsrdquo Journal of Differen-tial Equations vol 217 no 1 pp 26ndash53 2005

[31] X Fan and Z Wang ldquoStability analysis of an SEIR epidemicmodel with stochastic perturbation and numerical simulationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 14 no 2 pp 113ndash121 2013

[32] X Fan Z Wang and X Xu ldquoGlobal stability of two-groupepidemicmodels with distributed delays and random perturba-tionrdquoAbstract andApplied Analysis vol 2012 Article ID 13209512 pages 2012

[33] Z Wang X Fan and Q Han ldquoGlobal stability of deterministicand stochastic multigroup SEIQR models in computer net-workrdquo Applied Mathematical Modelling vol 37 no 20-21 pp8673ndash8686 2013

[34] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of