Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 703826 11 pageshttpdxdoiorg1011552013703826

Research ArticleGlobal Dynamic Behavior of a Multigroup Cholera Model withIndirect Transmission

Ming-Tao Li1 Gui-Quan Sun234 Juan Zhang1 and Zhen Jin12

1 Department of Mathematics North University of China Taiyuan Shanrsquoxi 030051 China2 Complex Systems Center Shanxi University Taiyuan Shanrsquoxi 030006 China3 Institute of Information Economy Hangzhou Normal University Hangzhou Zhejiang 310036 China4 School of Mathematical Sciences Fudan University Shanghai 200433 China

Correspondence should be addressed to Zhen Jin jinzhn263net

Received 4 October 2013 Accepted 7 November 2013

Academic Editor Antonia Vecchio

Copyright copy 2013 Ming-Tao Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For a multigroup cholera model with indirect transmission the infection for a susceptible person is almost invariably transmittedby drinking contaminated water in which pathogens V cholerae are present The basic reproduction numberR

0is identified and

global dynamics are completely determined byR0 It shows thatR

0is a globally threshold parameter in the sense that if it is less than

one the disease-free equilibrium is globally asymptotically stable whereas if it is larger than one there is a unique endemic equilib-riumwhich is global asymptotically stable For the proof of global stability with the disease-free equilibrium we use the comparisonprinciple and for the endemic equilibrium we use the classical method of Lyapunov function and the graph-theoretic approach

1 Introduction

Cholera a waterborne gastroenteric infection remains asignificant threat to public health in the developing worldOutbreaks of cholera occur cyclically usually twice per yearin endemic areas and the intensity of these outbreaks variesover longer periods [1] Hence in the last few decades enor-mous attention is being paid to the cholera disease and severalmathematical dynamic models have been developed to studythe transmission of cholera [1ndash7] In these papers theyconsider the population is uniformlymixed butmany factorscan lead to heterogeneity in a host population So in this paperwe divide different population into different groups whichcan be divided geographically into communities cities andcountries to incorporate differential infectivity of multiplestrains of the disease agent

In the case of cholera the transmission usually occursthrough ingestion of contaminated water or feces rather thanthrough casual human-human contact [1] Therefore directcontact of healthy person with an infected person is not arisk for contracting infection whereas a healthy person maycontract infection by drinking contaminated water in which

pathogens V cholerae are present [2] The members of thisbacterial genus (V cholerae) naturally colonize in lakes riversand estuaries Therefore we consider that cholera transmitsto other individuals via bacteria in the aquatic environmentand formulates a multi-group epidemic model for choleraLet 119873 be the total population which is divided into fourepidemiological compartments susceptible compartment 119878infectious compartment 119868 recovered compartment 119877 andvaccinated compartment 119881 Let 119882 be the density of Vcholerae in the aquatic environment As a consequence of theincrease in the density of virulent V cholerae in the aquaticenvironment humans become infected and begin to shedincreasing numbers of bacteria into the aquatic environmentfurther elevating bacterial density and exacerbating the out-break [1]The growth rate of density of bacteria in the aquaticenvironment is assumed to be proportional to the numberof infectious individuals We assume that the immunityinduced by vaccination is perfect therefore individuals invaccinated individuals 119881 cannot be infected The model iscalled a multi-group cholera SIRVW epidemic model

In recent years multi-group epidemic models havebeen used to describe the transmission dynamics of many

2 Discrete Dynamics in Nature and Society

infectious disease in heterogeneous individuals such asHIVAIDS [8] dengue [9] West-Nile virus [10] sexuallytransmitted diseases [11] and so on It is well knownthat global dynamics of multi-group models with higherdimensions especially the global stability of the endemicequilibrium is a very challenging problem Lajmanovichand York [12] proved global stability of the unique endemicequilibrium by using a quadratic global Lyapunov functionon a class of 119899-group SIS models for gonorrhea Hethcote[13] proved global stability of the endemic equilibrium formulti-group SIR model without vital dynamics Thieme [14]proved global stability of the endemic equilibrium of multi-group SEIRSmodel under certain restrictions However theyonly proved global stability of the endemic equilibrium formulti-group model under some special conditions In 2006Guo et al [15] have first succeeded to establish the completeglobal dynamics for a multi-group SIR model by making useof the theory of non-negative matrices Lyapunov functionsand a subtle grouping technique in estimating the derivativesof Lyapunov functions guided by graph theory By using theresults or ideas of [15] the papers [16 17] proved the globalstability of the endemic equilibrium for multi-group modelwith nonlinear incidence rates and the papers [18 19] provedthe global stability of the endemic equilibrium for multi-group model with distributed delays

Distinguishing from these multi-group models withdirect transmission from person to person a multi-groupcholeramodel with indirect transmission from the bacteria ofthe aquatic environment to person is proposed in this paperWe prove that the disease-free equilibrium is globally asymp-totically stable ifR

0lt 1 while an endemic equilibrium exists

uniquely and is globally asymptotically stable ifR0gt 1

The organization of this paper is as follows In Section 2we construct a multi-group cholera epidemiological and givesome dynamic analysis on the disease-free equilibrium andthe endemic equilibrium An example is given in Section 3and some conclusions are included in Section 4

2 Mathematical Modeling and Analysis

For a multi-group epidemic model with cholera the popula-tion of human is divided into 119899 discrete groups where 119899 isin

N Let 119878119894(119905) 119868119894(119905) 119877

119894(119905) and 119881

119894(119905) be the numbers of sus-

ceptible infectious recovered and vaccinated individuals ingroup 119894 = 1 2 119899 at time 119905 respectively Let 119882

119894(119905) be

the density of bacteria in the aquatic environment ingroup 119894 = 1 2 119899 at time 119905 Based on the assumptions inSection 1 the disease transmission rate of cholera betweencompartments 119878

119894and 119882

119895is denoted by 120573

119894119895 which means

the susceptible individuals in the 119894th group can contactthe bacteria of the aquatic environment in the 119895th (119895 =

1 2 119899) group So the new infection that occurred inthe 119894th group is given by sum

119899

119895=1120573119894119895119878119894119882119895 The recruitment rate

of individuals into 119878119894(119905) compartment with the 119894th group

is given by a constant 119860119894 Within the 119894th group it is

assumed that natural death of human is 119889119894 A simple immu-

nization policy is considered where the vaccination ratein 119878119894(119905) compartment is given by a constant 120574

119894and the losing

immunity rate from vaccination individuals is 120582119894 We assume

that individuals in 119868119894(119905) compartment recover with a rate

constant 119903119894 In 119882

119894(119905) compartment the brucella shedding

rate from 119868119894(119905) compartment is 119896

119894 and the decaying rate of

brucella is 120575119894 So a general multi-group SIRVW epidemic

model is described by the following system of differentialequations

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895 minus (119889119894+ 119903119894) 119868119894

119889119877119894

119889119905= 119903119894119868119894minus 119889119894119877119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(1)

The parameters 119860119894 119889119894 120582119894 120574119894 119896119894 and 120575

119894are positive for

all 119894 = 1 2 119899 which is made for the biological justifi-cation And we assume that 120573

119894119895is nonnegative for all 119894 119895 =

1 2 119899 and 119899-squarematrix 119861 = (120573119894119895)1le119894119895le119899

is irreduciblewhich implies that every pair of groups is joined by aninfectious path so that the presence of an infectious individualin the first group can cause infection in the second group

Observe that the variable 119877119894does not appear in the first

and last two equations of system (1) this allows us to considerthe following reduced system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(2)

For each group 119894 adding the four equations in system (2)gives

119889 (119878119894+ 119868119894+ 119881119894)

119889119905= 119860119894minus 119889119894(119878119894+ 119868119894+ 119881119894) minus 119903119894119868119894

le (119860119894minus 119889119894(119878119894+ 119868119894+ 119881119894))

(3)

Discrete Dynamics in Nature and Society 3

then it follows that

lim119905rarrinfin

sup (119878119894+ 119868119894+ 119881119894) le

119860119894

119889119894

lim119905rarrinfin

sup119882119894le

119896119894119860119894

119889119894120575119894

(4)

Therefore omega limit sets of system (2) are containedin the following bounded region in the nonnegative cone ofR4119899

119883 = (119878119894 119868119894 119881119894119882119894) | 119878119894 119868119894 119881119894119882119894ge 0 0 le (119878

119894+ 119868119894+ 119881119894)

le119860119894

119889119894

119882119894le

119896119894119860119894

119889119894120575119894

119894 = 1 2 119899

(5)

It can be verified that region 119883 is positively invariant withrespect to system (2) System (2) always has a disease-freeequilibrium

1198750= (1198780

1 0 1198810

1 0 119878

0

119894 0 1198810

119894 0 119878

0

119899 0 1198810

119899 0) (6)

on the boundary of 119883 where

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119881

0

119894=

119860119894120574119894

119889119894(120582119894+ 119889119894+ 120574119894) (7)

21 The Basic Reproduction Number According to the nextgeneration matrix formulated in papers [20ndash22] we definethe basic reproduction number R

0of system (2) In order to

formulate R0 we order the infected variables first by disease

state and then by group that is

11986811198821 11986821198822 119868

119899119882119899 (8)

Consider the following auxiliary system

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(9)

Follow the recipe from van denDriessche andWatmough[21] to obtain

119865 =

((((

(

0 120573111198780

10 120573121198780

1sdot sdot sdot 0 120573

11198991198780

1

0 0 0 0 sdot sdot sdot 0 0

0 120573211198780

20 120573221198780

2sdot sdot sdot 0 120573

21198991198780

2

0 0 0 0 sdot sdot sdot 0 0

0 12057311989911198780

1198990 12057311989921198780

119899sdot sdot sdot 0 120573

1198991198991198780

119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

119881 =

((((

(

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

minus1198961

1205751

0 0 sdot sdot sdot 0 0

0 0 1198892+ 1199032

0 sdot sdot sdot 0 0

0 0 minus1198962

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot 119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot minus1198961

120575119899

))))

)2119899times2119899

(10)We can get the inverse of 119881 which equals

119881minus1

=

(((((((((((((((((((((((((

(

1

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

1198961

1205751(1198891+ 1199031)

1

1205751

0 0 sdot sdot sdot 0 0

0 01

1198892+ 1199032

0 sdot sdot sdot 0 0

0 01198962

1205752(1198892+ 1199032)

1

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot1

119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot119896119899

120575119899(119889119899+ 119903119899)

1

120575119899

)))))))))))))))))))))))))

)2119899times2119899

(11)

