Hindawi Publishing CorporationBioMed Research InternationalVolume 2013 Article ID 150681 10 pageshttpdxdoiorg1011552013150681

Research ArticleTransmission Model of Hepatitis B Virus with theMigration Effect

Muhammad Altaf Khan1 Saeed Islam1 Muhammad Arif1 and Zahoor ul Haq2

1 Department of Mathematics Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan2Department of Management Sciences Abdul Wali Khan University Mardan Khyber Pakhtunkhwa Pakistan

Correspondence should be addressed to Muhammad Altaf Khan altafdirgmailcom

Received 13 February 2013 Revised 25 March 2013 Accepted 25 March 2013

Academic Editor Kanury Rao

Copyright copy 2013 Muhammad Altaf Khan et alThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Hepatitis B is a globally infectious diseaseMathematicalmodeling ofHBV transmission is an interesting research area In this paperwe present characteristics of HBV virus transmission in the form of a mathematical model We analyzed the effect of immigrantsin the model to study the effect of immigrants for the host population We added the following flow parameters ldquothe transmissionbetweenmigrated and exposed classrdquo and ldquothe transmission betweenmigrated and acute classrdquoWith these new features we obtaineda compartment model of six differential equations First we find the basic threshold quantity Ro and then find the local asymptoticstability of disease-free equilibrium and endemic equilibrium Furthermore we find the global stability of the disease-free andendemic equilibria Previous similar publications have not added the kind of information about the numerical results of the modelIn our case from numerical simulation a detailed discussion of the parameters and their numerical results is presented We claimthat with these assumptions and by adding the migrated class the model informs policy for governments to be aware of theimmigrants and subject them to tests about the disease status Immigrants for short visits and students should be subjected totests to reduce the number of immigrants with disease

1 Introduction

According to World Health Organization about 350 millionpeople are infected with the hepatitis B virus (HBV)1 andabout 170 million people are chronically infected with thehepatitis C virus (HCV)2 The majority of those infected livein developing countries with few incidences inWestern coun-tries Migration is one of the defining issues of our time Forexample more than 5 million Canadians migrate out of thecountry each year and over 250000 new immigrants arrivein Canada each year Countries of the world are increasinglyconnected through travel andmigration and thus migrationhas health implications in one location for both local andglobal migrations since infectious diseases do not remainisolated geographically The UK Hepatitis Foundation esti-mated in 2007 that the number of hepatitis B cases in the UKdoubled in the previous 6 years chiefly due to immigrationof infected people many from the new member states of theEuropean Union where the prevalence of viral hepatitis ishigher The United Arab Emirates mandated that hepatitis C

testing is to be done at the time of residence visa renewalsand added hepatitis C to the list of diseases such as HIV andhepatitis B as diseases warranting deportation [1] In Chinahepatitis B virus infection is a major public health problemHepatitis B is the first one among the diseases with legalmanagement measures In China an estimated 93 millionpeople have been infected with the hepatitis B virus [2] Theseroepidemiological survey on HBV infection conducted in2006 showed that HBsAg carrier rate was 718 percent in theoverall dynamics of HBV In this paper we consider a systemof ordinary differential equations which describes the trans-mission ofHBV transmission in China Severalmathematicalmodels have been formulated on the HBV transmission inChina Medley and coauthors used a mathematical modeland developed the strategies to eliminate the HBV in NewZealand in 2008 [3 4] Anderson and May [5] used a mathe-matical model which illustrated the effects of carriers on thetransmission of HBV An age structure model was pro-posed by Zhao et al [6] to predict the dynamics of HBVtransmission and evaluate the long-term effectiveness of the

2 BioMed Research International

vaccination program in China Wang et al [7] proposed andanalyzed the hepatitis B virus infection in a diffusion modelconfined to a finite domain A hepatitis B virus (HBV) modelwith spatial diffusion and saturation response of the infectionrate is investigated by Xu and Ma [8] Also Zou et al [9]proposed a mathematical model to understand the transmis-sion dynamics and prevalence of HBV in mainland ChinaA model to describe waning of immunity after sometime hasbeen studied by a number of authors [10ndash16] In the context ofrapid global migration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date This isespecially important given that the scope and speed ofmigration is expected to grow in coming years In the contextof rapid globalmigration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date Thisis especially important given that the scope and speed ofmigration is expected to grow in coming years

In this paper we construct the compartmental model ofhepatitis B transmission We have categorized the model intosix compartments Susceptible-119878(119905) Exposed-119864(119905) Acute-119860(119905) Carrier-119862(119905) Vaccinated-119881(119905) and Migrated-119872(119905)individuals The migrated class of individuals comes fromdifferent parts of the world to the host country and theirinteraction occurs in the form of sexual interactions bloodtransportation and transfusion We modify the model fromPang et al [10] by adding some new transmission dynamicsand introduce the migrated class in the