4 Discrete Dynamics in Nature and Society

Thus the next generation matrix is 119865119881minus1

119865119881minus1

=(((

(

11986011

sdot sdot sdot 1198601119899

11986111

sdot sdot sdot 1198611119899

1198601198991

sdot sdot sdot 119860119899119899

1198611198991

sdot sdot sdot 119861119899119899

0 sdot sdot sdot 0 0 sdot sdot sdot 0

0 sdot sdot sdot 0 0 sdot sdot sdot 0

)))

)2119899times2119899

119860 = (

11986011

11986012

sdot sdot sdot 1198601119899

11986021

11986022

sdot sdot sdot 1198602119899

1198601198991

1198601198992

sdot sdot sdot 119860119899119899

)

119899times119899

(12)

So we can calculate the basic reproduction number of system(2)

R0= 120588 (119865119881

minus1) = 120588 (119860) (13)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894)

119894 = 1 2 119899

(14)

and 120588 denotes the spectral radius As we will show R0is the

key threshold parameters whose values completely character-ize the global dynamics of system (2)

22 Global Stability of the Disease-Free Equilibrium of System(2) For the disease-free equilibrium 119875

0of system (2) we

have the following property

Theorem 1 If R0

lt 1 the disease-free equilibrium 1198750of

system (2) is globally asymptotically stable in the region 119883

Proof Let 119872 = 119865minus119881 and define 119904(119872) = maxRe 120582 120582 is aneigenvalue of 119872 so 119904(119872) is a simple eigenvalue of 119872 witha positive eigenvector [23] By Theorem 2 in [21] there holdtwo equivalences

R0gt 1 lArrrArr 119904 (119872) gt 0 R

0lt 1 lArrrArr 119904 (119872) lt 0 (15)

To prove the locally stability of disease-free equilibriumwe check the hypotheses (A1)ndash(A5) in [21] Hypotheses (A1)ndash(A4) are easily verified while (A5) is satisfied if all eigenvaluesof the 4119899 times 4119899 matrix

119869|1198750= (

119872 0

1198693

1198694

)

4119899times4119899

(16)

have negative real parts where 1198693= minus119865

1198694=

((((

(

minus(1198891+ 1205741) 120582

10 0 sdot sdot sdot 0 0

1205741

minus (1198891+ 1205821) 0 0 sdot sdot sdot 0 0

0 0 minus (1198892+ 1205742) 120582

2sdot sdot sdot 0 0

0 0 1205742

minus (1198892+ 1205822) sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot minus (119889119899+ 120574119899) 120582

119899

0 0 0 0 sdot sdot sdot 120574119899

minus(119889119899+ 120582119899)

))))

)2119899times2119899

(17)

Calculate the eigenvalues of 1198694

119904 (1198694) = max minus119889

1 minus119889

119899 minus (1198891+ 1205821+ 1205741)

minus (119889119899+ 120582119899+ 120574119899) lt 0

(18)

If R0lt 1 then 119904(119872) lt 0 and 119904(119869|

1198750) lt 0 and the disease-

free equilibrium 1198750of (2) is locally asymptotically stable

Nowwewill prove that the disease-free equilibrium 1198750of

system (2) is globally attractive when R0lt 1 From the third

equation of system (2) we have

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

= 120574119894(119873119894minus (119868119894+ 119881119894)) minus (120582

119894+ 119889119894) 119881119894

le 120574119894

119860119894

119889119894

minus (120582119894+ 120574119894+ 119889119894) 119881119894

(19)

So we can have that for a small enough positive number 1205981

there exists 119905119894gt 0 119894 = 1 2 119899 such that for all 119905 gt 119905

119894

119881119894le

119860119894120574119894

119889119894(120582119894+ 120574119894+ 119889119894)+ 1205981= 1198810

119894+ 1205981 (20)

Also from the equations of system (2) we have

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

le 119860119894+ 120582119894(1198810

119894+ 1205981) minus (119889

119894+ 120574119894) 119878119900

(21)

Then

lim119905rarrinfin

sup 119878119894=

119860119894+ 120582119894(1198810

119894+ 1205981)

119889119894+ 120574119894

=1198780

119894+1205982 (120598

2=

1205821198941205981

119889119894+ 120574119894

)

(22)

Discrete Dynamics in Nature and Society 5

From system (9) and 119878119894le 1198780

119894+ 1205982with all 119905 gt 119905

119894 Thus

when 119905 gt 119905119894 we derive

119889119868119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(23)

Consider the following auxiliary system

1198891198681015840

119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(24)

Let 1198720be the matrix defined by

1198720=

((((

(

0 12057311

0 12057312

sdot sdot sdot 0 1205731119899

0 0 0 0 sdot sdot sdot 0 0

0 12057321

0 12057322

sdot sdot sdot 0 1205732119899

0 0 0 0 sdot sdot sdot 0 0

0 1205731198991

0 1205731198992

sdot sdot sdot 0 120573119899119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

(25)

and set 1198721

= 119872 + 12059821198720 It follows from Theorem 2 in

[21] that R0

lt 1 if and only if 119904(119872) lt 0 Thus thereexists an 120598

2gt 0 small enough such that 119904(119872

1) lt 0 Using

the Perron-Frobenius theorem all eigenvalues of the mat-rix 119872

1have negative real parts when 119904(119872

1) lt 0 Therefore

it has

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(26)

which implies that the zero solution of system (24) is globallyasymptotically stable Using the comparison principle ofSmith and Waltman [23] we know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(27)

By the theory of asymptotic autonomous system of Thieme[24] it is also known that

(1198781 (119905) 1198811 (119905) 119878119899 (119905) 119881119899 (119905))

997888rarr (1198781(0) 119881

1(0) 119878

119899(0) 119881

119899(0)) 119905 997888rarr infin

(28)

So 1198750is globally attractive when R

0lt 1 It follows that the

disease-free equilibrium 1198750of (2) is globally asymptotically

stable when R0lt 1 This completes the proof

23 The Uniform Persistence and Unique Positive Solution ofSystem (2) In this section we give the proof of the uniformpersistence and the unique positive solution of system (2)Define

1198830= (119878119894 119868119894 119881119894119882119894) isin 119883 | 119868

119894119882119894gt 0 119894 = 1 2 119899

1205971198830= 119883 | 119883

0

(29)

Theorem 2 When R0

gt 1 there exists a positive constant1205761such that when 119868

119894(0) lt 120576

1 119882119894(0) lt 120576

1for (119878

119894(0)

119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0

lim sup119905rarrinfin

max 119868119894(119905) 119882

119894(119905) gt 120576

1 119894 = 1 2 119899 (30)

Proof Consider the following system

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(31)

Using Corollary 32 in Zhao and Jing [25] it then fol-lows that system (31) has a unique positive equilibrium(1198780

1 1198810

1 119878

0

119899 1198810

119899) which is globally asymptotically stable

As to R0gt 1 hArr 119904(119872) gt 0 choose 120576 gt 0 small enough

such that 119904(1198722) gt 0 where 119872

2= 119872 minus 120576119872

0 Let us consider

a perturbed system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(32)

From our previous analysis of system (32) we canrestrict 120576

1gt 0 small enough such that (32) admits a unique

positive equilibrium (1198780

119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) which is

globally asymptotically stable 1198780119894(1205761) is continuous in 120576

1 so

we can further restrict 1205761small enough such that 1198780

119894(1205761) gt

1198780

119894minus 120576 119894 = 1 2 119899For the sake of contradiction ofTheorem 2 there is a 119879 gt

0 such that 119868119894(119905) lt 120576

1119882119894(119905) lt 120576

1 119894 = 1 2 119899 for all 119905 ge 119879

Then for 119905 ge 119879 we have

119889119878119894

119889119905ge 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119877119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119877119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119877119894

119894 = 1 2 119899

(33)

Since the equilibrium (1198780119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) of

(32) is globally asymptotically stable and 1198780

119894(1205761) gt 119878

0

119894minus 120576

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

2 Discrete Dynamics in Nature and Society

infectious disease in heterogeneous individuals such asHIVAIDS [8] dengue [9] West-Nile virus [10] sexuallytransmitted diseases [11] and so on It is well knownthat global dynamics of multi-group models with higherdimensions especially the global stability of the endemicequilibrium is a very challenging problem Lajmanovichand York [12] proved global stability of the unique endemicequilibrium by using a quadratic global Lyapunov functionon a class of 119899-group SIS models for gonorrhea Hethcote[13] proved global stability of the endemic equilibrium formulti-group SIR model without vital dynamics Thieme [14]proved global stability of the endemic equilibrium of multi-group SEIRSmodel under certain restrictions However theyonly proved global stability of the endemic equilibrium formulti-group model under some special conditions In 2006Guo et al [15] have first succeeded to establish the completeglobal dynamics for a multi-group SIR model by making useof the theory of non-negative matrices Lyapunov functionsand a subtle grouping technique in estimating the derivativesof Lyapunov functions guided by graph theory By using theresults or ideas of [15] the papers [16 17] proved the globalstability of the endemic equilibrium for multi-group modelwith nonlinear incidence rates and the papers [18 19] provedthe global stability of the endemic equilibrium for multi-group model with distributed delays

Distinguishing from these multi-group models withdirect transmission from person to person a multi-groupcholeramodel with indirect transmission from the bacteria ofthe aquatic environment to person is proposed in this paperWe prove that the disease-free equilibrium is globally asymp-totically stable ifR

0lt 1 while an endemic equilibrium exists

uniquely and is globally asymptotically stable ifR0gt 1

The organization of this paper is as follows In Section 2we construct a multi-group cholera epidemiological and givesome dynamic analysis on the disease-free equilibrium andthe endemic equilibrium An example is given in Section 3and some conclusions are included in Section 4

2 Mathematical Modeling and Analysis

For a multi-group epidemic model with cholera the popula-tion of human is divided into 119899 discrete groups where 119899 isin

N Let 119878119894(119905) 119868119894(119905) 119877

119894(119905) and 119881

119894(119905) be the numbers of sus-

ceptible infectious recovered and vaccinated individuals ingroup 119894 = 1 2 119899 at time 119905 respectively Let 119882

119894(119905) be

the density of bacteria in the aquatic environment ingroup 119894 = 1 2 119899 at time 119905 Based on the assumptions inSection 1 the disease transmission rate of cholera betweencompartments 119878

119894and 119882

119895is denoted by 120573

119894119895 which means

the susceptible individuals in the 119894th group can contactthe bacteria of the aquatic environment in the 119895th (119895 =

1 2 119899) group So the new infection that occurred inthe 119894th group is given by sum

119899

119895=1120573119894119895119878119894119882119895 The recruitment rate

of individuals into 119878119894(119905) compartment with the 119894th group

is given by a constant 119860119894 Within the 119894th group it is

assumed that natural death of human is 119889119894 A simple immu-

nization policy is considered where the vaccination ratein 119878119894(119905) compartment is given by a constant 120574

119894and the losing

immunity rate from vaccination individuals is 120582119894 We assume

that individuals in 119868119894(119905) compartment recover with a rate

constant 119903119894 In 119882

119894(119905) compartment the brucella shedding

rate from 119868119894(119905) compartment is 119896

119894 and the decaying rate of

brucella is 120575119894 So a general multi-group SIRVW epidemic

model is described by the following system of differentialequations

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895 minus (119889119894+ 119903119894) 119868119894

119889119877119894

119889119905= 119903119894119868119894minus 119889119894119877119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(1)

The parameters 119860119894 119889119894 120582119894 120574119894 119896119894 and 120575

119894are positive for

all 119894 = 1 2 119899 which is made for the biological justifi-cation And we assume that 120573

119894119895is nonnegative for all 119894 119895 =

1 2 119899 and 119899-squarematrix 119861 = (120573119894119895)1le119894119895le119899

is irreduciblewhich implies that every pair of groups is joined by aninfectious path so that the presence of an infectious individualin the first group can cause infection in the second group

Observe that the variable 119877119894does not appear in the first

and last two equations of system (1) this allows us to considerthe following reduced system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(2)

For each group 119894 adding the four equations in system (2)gives

119889 (119878119894+ 119868119894+ 119881119894)

119889119905= 119860119894minus 119889119894(119878119894+ 119868119894+ 119881119894) minus 119903119894119868119894

le (119860119894minus 119889119894(119878119894+ 119868119894+ 119881119894))

(3)

Discrete Dynamics in Nature and Society 3

then it follows that

lim119905rarrinfin

sup (119878119894+ 119868119894+ 119881119894) le

119860119894

119889119894

lim119905rarrinfin

sup119882119894le

119896119894119860119894

119889119894120575119894

(4)

Therefore omega limit sets of system (2) are containedin the following bounded region in the nonnegative cone ofR4119899

119883 = (119878119894 119868119894 119881119894119882119894) | 119878119894 119868119894 119881119894119882119894ge 0 0 le (119878

119894+ 119868119894+ 119881119894)

le119860119894

119889119894

119882119894le

119896119894119860119894

119889119894120575119894

119894 = 1 2 119899

(5)

It can be verified that region 119883 is positively invariant withrespect to system (2) System (2) always has a disease-freeequilibrium

1198750= (1198780

1 0 1198810

1 0 119878

0

119894 0 1198810

119894 0 119878

0

119899 0 1198810

119899 0) (6)

on the boundary of 119883 where

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119881

0

119894=

119860119894120574119894

119889119894(120582119894+ 119889119894+ 120574119894) (7)

21 The Basic Reproduction Number According to the nextgeneration matrix formulated in papers [20ndash22] we definethe basic reproduction number R

0of system (2) In order to

formulate R0 we order the infected variables first by disease

state and then by group that is

11986811198821 11986821198822 119868

119899119882119899 (8)

Consider the following auxiliary system

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(9)

Follow the recipe from van denDriessche andWatmough[21] to obtain

119865 =

((((

(

0 120573111198780

10 120573121198780

1sdot sdot sdot 0 120573

11198991198780

1

0 0 0 0 sdot sdot sdot 0 0

0 120573211198780

20 120573221198780

2sdot sdot sdot 0 120573

21198991198780

2

0 0 0 0 sdot sdot sdot 0 0

0 12057311989911198780

1198990 12057311989921198780

119899sdot sdot sdot 0 120573

1198991198991198780

119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

119881 =

((((

(

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

minus1198961

1205751

0 0 sdot sdot sdot 0 0

0 0 1198892+ 1199032

0 sdot sdot sdot 0 0

0 0 minus1198962

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot 119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot minus1198961

120575119899

))))

)2119899times2119899

(10)We can get the inverse of 119881 which equals

119881minus1

=

(((((((((((((((((((((((((

(

1

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

1198961

1205751(1198891+ 1199031)

1

1205751

0 0 sdot sdot sdot 0 0

0 01

1198892+ 1199032

0 sdot sdot sdot 0 0

0 01198962

1205752(1198892+ 1199032)

1

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot1

119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot119896119899

120575119899(119889119899+ 119903119899)

1

120575119899

)))))))))))))))))))))))))

)2119899times2119899

(11)

4 Discrete Dynamics in Nature and Society

Thus the next generation matrix is 119865119881minus1

119865119881minus1

=(((

(

11986011

sdot sdot sdot 1198601119899

11986111

sdot sdot sdot 1198611119899

1198601198991

sdot sdot sdot 119860119899119899

1198611198991

sdot sdot sdot 119861119899119899

0 sdot sdot sdot 0 0 sdot sdot sdot 0

0 sdot sdot sdot 0 0 sdot sdot sdot 0

)))

)2119899times2119899

119860 = (

11986011

11986012

sdot sdot sdot 1198601119899

11986021

11986022

sdot sdot sdot 1198602119899

1198601198991

1198601198992

sdot sdot sdot 119860119899119899

)

119899times119899

(12)

So we can calculate the basic reproduction number of system(2)

R0= 120588 (119865119881

minus1) = 120588 (119860) (13)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894)

119894 = 1 2 119899

(14)

and 120588 denotes the spectral radius As we will show R0is the

key threshold parameters whose values completely character-ize the global dynamics of system (2)

22 Global Stability of the Disease-Free Equilibrium of System(2) For the disease-free equilibrium 119875

0of system (2) we

have the following property

Theorem 1 If R0

lt 1 the disease-free equilibrium 1198750of

system (2) is globally asymptotically stable in the region 119883

Proof Let 119872 = 119865minus119881 and define 119904(119872) = maxRe 120582 120582 is aneigenvalue of 119872 so 119904(119872) is a simple eigenvalue of 119872 witha positive eigenvector [23] By Theorem 2 in [21] there holdtwo equivalences

R0gt 1 lArrrArr 119904 (119872) gt 0 R

0lt 1 lArrrArr 119904 (119872) lt 0 (15)

To prove the locally stability of disease-free equilibriumwe check the hypotheses (A1)ndash(A5) in [21] Hypotheses (A1)ndash(A4) are easily verified while (A5) is satisfied if all eigenvaluesof the 4119899 times 4119899 matrix

119869|1198750= (

119872 0

1198693

1198694

)

4119899times4119899

(16)

have negative real parts where 1198693= minus119865

1198694=

((((

(

minus(1198891+ 1205741) 120582

10 0 sdot sdot sdot 0 0

1205741

minus (1198891+ 1205821) 0 0 sdot sdot sdot 0 0

0 0 minus (1198892+ 1205742) 120582

2sdot sdot sdot 0 0

0 0 1205742

minus (1198892+ 1205822) sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot minus (119889119899+ 120574119899) 120582

119899

0 0 0 0 sdot sdot sdot 120574119899

minus(119889119899+ 120582119899)

))))

)2119899times2119899

(17)

Calculate the eigenvalues of 1198694

119904 (1198694) = max minus119889

1 minus119889

119899 minus (1198891+ 1205821+ 1205741)

minus (119889119899+ 120582119899+ 120574119899) lt 0

(18)

If R0lt 1 then 119904(119872) lt 0 and 119904(119869|

1198750) lt 0 and the disease-

free equilibrium 1198750of (2) is locally asymptotically stable

Nowwewill prove that the disease-free equilibrium 1198750of

system (2) is globally attractive when R0lt 1 From the third

equation of system (2) we have

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

= 120574119894(119873119894minus (119868119894+ 119881119894)) minus (120582

119894+ 119889119894) 119881119894

le 120574119894

119860119894

119889119894

minus (120582119894+ 120574119894+ 119889119894) 119881119894

(19)

So we can have that for a small enough positive number 1205981

there exists 119905119894gt 0 119894 = 1 2 119899 such that for all 119905 gt 119905

119894

119881119894le

119860119894120574119894

119889119894(120582119894+ 120574119894+ 119889119894)+ 1205981= 1198810

119894+ 1205981 (20)

Also from the equations of system (2) we have

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

le 119860119894+ 120582119894(1198810

119894+ 1205981) minus (119889

119894+ 120574119894) 119878119900

(21)

Then

lim119905rarrinfin

sup 119878119894=

119860119894+ 120582119894(1198810

119894+ 1205981)

119889119894+ 120574119894

=1198780

119894+1205982 (120598

2=

1205821198941205981

119889119894+ 120574119894

)

(22)

Discrete Dynamics in Nature and Society 5

From system (9) and 119878119894le 1198780

119894+ 1205982with all 119905 gt 119905

119894 Thus

when 119905 gt 119905119894 we derive

119889119868119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(23)

Consider the following auxiliary system

1198891198681015840

119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(24)

Let 1198720be the matrix defined by

1198720=

((((

(

0 12057311

0 12057312

sdot sdot sdot 0 1205731119899

0 0 0 0 sdot sdot sdot 0 0

0 12057321

0 12057322

sdot sdot sdot 0 1205732119899

0 0 0 0 sdot sdot sdot 0 0

0 1205731198991

0 1205731198992

sdot sdot sdot 0 120573119899119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

(25)

and set 1198721

= 119872 + 12059821198720 It follows from Theorem 2 in

[21] that R0

lt 1 if and only if 119904(119872) lt 0 Thus thereexists an 120598

2gt 0 small enough such that 119904(119872

1) lt 0 Using

the Perron-Frobenius theorem all eigenvalues of the mat-rix 119872

1have negative real parts when 119904(119872

1) lt 0 Therefore

it has

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(26)

which implies that the zero solution of system (24) is globallyasymptotically stable Using the comparison principle ofSmith and Waltman [23] we know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(27)

By the theory of asymptotic autonomous system of Thieme[24] it is also known that

(1198781 (119905) 1198811 (119905) 119878119899 (119905) 119881119899 (119905))

997888rarr (1198781(0) 119881

1(0) 119878

119899(0) 119881

119899(0)) 119905 997888rarr infin

(28)

So 1198750is globally attractive when R

0lt 1 It follows that the

disease-free equilibrium 1198750of (2) is globally asymptotically

stable when R0lt 1 This completes the proof

23 The Uniform Persistence and Unique Positive Solution ofSystem (2) In this section we give the proof of the uniformpersistence and the unique positive solution of system (2)Define

1198830= (119878119894 119868119894 119881119894119882119894) isin 119883 | 119868

119894119882119894gt 0 119894 = 1 2 119899

1205971198830= 119883 | 119883

0

(29)

Theorem 2 When R0

gt 1 there exists a positive constant1205761such that when 119868

119894(0) lt 120576

1 119882119894(0) lt 120576

1for (119878

119894(0)

119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0

lim sup119905rarrinfin

max 119868119894(119905) 119882

119894(119905) gt 120576

1 119894 = 1 2 119899 (30)

Proof Consider the following system

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(31)

Using Corollary 32 in Zhao and Jing [25] it then fol-lows that system (31) has a unique positive equilibrium(1198780

1 1198810

1 119878

0

119899 1198810

119899) which is globally asymptotically stable

As to R0gt 1 hArr 119904(119872) gt 0 choose 120576 gt 0 small enough

such that 119904(1198722) gt 0 where 119872

2= 119872 minus 120576119872

0 Let us consider

a perturbed system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(32)

From our previous analysis of system (32) we canrestrict 120576

1gt 0 small enough such that (32) admits a unique

positive equilibrium (1198780

119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) which is

globally asymptotically stable 1198780119894(1205761) is continuous in 120576

1 so

we can further restrict 1205761small enough such that 1198780

119894(1205761) gt

1198780

119894minus 120576 119894 = 1 2 119899For the sake of contradiction ofTheorem 2 there is a 119879 gt

0 such that 119868119894(119905) lt 120576

1119882119894(119905) lt 120576

1 119894 = 1 2 119899 for all 119905 ge 119879

Then for 119905 ge 119879 we have

119889119878119894

119889119905ge 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119877119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119877119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119877119894

119894 = 1 2 119899

(33)

Since the equilibrium (1198780119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) of

(32) is globally asymptotically stable and 1198780

119894(1205761) gt 119878

0

119894minus 120576

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

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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

Discrete Dynamics in Nature and Society 3

then it follows that

lim119905rarrinfin

sup (119878119894+ 119868119894+ 119881119894) le

119860119894

119889119894

lim119905rarrinfin

sup119882119894le

119896119894119860119894

119889119894120575119894

(4)

Therefore omega limit sets of system (2) are containedin the following bounded region in the nonnegative cone ofR4119899

119883 = (119878119894 119868119894 119881119894119882119894) | 119878119894 119868119894 119881119894119882119894ge 0 0 le (119878

119894+ 119868119894+ 119881119894)

le119860119894

119889119894

119882119894le

119896119894119860119894

119889119894120575119894

119894 = 1 2 119899

(5)

It can be verified that region 119883 is positively invariant withrespect to system (2) System (2) always has a disease-freeequilibrium

1198750= (1198780

1 0 1198810

1 0 119878

0

119894 0 1198810

119894 0 119878

0

119899 0 1198810

119899 0) (6)

on the boundary of 119883 where

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119881

0

119894=

119860119894120574119894

119889119894(120582119894+ 119889119894+ 120574119894) (7)

21 The Basic Reproduction Number According to the nextgeneration matrix formulated in papers [20ndash22] we definethe basic reproduction number R

0of system (2) In order to

formulate R0 we order the infected variables first by disease

state and then by group that is

11986811198821 11986821198822 119868

119899119882119899 (8)

Consider the following auxiliary system

119889119868119894

119889119905=

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(9)

Follow the recipe from van denDriessche andWatmough[21] to obtain

119865 =

((((

(

0 120573111198780

10 120573121198780

1sdot sdot sdot 0 120573

11198991198780

1

0 0 0 0 sdot sdot sdot 0 0

0 120573211198780

20 120573221198780

2sdot sdot sdot 0 120573

21198991198780

2

0 0 0 0 sdot sdot sdot 0 0

0 12057311989911198780

1198990 12057311989921198780

119899sdot sdot sdot 0 120573

1198991198991198780

119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

119881 =

((((

(

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

minus1198961

1205751

0 0 sdot sdot sdot 0 0

0 0 1198892+ 1199032

0 sdot sdot sdot 0 0

0 0 minus1198962

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot 119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot minus1198961

120575119899

))))

)2119899times2119899

(10)We can get the inverse of 119881 which equals

119881minus1

=

(((((((((((((((((((((((((

(

1

1198891+ 1199031

0 0 0 sdot sdot sdot 0 0

1198961

1205751(1198891+ 1199031)

1

1205751

0 0 sdot sdot sdot 0 0

0 01

1198892+ 1199032

0 sdot sdot sdot 0 0

0 01198962

1205752(1198892+ 1199032)

1

1205752

sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot1

119889119899+ 119903119899

0

0 0 0 0 sdot sdot sdot119896119899

120575119899(119889119899+ 119903119899)

1

120575119899

)))))))))))))))))))))))))

)2119899times2119899

(11)

4 Discrete Dynamics in Nature and Society

Thus the next generation matrix is 119865119881minus1

119865119881minus1

=(((

(

11986011

sdot sdot sdot 1198601119899

11986111

sdot sdot sdot 1198611119899

1198601198991

sdot sdot sdot 119860119899119899

1198611198991

sdot sdot sdot 119861119899119899

0 sdot sdot sdot 0 0 sdot sdot sdot 0

0 sdot sdot sdot 0 0 sdot sdot sdot 0

)))

)2119899times2119899

119860 = (

11986011

11986012

sdot sdot sdot 1198601119899

11986021

11986022

sdot sdot sdot 1198602119899

1198601198991

1198601198992

sdot sdot sdot 119860119899119899

)

119899times119899

(12)

So we can calculate the basic reproduction number of system(2)

R0= 120588 (119865119881

minus1) = 120588 (119860) (13)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894)

119894 = 1 2 119899

(14)

and 120588 denotes the spectral radius As we will show R0is the

key threshold parameters whose values completely character-ize the global dynamics of system (2)

22 Global Stability of the Disease-Free Equilibrium of System(2) For the disease-free equilibrium 119875

0of system (2) we

have the following property

Theorem 1 If R0

lt 1 the disease-free equilibrium 1198750of

system (2) is globally asymptotically stable in the region 119883

Proof Let 119872 = 119865minus119881 and define 119904(119872) = maxRe 120582 120582 is aneigenvalue of 119872 so 119904(119872) is a simple eigenvalue of 119872 witha positive eigenvector [23] By Theorem 2 in [21] there holdtwo equivalences

R0gt 1 lArrrArr 119904 (119872) gt 0 R

0lt 1 lArrrArr 119904 (119872) lt 0 (15)

To prove the locally stability of disease-free equilibriumwe check the hypotheses (A1)ndash(A5) in [21] Hypotheses (A1)ndash(A4) are easily verified while (A5) is satisfied if all eigenvaluesof the 4119899 times 4119899 matrix

119869|1198750= (

119872 0

1198693

1198694

)

4119899times4119899

(16)

have negative real parts where 1198693= minus119865

1198694=

((((

(

minus(1198891+ 1205741) 120582

10 0 sdot sdot sdot 0 0

1205741

minus (1198891+ 1205821) 0 0 sdot sdot sdot 0 0

0 0 minus (1198892+ 1205742) 120582

2sdot sdot sdot 0 0

0 0 1205742

minus (1198892+ 1205822) sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot minus (119889119899+ 120574119899) 120582

119899

0 0 0 0 sdot sdot sdot 120574119899

minus(119889119899+ 120582119899)

))))

)2119899times2119899

(17)

Calculate the eigenvalues of 1198694

119904 (1198694) = max minus119889

1 minus119889

119899 minus (1198891+ 1205821+ 1205741)

minus (119889119899+ 120582119899+ 120574119899) lt 0

(18)

If R0lt 1 then 119904(119872) lt 0 and 119904(119869|

1198750) lt 0 and the disease-

free equilibrium 1198750of (2) is locally asymptotically stable

Nowwewill prove that the disease-free equilibrium 1198750of

system (2) is globally attractive when R0lt 1 From the third

equation of system (2) we have

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

= 120574119894(119873119894minus (119868119894+ 119881119894)) minus (120582

119894+ 119889119894) 119881119894

le 120574119894

119860119894

119889119894

minus (120582119894+ 120574119894+ 119889119894) 119881119894

(19)

So we can have that for a small enough positive number 1205981

there exists 119905119894gt 0 119894 = 1 2 119899 such that for all 119905 gt 119905

119894

119881119894le

119860119894120574119894

119889119894(120582119894+ 120574119894+ 119889119894)+ 1205981= 1198810

119894+ 1205981 (20)

Also from the equations of system (2) we have

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

le 119860119894+ 120582119894(1198810

119894+ 1205981) minus (119889

119894+ 120574119894) 119878119900

(21)

Then

lim119905rarrinfin

sup 119878119894=

119860119894+ 120582119894(1198810

119894+ 1205981)

119889119894+ 120574119894

=1198780

119894+1205982 (120598

2=

1205821198941205981

119889119894+ 120574119894

)

(22)

Discrete Dynamics in Nature and Society 5

From system (9) and 119878119894le 1198780

119894+ 1205982with all 119905 gt 119905

119894 Thus

when 119905 gt 119905119894 we derive

119889119868119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(23)

Consider the following auxiliary system

1198891198681015840

119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(24)

Let 1198720be the matrix defined by

1198720=

((((

(

0 12057311

0 12057312

sdot sdot sdot 0 1205731119899

0 0 0 0 sdot sdot sdot 0 0

0 12057321

0 12057322

sdot sdot sdot 0 1205732119899

0 0 0 0 sdot sdot sdot 0 0

0 1205731198991

0 1205731198992

sdot sdot sdot 0 120573119899119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

(25)

and set 1198721

= 119872 + 12059821198720 It follows from Theorem 2 in

[21] that R0

lt 1 if and only if 119904(119872) lt 0 Thus thereexists an 120598

2gt 0 small enough such that 119904(119872

1) lt 0 Using

the Perron-Frobenius theorem all eigenvalues of the mat-rix 119872

1have negative real parts when 119904(119872

1) lt 0 Therefore

it has

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(26)

which implies that the zero solution of system (24) is globallyasymptotically stable Using the comparison principle ofSmith and Waltman [23] we know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(27)

By the theory of asymptotic autonomous system of Thieme[24] it is also known that

(1198781 (119905) 1198811 (119905) 119878119899 (119905) 119881119899 (119905))

997888rarr (1198781(0) 119881

1(0) 119878

119899(0) 119881

119899(0)) 119905 997888rarr infin

(28)

So 1198750is globally attractive when R

0lt 1 It follows that the

disease-free equilibrium 1198750of (2) is globally asymptotically

stable when R0lt 1 This completes the proof

23 The Uniform Persistence and Unique Positive Solution ofSystem (2) In this section we give the proof of the uniformpersistence and the unique positive solution of system (2)Define

1198830= (119878119894 119868119894 119881119894119882119894) isin 119883 | 119868

119894119882119894gt 0 119894 = 1 2 119899

1205971198830= 119883 | 119883

0

(29)

Theorem 2 When R0

gt 1 there exists a positive constant1205761such that when 119868

119894(0) lt 120576

1 119882119894(0) lt 120576

1for (119878

119894(0)

119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0

lim sup119905rarrinfin

max 119868119894(119905) 119882

119894(119905) gt 120576

1 119894 = 1 2 119899 (30)

Proof Consider the following system

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(31)

Using Corollary 32 in Zhao and Jing [25] it then fol-lows that system (31) has a unique positive equilibrium(1198780

1 1198810

1 119878

0

119899 1198810

119899) which is globally asymptotically stable

As to R0gt 1 hArr 119904(119872) gt 0 choose 120576 gt 0 small enough

such that 119904(1198722) gt 0 where 119872

2= 119872 minus 120576119872

0 Let us consider

a perturbed system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(32)

From our previous analysis of system (32) we canrestrict 120576

1gt 0 small enough such that (32) admits a unique

positive equilibrium (1198780

119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) which is

globally asymptotically stable 1198780119894(1205761) is continuous in 120576

1 so

we can further restrict 1205761small enough such that 1198780

119894(1205761) gt

1198780

119894minus 120576 119894 = 1 2 119899For the sake of contradiction ofTheorem 2 there is a 119879 gt

0 such that 119868119894(119905) lt 120576

1119882119894(119905) lt 120576

1 119894 = 1 2 119899 for all 119905 ge 119879

Then for 119905 ge 119879 we have

119889119878119894

119889119905ge 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119877119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119877119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119877119894

119894 = 1 2 119899

(33)

Since the equilibrium (1198780119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) of

(32) is globally asymptotically stable and 1198780

119894(1205761) gt 119878

0

119894minus 120576

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Thus the next generation matrix is 119865119881minus1

119865119881minus1

=(((

(

11986011

sdot sdot sdot 1198601119899

11986111

sdot sdot sdot 1198611119899

1198601198991

sdot sdot sdot 119860119899119899

1198611198991

sdot sdot sdot 119861119899119899

0 sdot sdot sdot 0 0 sdot sdot sdot 0

0 sdot sdot sdot 0 0 sdot sdot sdot 0

)))

)2119899times2119899

119860 = (

11986011

11986012

sdot sdot sdot 1198601119899

11986021

11986022

sdot sdot sdot 1198602119899

1198601198991

1198601198992

sdot sdot sdot 119860119899119899

)

119899times119899

(12)

So we can calculate the basic reproduction number of system(2)

R0= 120588 (119865119881

minus1) = 120588 (119860) (13)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894)

119894 = 1 2 119899

(14)

and 120588 denotes the spectral radius As we will show R0is the

key threshold parameters whose values completely character-ize the global dynamics of system (2)

22 Global Stability of the Disease-Free Equilibrium of System(2) For the disease-free equilibrium 119875

0of system (2) we

have the following property

Theorem 1 If R0

lt 1 the disease-free equilibrium 1198750of

system (2) is globally asymptotically stable in the region 119883

Proof Let 119872 = 119865minus119881 and define 119904(119872) = maxRe 120582 120582 is aneigenvalue of 119872 so 119904(119872) is a simple eigenvalue of 119872 witha positive eigenvector [23] By Theorem 2 in [21] there holdtwo equivalences

R0gt 1 lArrrArr 119904 (119872) gt 0 R

0lt 1 lArrrArr 119904 (119872) lt 0 (15)

To prove the locally stability of disease-free equilibriumwe check the hypotheses (A1)ndash(A5) in [21] Hypotheses (A1)ndash(A4) are easily verified while (A5) is satisfied if all eigenvaluesof the 4119899 times 4119899 matrix

119869|1198750= (

119872 0

1198693

1198694

)

4119899times4119899

(16)

have negative real parts where 1198693= minus119865

1198694=

((((

(

minus(1198891+ 1205741) 120582

10 0 sdot sdot sdot 0 0

1205741

minus (1198891+ 1205821) 0 0 sdot sdot sdot 0 0

0 0 minus (1198892+ 1205742) 120582

2sdot sdot sdot 0 0

0 0 1205742

minus (1198892+ 1205822) sdot sdot sdot 0 0

0 0 0 0 sdot sdot sdot minus (119889119899+ 120574119899) 120582

119899

0 0 0 0 sdot sdot sdot 120574119899

minus(119889119899+ 120582119899)

))))

)2119899times2119899

(17)

Calculate the eigenvalues of 1198694

119904 (1198694) = max minus119889

1 minus119889

119899 minus (1198891+ 1205821+ 1205741)

minus (119889119899+ 120582119899+ 120574119899) lt 0

(18)

If R0lt 1 then 119904(119872) lt 0 and 119904(119869|

1198750) lt 0 and the disease-

free equilibrium 1198750of (2) is locally asymptotically stable

Nowwewill prove that the disease-free equilibrium 1198750of

system (2) is globally attractive when R0lt 1 From the third

equation of system (2) we have

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

= 120574119894(119873119894minus (119868119894+ 119881119894)) minus (120582

119894+ 119889119894) 119881119894

le 120574119894

119860119894

119889119894

minus (120582119894+ 120574119894+ 119889119894) 119881119894

(19)

So we can have that for a small enough positive number 1205981

there exists 119905119894gt 0 119894 = 1 2 119899 such that for all 119905 gt 119905

119894

119881119894le

119860119894120574119894

119889119894(120582119894+ 120574119894+ 119889119894)+ 1205981= 1198810

119894+ 1205981 (20)

Also from the equations of system (2) we have

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

le 119860119894+ 120582119894(1198810

119894+ 1205981) minus (119889

119894+ 120574119894) 119878119900

(21)

Then

lim119905rarrinfin

sup 119878119894=

119860119894+ 120582119894(1198810

119894+ 1205981)

119889119894+ 120574119894

=1198780

119894+1205982 (120598

2=

1205821198941205981

119889119894+ 120574119894

)

(22)

Discrete Dynamics in Nature and Society 5

From system (9) and 119878119894le 1198780

119894+ 1205982with all 119905 gt 119905

119894 Thus

when 119905 gt 119905119894 we derive

119889119868119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(23)

Consider the following auxiliary system

1198891198681015840

119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(24)

Let 1198720be the matrix defined by

1198720=

((((

(

0 12057311

0 12057312

sdot sdot sdot 0 1205731119899

0 0 0 0 sdot sdot sdot 0 0

0 12057321

0 12057322

sdot sdot sdot 0 1205732119899

0 0 0 0 sdot sdot sdot 0 0

0 1205731198991

0 1205731198992

sdot sdot sdot 0 120573119899119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

(25)

and set 1198721

= 119872 + 12059821198720 It follows from Theorem 2 in

[21] that R0

lt 1 if and only if 119904(119872) lt 0 Thus thereexists an 120598

2gt 0 small enough such that 119904(119872

1) lt 0 Using

the Perron-Frobenius theorem all eigenvalues of the mat-rix 119872

1have negative real parts when 119904(119872

1) lt 0 Therefore

it has

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(26)

which implies that the zero solution of system (24) is globallyasymptotically stable Using the comparison principle ofSmith and Waltman [23] we know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(27)

By the theory of asymptotic autonomous system of Thieme[24] it is also known that

(1198781 (119905) 1198811 (119905) 119878119899 (119905) 119881119899 (119905))

997888rarr (1198781(0) 119881

1(0) 119878

119899(0) 119881

119899(0)) 119905 997888rarr infin

(28)

So 1198750is globally attractive when R

0lt 1 It follows that the

disease-free equilibrium 1198750of (2) is globally asymptotically

stable when R0lt 1 This completes the proof

23 The Uniform Persistence and Unique Positive Solution ofSystem (2) In this section we give the proof of the uniformpersistence and the unique positive solution of system (2)Define

1198830= (119878119894 119868119894 119881119894119882119894) isin 119883 | 119868

119894119882119894gt 0 119894 = 1 2 119899

1205971198830= 119883 | 119883

0

(29)

Theorem 2 When R0

gt 1 there exists a positive constant1205761such that when 119868

119894(0) lt 120576

1 119882119894(0) lt 120576

1for (119878

119894(0)

119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0

lim sup119905rarrinfin

max 119868119894(119905) 119882

119894(119905) gt 120576

1 119894 = 1 2 119899 (30)

Proof Consider the following system

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(31)

Using Corollary 32 in Zhao and Jing [25] it then fol-lows that system (31) has a unique positive equilibrium(1198780

1 1198810

1 119878

0

119899 1198810

119899) which is globally asymptotically stable

As to R0gt 1 hArr 119904(119872) gt 0 choose 120576 gt 0 small enough

such that 119904(1198722) gt 0 where 119872

2= 119872 minus 120576119872

0 Let us consider

a perturbed system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(32)

From our previous analysis of system (32) we canrestrict 120576

1gt 0 small enough such that (32) admits a unique

positive equilibrium (1198780

119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) which is

globally asymptotically stable 1198780119894(1205761) is continuous in 120576

1 so

we can further restrict 1205761small enough such that 1198780

119894(1205761) gt

1198780

119894minus 120576 119894 = 1 2 119899For the sake of contradiction ofTheorem 2 there is a 119879 gt

0 such that 119868119894(119905) lt 120576

1119882119894(119905) lt 120576

1 119894 = 1 2 119899 for all 119905 ge 119879

Then for 119905 ge 119879 we have

119889119878119894

119889119905ge 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119877119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119877119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119877119894

119894 = 1 2 119899

(33)

Since the equilibrium (1198780119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) of

(32) is globally asymptotically stable and 1198780

119894(1205761) gt 119878

0

119894minus 120576

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

From system (9) and 119878119894le 1198780

119894+ 1205982with all 119905 gt 119905

119894 Thus

when 119905 gt 119905119894 we derive

119889119868119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(23)

Consider the following auxiliary system

1198891198681015840

119894

119889119905= (1198780

119894+ 1205982)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(24)

Let 1198720be the matrix defined by

1198720=

((((

(

0 12057311

0 12057312

sdot sdot sdot 0 1205731119899

0 0 0 0 sdot sdot sdot 0 0

0 12057321

0 12057322

sdot sdot sdot 0 1205732119899

0 0 0 0 sdot sdot sdot 0 0

0 1205731198991

0 1205731198992

sdot sdot sdot 0 120573119899119899

0 0 0 0 sdot sdot sdot 0 0

))))

)2119899times2119899

(25)

and set 1198721

= 119872 + 12059821198720 It follows from Theorem 2 in

[21] that R0

lt 1 if and only if 119904(119872) lt 0 Thus thereexists an 120598

2gt 0 small enough such that 119904(119872

1) lt 0 Using

the Perron-Frobenius theorem all eigenvalues of the mat-rix 119872

1have negative real parts when 119904(119872

1) lt 0 Therefore

it has

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(26)

which implies that the zero solution of system (24) is globallyasymptotically stable Using the comparison principle ofSmith and Waltman [23] we know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (0 0 0 0 0 0) 119905 997888rarr infin

(27)

By the theory of asymptotic autonomous system of Thieme[24] it is also known that

(1198781 (119905) 1198811 (119905) 119878119899 (119905) 119881119899 (119905))

997888rarr (1198781(0) 119881

1(0) 119878

119899(0) 119881

119899(0)) 119905 997888rarr infin

(28)

So 1198750is globally attractive when R

0lt 1 It follows that the

disease-free equilibrium 1198750of (2) is globally asymptotically

stable when R0lt 1 This completes the proof

23 The Uniform Persistence and Unique Positive Solution ofSystem (2) In this section we give the proof of the uniformpersistence and the unique positive solution of system (2)Define

1198830= (119878119894 119868119894 119881119894119882119894) isin 119883 | 119868

119894119882119894gt 0 119894 = 1 2 119899

1205971198830= 119883 | 119883

0

(29)

Theorem 2 When R0

gt 1 there exists a positive constant1205761such that when 119868

119894(0) lt 120576

1 119882119894(0) lt 120576

1for (119878

119894(0)

119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0

lim sup119905rarrinfin

max 119868119894(119905) 119882

119894(119905) gt 120576

1 119894 = 1 2 119899 (30)

Proof Consider the following system

119889119878119894

119889119905= 119860119894+ 120582119894119881119894minus (120574119894+ 119889119894) 119878119894

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(31)

Using Corollary 32 in Zhao and Jing [25] it then fol-lows that system (31) has a unique positive equilibrium(1198780

1 1198810

1 119878

0

119899 1198810

119899) which is globally asymptotically stable

As to R0gt 1 hArr 119904(119872) gt 0 choose 120576 gt 0 small enough

such that 119904(1198722) gt 0 where 119872

2= 119872 minus 120576119872

0 Let us consider

a perturbed system

119889119878119894

119889119905= 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119881119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

119894 = 1 2 119899

(32)

From our previous analysis of system (32) we canrestrict 120576

1gt 0 small enough such that (32) admits a unique

positive equilibrium (1198780

119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) which is

globally asymptotically stable 1198780119894(1205761) is continuous in 120576

1 so

we can further restrict 1205761small enough such that 1198780

119894(1205761) gt

1198780

119894minus 120576 119894 = 1 2 119899For the sake of contradiction ofTheorem 2 there is a 119879 gt

0 such that 119868119894(119905) lt 120576

1119882119894(119905) lt 120576

1 119894 = 1 2 119899 for all 119905 ge 119879

Then for 119905 ge 119879 we have

119889119878119894

119889119905ge 119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119877119894minus 1205761119878119894

119899

sum

119895=1

120573119894119895

119889119877119894

119889119905= 120574119894119878119894minus (120582119894+ 119889119894) 119877119894

119894 = 1 2 119899

(33)

Since the equilibrium (1198780119894(1205761) 1198810

119894(1205761) 119894 = 1 2 119899) of

(32) is globally asymptotically stable and 1198780

119894(1205761) gt 119878

0

119894minus 120576

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

6 Discrete Dynamics in Nature and Society

119894 = 1 2 119899 There exists a 1198791gt 119879 gt 0 such that 119878

119894(119905) gt

1198780

119894minus 120576 119894 = 1 2 119899 for 119905 gt 119879

1 As a consequence for 119905 gt 119879

1

there holds119889119868119894

119889119905ge (1198780

119894minus 120576)

119899

sum

119895=1

120573119894119895119882119895minus (119889119894+ 119903119894) 119868119894

119889119882119894

119889119905= 119896119894119868119894minus 120575119894119882119894

119894 = 1 2 119899

(34)

Consider the following system

1198891198681015840

119894

119889119905= (1198780

119894minus 120576)

119899

sum

119895=1

1205731198941198951198821015840

119895minus (119889119894+ 119903119894) 1198681015840

119894

1198891198821015840

119894

119889119905= 1198961198941198681015840

119894minus 1205751198941198821015840

119894

119894 = 1 2 119899

(35)

Since the matrix 1198722has positive eigenvalue 119904(119872

2) with a

positive eigenvector It is easy to see that

(1198681015840

1(119905) 119882

1015840

1(119905) 1198681015840

2(119905) 119882

1015840

2(119905) 119868

1015840

119899(119905) 119882

1015840

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(36)

Using the comparison principle of Smith and Waltman [23]we also know that

(1198681(119905) 119882

1(119905) 1198682(119905) 119882

2(119905) 119868

119899(119905) 119882

119899(119905))

997888rarr (infininfininfininfin infininfin) 119905 997888rarr infin

(37)

which leads to a contradiction therefore we claim thatlim sup119905rarrinfin

max 119868119894 (119905) 119882119894 (119905) gt 120576

1 119894 = 1 2 119899 (38)

This completes the proof

We also have the following result of system (2)

Theorem 3 If R0

gt 1 then system (2) admits at least onepositive equilibrium and there is a positive constant 120576 suchthat every solution (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905)) of the system (2)

with (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0)) isin 119883

0satisfies

min lim inf119905rarrinfin

119868119894(119905) lim inf119905rarrinfin

119882119894(119905) ge 120576 119894 = 1 2 119899

(39)which implies that system (2) is uniformly persistent

Proof Now we prove that system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the form of (2) it is easy to

see that both 119883 and 1198830are positively invariant and 120597119883

0is

relatively closed in 119883 Furthermore system (2) is pointdissipative Let119872120597

= (119878119894 (0) 119868119894 (0) 119881119894 (0) 119882119894 (0)) | (119878119894 (119905) 119868119894 (119905) 119881119894 (119905) 119882119894 (119905))

isin 1205971198830 forall119905 ge 0 119894 = 1 2 119899

(40)

It is easy to show that

119872120597= (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881119894(119905) ge 0 119894 = 1 2 119899

(41)

Noting that (119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 =

1 2 119899 sube 119872120597 We only need to prove 119872

120597sube

(119878119894(119905) 0 119881

119894(119905) 0) | 119878

119894(119905) 119881

119894(119905) ge 0 119894 = 1 2 119899

Assume (119878119894(0) 119868119894(0) 119881

119894(0) 119882

119894(0) 119894 = 1 2 119899) isin 119872

120597 It

suffices to show that 119868119894(119905) = 0 119882

119894(119905) = 0 for all 119905 ge 0

119894 = 1 2 119899 Suppose not Then there exist an 1198940 1 le 119894

0le

119899 and 1199050

ge 0 such that 1198681198940(1199050) gt 0 119882

1198940(1199050) gt 0 and we

partition 1 2 119899 into two sets 1198761and 119876

2such that

(119868119894(1199050) 119882119894(1199050))119879= 0 forall119894 isin 119876

1

(119868119894(1199050) 119882119894(1199050))119879gt 0 forall119894 isin 119876

2

(42)

1198761is nonempty due to the definition of 119872

120597 1198762is non-

empty since 1198681198940(1199050) gt 0119882

1198940(1199050) gt 0 For any 119894 isin 119876

2and we

have that

119889119882119894(1199050)

1198891199050

= 119896119894119868119894(1199050) minus 120575119894119882119894(1199050) gt 119896119894119868119894(1199050) 119894 isin 119876

2 (43)

It follows that there is an 120578 gt 0 such that 119868119894(119905) gt 0 for 119905

0lt

119905 lt 1199050+120578 119894 isin 119876

2 Thismeans that (119878

119894(119905) 119868119894(119905) 119881119894(119905)119882

119894(119905) 119894 =

1 2 119899) does not belong to 1205971198830for 1199050lt 119905 lt 119905

0+ 120578 which

contradicts the assumption that (119878119894(0) 119868119894(0) 119881119894(0)119882

119894(0) 119894 =

1 2 119899) isin 119872120597 This proves the system (41)

1198750is globally asymptotically stable for system (2) It is

clear that there is only an equilibriaum1198750in119872120597and by afore-

mentioned claim it then follows that 1198750is isolated invariant

set in119883119882119904(1198750)cap1198830= 0 Clearly every orbit in119872

120597converges

to 1198750 1198750is acyclic in 119872

120597 Using Theorem 46 in Thieme

[26] we conclude that the system (2) is uniformly persistentwith respect to (119883

0 1205971198830) By the result of [27 28] system

(2) has an equilibrium (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830 We further claim that 119878

lowast

119894 119881lowast

119894gt 0 119894 = 1 2 119899

Suppose that 119878lowast

119894= 119881lowast

119894= 0 119894 = 1 2 119899 from of

(2) we can get 119868lowast

119894= 119882

lowast

119894= 0 119894 = 1 2 119899 It is

a contradiction Then (119878lowast

1 119868lowast

1 119881lowast

1119882lowast

1 119878

lowast

119899 119868lowast

119899 119881lowast

119899119882lowast

119899) isin

1198830is a componentwise positive equilibrium of system (2)

This completes the proof

The following theorem shows that there exists a uniquepositive solution for system (2) whenR

0gt 1

Theorem4 If R0gt 1 then there only exists a unique positive

equilibrium 119875lowast for system (2)

Proof Consider the following system

119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895= 0

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894= 0

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

Discrete Dynamics in Nature and Society 7

120574119894119878119894minus (120582119894+ 119889119894) 119881119894= 0

119896119894119868119894minus 120575119894119882119894= 0

119894 = 1 2 119899

(44)

We have that

119878119894=

119889119894+ 120582119894

119889119894(119889119894+ 120582119894+ 120574119894)(119860119894minus (119889119894+ 119903119894) 119868119894)

119882119894=

119896119894119868119894

120575119894

119881119894=

120574119894119878119894

119889119894+ 120582119894

119894 = 1 2 119899

(45)

Hence the equilibrium of system (2) is equal to thefollowing system

119861119894(119860119894minus 119899119894119868119894)

119899

sum

119895=1

120573119894119895119868119895minus 119899119894119868119894= 0 119894 = 1 2 119899 (46)

where

119861119894=

119896119894(119889119894+ 120582119894)

119889119894120575119894(119889119894+ 120582119894+ 120574119894) 119899119894= 119889119894+ 119903119894 119894 = 1 2 119899

(47)

Therefore we only need to prove that (46) has a uniquepositive equilibrium when R

0gt 1 Use the method in

[12] to demonstrate the unique positive equilibrium of (46)First we prove that 119868

lowast

119894= ℎ 119894 = 1 2 119899 is the only

positive solution of (46) Assume that 119868lowast

119894= ℎ and 119868

lowast

119894=

119896 are two positive solutions of (46) both nonzero If ℎ = 119896then ℎ

119894= 119896119894for some 119894 (119894 = 1 2 119899) Assume without

loss of generality that ℎ1

gt 1198961and moreover that ℎ

11198961

ge

ℎ119894119896119894for all 119894 (119894 = 1 2 119899) Since ℎ and 119896 are positive

solutions of (46) we substitute them into (46) We obtain

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895minus 1198991ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

(48)

so

0 = 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

minus 11989911198961

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895minus 11989911198961

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

= 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(49)

But (ℎ119894ℎ1)1198961le 119896119894and 119861

1(1198601minus 1198991ℎ1) lt 1198611(1198601minus 11989911198961) thus

from the above equalities we get

1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895ℎ119895

1198961

ℎ1

le 1198611(1198601minus 1198991ℎ1)

119899

sum

119895=1

1205731119895119896119895

lt 1198611(1198601minus 11989911198961)

119899

sum

119895=1

1205731119895119896119895

(50)

This is a contradiction so there is only one positivesolution 119868

lowast

119894= ℎ 119894 = 1 2 119899 of (46) So when R

0gt 1

there only exists a unique positive equilibrium for system(2)

24 Global Stability of the Unique Endemic Solution of System(2) In this section we prove that the unique endemicequilibrium of system (2) is globally asymptotically stablein 1198830 In order to prove global stability of the endemic

equilibrium the Lyapunov function will be used In thefollowing we also use a Lyapunov function to prove globalstability of the endemic equilibrium

Theorem 5 If R0gt 1 the unique positive equilibrium 119875

lowast ofsystem (2) is globally asymptotically stable in 119883

0

Proof Following [15] we define

120585119894119895= 120573119894119895119878lowast

119894119882lowast

119895 1 le 119894 119895 le 119899 119899 ge 2 (51)

B =

(((((((

(

119899

sum

119895 = 1

1205851119895

minus12058521

sdot sdot sdot minus1205851198991

minus12058512

119899

sum

119895 = 2

1205852119895

sdot sdot sdot minus1205851198992

d

minus1205851119899

minus1205852119899

sdot sdot sdot

119899

sum

119895 = 119899

120585119899119895

)))))))

)119899times119899

(52)

which is a Laplacian matrix whose column sums are zero andwhich is irreducible Therefore it follows from Lemma 21 of[15] that the solution space of linear system

B120577 = 0 (53)

has dimension 1 with a basis

120577 = (1205771 1205772 120577

119899)119879= (1198881 1198882 119888

119899)119879 (54)

where 119888119894denotes the cofactor of the 119894th diagonal entry of B

Note that from (53) we have that

119899

sum

119895=1

120577119894120585119894119895=

119899

sum

119895=1

120577119895120585119895119894 119894 = 1 2 119899 (55)

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

8 Discrete Dynamics in Nature and Society

For such 120577 = (1205771 1205772 120577

119899) we define a Lyapunov func-

tion

119871 (S IVW)

=

119899

sum

119894=1

120577119894(119878119894minus 119878lowast

119894minus 119878lowast

119894ln

119878lowast

119894

119878119894

+ 119868119894minus 119868lowast

119894minus 119868lowast

119894ln

119868lowast

119894

119868119894

+ 119881119894minus 119881lowast

119894minus 119881lowast

119894ln

119881lowast

119894

119881119894

+119889119894+ 119903119894

119896119894

(119882119894minus 119882lowast

119894minus 119882lowast

119894ln

119882lowast

119894

119882119894

))

(56)

where S = (1198781 1198782 119878

119899) I = (119868

1 1198682 119868

119899) V =

(1198811 1198812 119881

119899) and W = (119882

11198822 119882

119899) It is easy to

see that 119871(S IVW) ge 0 for all (S IVW) ge 0 and theequality 119871(S IVW) = 0 holds if and only if (S IVW) =

(Slowast IlowastVlowastWlowast) The derivative along the trajectories ofsystem (2) is

1198711015840(S IVW)

=

119899

sum

119894=1

120577119894(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895

minus119878lowast

119894

119878119894

(119860119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894minus

119899

sum

119895=1

120573119894119895119878119894119882119895)

+

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894

minus119868lowast

119894

119868119894

(

119899

sum

119895=1

120573119894119895119878119894119882119895minus (119889119894+ 119903119894) 119868119894) + 120574

119894119878119894

minus (120582119894+ 119889119894) 119881119894minus

119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894)

+119889119894+ 119903119894

119896119894

(119896119894119868119894minus 120575119894119882119894minus

119882lowast

119894

119882119894

(119896119894119868119894minus 120575119894119882119894)))

= 1198711+ 1198712+ 1198713

(57)

From system (44) we have

119860119894= (119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894+

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895 (58)

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895= (119889119894+ 119903119894) 119868lowast

119894=

120575119894(119889119894+ 119903119894)119882lowast

119894

119896119894

(59)

So

1198711=

119899

sum

119894=1

120577119894(

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895minus

120575119894(119889119894+ 119903119894)119882119894

119896119894

)

1198712=

119899

sum

119894=1

120577119894((119889119894+ 120574119894) 119878lowast

119894minus 120582119894119881lowast

119894minus (119889119894+ 120574119894) 119878119894+ 120582119894119881119894

+119878lowast

119894

119878119894

((119889119894+120574119894) 119878lowast

119894minus120582119894119881lowastminus(119889119894+120574119894) 119878119894+120582119894119881119894)

+ 120574119894119878119894minus (120582119894+ 119889119894) 119881119894

+119881lowast

119894

119881119894

(120574119894119878119894minus (120582119894+ 119889119894) 119881119894))

=

119899

sum

119894=1

120577119894(119889119894119878lowast

119894(2 minus

119878119894

119878lowast

119894

minus119878lowast

119894

119878i)

+ 120582119894119881lowast

119894(2 minus

119878119894119881lowast

119894

119878lowast

119894119881119894

minus119878lowast

119894119881119894

119878119894119881lowast

119894

)

+119889119894119881lowast

119894(3 minus

119881119894

119881lowast

119894

minus119878lowast

119894

119878119894

minus119878119894119881lowast

119894

119878lowast

119894119881119894

)) le 0

1198713=

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

(60)

Now we claim that

119899

sum

119894=1

120577119894

119899

sum

119895=1

120573119894119895119878lowast

119894119882119895=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(61)

Appealing to (51) (55) and (59)

119899

sum

119894=1

119899

sum

119895=1

120577119894120573119894119895119878lowast

119894119882119895

=

119899

sum

119894=1

119899

sum

119895=1

120577119895120573119895119894119878lowast

119895119882119894=

119899

sum

119894=1

119899

sum

119895=1

119882119894

119882lowast

119894

120577119895120573119895119894119878lowast

119895119882lowast

119894

=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119895120585119895119894=

119899

sum

119894=1

119882119894

119882lowast

119894

119899

sum

119895=1

120577119894120585119894119895

=

119899

sum

119894=1

120577119894

120575119894(119889119894+ 119903119894)119882119894

119896119894

(62)

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

Discrete Dynamics in Nature and Society 9

From (61) we have

1198711015840(S IVW)

le

119899

sum

119894=1

120577119894(3

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895minus

119899

sum

119895=1

120573119894119895119878lowast

119894119882lowast

119895

119878lowast

119894

119878119894

minus

119899

sum

119895=1

120573119894119895119878119894119882119895

119868lowast

119894

119868119894

minus (119889119894+ 119903119894) 119868119894

119882lowast

119894

119882119894

)

=

119899

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119894119868lowast

119894

)

= 119867119899(1198781 11986811198821 119878

119899 119868119899119882119899)

(63)

Next we show that 119867119899

le 0 for all (1198781 11986811198821 119878

119899

119868119899119882119899) isin 119883

0by applying the graph-theoretic approach

developed in [29ndash31] As in [29] 119871 = 119866(119861) denotesthe directed graph associated with matrix B 119876 presents asubgraph of 119871 119862119876 denotes the unique elementary cycle of119876 119864(119862119876) presents the set of directed arcs in 119862119876 and 119897 =

119897(119876) denotes the number of arcs in 119862119876 Then 119867119899can be

rewritten as

119867119899= sum

119876

119867119899119876

(64)

where

119867119899119876

= prod

(119903119898)isin119864(119876)

120585119903119898

times (3119897 minus sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

))

(65)

For instance

1198671= 1198671(1198781 11986811198821)

= sum

119894=119895=1

120577112058511

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

) le 0

1198672= 1198672(1198781 11986811198821 1198782 11986821198822)

=

2

sum

119894119895=1

120577119894120585119894119895(3 minus

119878lowast

119894

119878119894

minus

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

minus119882lowast

119894119868119894

119882119895119868lowast

119894

)

= 1205851112058521

(3 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

)

+ 1205852212058512

(3 minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

)

+ 1205851212058521

(6 minus119878lowast

1

1198781

minus11987811198821119868lowast

1

119878lowast

1119882lowast

11198681

minus119882lowast

11198681

1198821119868lowast

1

minus119878lowast

2

1198782

minus11987821198822119868lowast

2

119878lowast

2119882lowast

21198682

minus119882lowast

21198682

1198822119868lowast

2

) le 0

(66)

Note that for each unicycle graph 119876 it is easy to see that

prod

(119894119895)isin119864(119862119876)

119878lowast

119894

119878119894

sdot

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

sdot119882lowast

119894119868119894

119882119894119868lowast

119894

= prod

(119894119895)isin119864(119862119876)

119882lowast

119894119882119895

119882119894119882lowast

119895

= 1 (67)

Therefore

sum

(119894119895)isin119864(119862119876)

(119878lowast

119894

119878119894

+

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

+119882lowast

119894119868119894

119882119894119868lowast

119894

) ge 3119897 (68)

and hence 119867119899119876

le 0 for each 119876 and 119867119899119876

= 0 if and only if

119878lowast

119894

119878119894

=

119878119894119882119895119868lowast

119894

119878lowast

119894119882lowast

119895119868119894

=119882lowast

119894119868119894

119882119894119868lowast

119894

(119894 119895) isin 119864 (119862119876) (69)

Thus

1198711015840(S IVW) le 119867

119899le 0 (70)

The equality 1198711015840(S IVW) = 0 holds if and only if 119878

119894=

119878lowast

119894 119868119894= 119868lowast

119894 119881119894= 119881lowast

119894 and 119882

119894= 119882lowast

119894for all 119894 = 1 2 119899

Therefore following from LaSallersquos Invariance Principle [32]the unique endemic equilibrium 119875

lowast of system (2) is globallyasymptotically stable This completes the proof

3 A Numerical Example

Consider the system (1) when 119894 = 2 one has the two-groupmodel as follows

1198891198781

119889119905= 1198601minus (1198891+ 1205741) 1198781+ 12058211198811minus (1205731111987811198821+ 1205731211987811198822)

1198891198681

119889119905= 1205731111987811198821+ 1205731211987811198822minus (1198891+ 1199031) 1198681

1198891198771

119889119905= 1199031119868119894minus 11988911198771

1198891198811

119889119905= 12057411198781minus (1205821+ 1198891) 1198811

1198891198821

119889119905= 11989611198681minus 12057511198821

1198891198782

119889119905= 1198602minus (1198892+ 1205742) 1198782+ 12058221198812minus (1205732111987821198821+ 1205732211987821198822)

1198891198682

119889119905= 1205732111987821198821+ 1205732211987821198822minus (1198892+ 1199032) 1198682

1198891198772

119889119905= 11990321198682minus 11988921198772

1198891198812

119889119905= 12057421198782minus (1205822+ 1198892) 1198812

1198891198822

119889119905= 11989621198682minus 12057521198822

(71)

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

10 Discrete Dynamics in Nature and Society

We can give the basic reproduction number of system(71) which is

R1015840

0=

11986011

+ 11986022

+ radic(11986011

minus 11986022)2+ 41198601211986021

2

(72)

where

119860119894119895=

1205731198941198951198961198951198780

119894

120575119895(119889119895+ 119903119895)

1198780

119894=

119860119894(120582119894+ 119889119894)

119889119894(120582119894+ 119889119894+ 120574119894) 119894 = 1 2

(73)

Taking 1198601= 150 119860

2= 220 119889

1= 01 119889

2= 01 120582

1= 04

1205822

= 06 1205821

= 05 1205822

= 05 1199031

= 1 1199032

= 1 1198961

= 101198962= 10 120575

1= 8 120575

2= 8 and using Matlab ODE solver we run

numerical simulations for two casesIf 12057311

= 000048 12057312

= 00004 12057321

= 00004 and 12057322

=

000045 we have R10158400asymp 09804 lt 1 Hence the disease-free

equilibrium of system (71) is globally asymptotically stable(see Figure 1(a)) If 120573

11= 00025 120573

12= 0001 120573

21= 0001

and 12057322

= 00020 we have R10158400

asymp 36594 gt 1 Hence theendemic equilibrium of system (71) is globally asymptoticallystable (see Figure 1(b))

4 Conclusion

Cholera epidemic has become a major health problem formany developing countries From good understanding ofthe transmission dynamics of cholera in many emergentepidemic regions the heterogeneous host population canbe divided into several homogeneous groups accordingto modes of transmission contact patterns or geographicdistributions Hence in this paper we proposed a multi-group cholera SIRVW epidemiological model In order todistinguish many multi-group models with direct transmis-sion from person to person we only considered this multi-group cholera model with indirect transmission from thebacteria of the aquatic environment to person Firstly thebasic reproduction numberR

0of this model is given Then

it is found that the model has two non-negative equilibriathe disease-free equilibrium and the endemic equilibriumThe disease-free equilibrium exists without any conditionwhereas the endemic equilibrium exists provided R

0gt 1

Finally through the analysis of the model it has been foundthat the global asymptotic behavior of multi-group SIRVWmodel is completely determined by the size of R

0 That is

the disease-free equilibrium is globally asymptotically stableifR0lt 1 while an endemic equilibrium exists uniquely and

is globally asymptotically stable ifR0gt 1 By running num-

erical simulations for the cases of two-groups model we cansee that the disease-free equilibrium of system (71) is globallystable when R1015840

0lt 1 and the unique endemic equilibrium of

system (71) is globally stable whenR10158400gt 1

Acknowledgments

This research is supported by the National Natural Sci-ence Foundation of China under Grants 11301490 1130149111331009 11171314 and 11147015 Natural Science Foundation

0 5 10 15 20 250

5

10

15

20

25

30

35

40

Time t

I1

andI2

I1

I2

(a)

Time t0 5 10 15 20 25

20

40

60

80

100

120

140

160

I1

andI2

I1

I2

(b)

Figure 1 (a) The disease dies out in both groups (b) The diseasepersists in both groups Initial conditions are 119878

1(0) = 280 119868

1(0) =

40 1198771(0) = 10 119881

1(0) = 130 119882

1(0) = 250 119878

2(0) = 260 119868

2(0) = 20

1198772(0) = 10 119881

2(0) = 130119882

2(0) = 300

of ShanrsquoXi Province Grant no 2012021002-1 the specializedresearch fund for the doctoral program of higher educationpreferential development no 20121420130001 China Post-doctoral Science Foundation under Grant no 2012M520814Shanghai Postdoctoral Science Foundation under Grants no13R21410100 and IDRC104519-010

References

[1] M A Jensen S M Faruque J J Mekalanos and B R LevinldquoModeling the role of bacteriophage in the control of choleraoutbreaksrdquo Proceedings of the National Academy of Sciences ofthe United States of America vol 103 no 12 pp 4652ndash46572006

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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

Discrete Dynamics in Nature and Society 11

[2] A K Misra and V Singh ldquoA delay mathematical model for thespread and control of water borne diseasesrdquo Journal of Theo-retical Biology vol 301 pp 49ndash56 2012

[3] C Torres Codeco ldquoEndemic and epidemic dynamics of cholerathe role of the aquatic reservoirrdquo BMC Infectious Diseases vol1 article 1 2001

[4] M Pascual M J Bouma and A P Dobson ldquoCholera and cli-mate revisiting the quantitative evidencerdquo Microbes and Infec-tion vol 4 no 2 pp 237ndash245 2002

[5] D M Hartley J G Morris Jr and D L Smith ldquoHyperinfec-tivity a critical element in the ability of V cholerae to causeepidemicsrdquo PLoS Medicine vol 3 no 1 pp 63ndash69 2006

[6] Z Mukandavire S Liao J Wang H Gaff D L Smith andJ G Morris Jr ldquoEstimating the reproductive numbers for the2008-2009 cholera outbreaks in Zimbabwerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 108 no 21 pp 8767ndash8772 2011

[7] Z Mukandavire D L Smith and J G Morris Jr ldquoCholerain Haiti reproductive numbers and vaccination coverage esti-matesrdquo Scientific Reports vol 3 article 997 2013

[8] W Z Huang K L Cooke and C Castillo-Chavez ldquoStabilityand bifurcation for a multiple-group model for the dynamics ofHIVAIDS transmissionrdquo SIAM Journal on Applied Mathemat-ics vol 52 no 3 pp 835ndash854 1992

[9] Z Feng and J X Velasco-Hernandez ldquoCompetitive exclusion ina vector-host model for the dengue feverrdquo Journal of Mathemat-ical Biology vol 35 no 5 pp 523ndash544 1997

[10] C Bowman A B Gumel P Van den Driessche J Wu andH Zhu ldquoA mathematical model for assessing control strategiesagainst West Nile virusrdquo Bulletin of Mathematical Biology vol67 pp 1107ndash1133 2005

[11] R Edwards S Kim and P van den Driessche ldquoA multigroupmodel for a heterosexually transmitted diseaserdquo MathematicalBiosciences vol 224 pp 87ndash94 2010

[12] A Lajmanovich and J A York ldquoA deterministic model for gon-orrhea in a nonhomogeneous populationrdquo Mathematical Bio-sciences vol 28 pp 221ndash236 1976

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

[14] 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 1985

[15] H Guo M Y Li and Z Shuai ldquoGlobal stability of the endemicequilibrium of multigroup SIR epidemic modelsrdquo CanadianApplied Mathematics Quarterly vol 14 pp 259ndash284 2006

[16] Z Yuan and LWang ldquoGlobal stability of epidemiological mod-els with groupmixing and nonlinear incidence ratesrdquoNonlinearAnalysis Real World Applications vol 11 no 2 pp 995ndash10042010

[17] R Sun and J Shi ldquoGlobal stability of multigroup epidemicmodel with group mixing and nonlinear incidence ratesrdquoApplied Mathematics and Computation vol 218 pp 280ndash2862011

[18] M Y Li Z Shuai and CWang ldquoGlobal stability of multi-groupepidemic models with distributed delaysrdquo Journal of Mathe-matical Analysis and Applications vol 361 pp 38ndash47 2010

[19] H Shu D Fan and JWei ldquoGlobal stability of multi-group SEIRepidemic models with distributed delays and nonlinear trans-missionrdquo Nonlinear Analysis Real World Applications vol 13no 4 pp 1581ndash1592 2012

[20] O Diekmann J A Heesterbeek and J A Metz ldquoOn the defi-nition and the computation of the basic reproduction ratio R0inmodels for infectious diseases in heterogeneous populationsrdquoJournal of Mathematical Biology vol 28 no 4 pp 365ndash3821990

[21] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[22] O Diekmann J A P Heesterbeek and M G Roberts ldquoTheconstruction of next-generation matrices for compartmentalepidemic modelsrdquo Journal of the Royal Society Interface vol 7no 47 pp 873ndash885 2010

[23] H L Smith and PWaltmanTheTheory of the Chemostat Cam-bridge University Press 1995

[24] HR Thieme ldquoConvergence results and a Poincare-Bendixsontrichotomy for asymptotically autonomous differential equa-tionsrdquo Journal of Mathematical Biology vol 30 pp 755ndash7631992

[25] X Q Zhao and Z J Jing ldquoGlobal asymptotic behavior in somecooperative systems of functional differential equationsrdquo Can-adian Applied Mathematics Quarterly vol 4 pp 421ndash444 1996

[26] H R Thieme ldquoPersistence under relaxed point-dissipativity(with application to an endemic model)rdquo Mathematical Bio-sciences vol 166 pp 407ndash435 1993

[27] X Q Zhao ldquoUniform persistence and periodic coexistencestates in infinitedimensional periodic semiflows with applica-tionsrdquoCanadianAppliedMathematics Quarterly vol 3 pp 473ndash495 1995

[28] W D Wang and X-Q Zhao ldquoAn epidemic model in a patchyenvironmentrdquoMathematical Biosciences vol 190 no 1 pp 97ndash112 2004

[29] 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

[30] J W Moon Counting Labelled Trees Canadian MathematicalCongress Montreal Canada 1970

[31] D E KnuthTheArt of Computer Programming vol 1 Addison-Wesley Reading Mass USA 1997

[32] J P Lasalle ldquoThe stability of dynamical systemsrdquo in Proceedingsof the Regional Conference Series in AppliedMathematics SIAMPhiladelphia Pa USA 1976

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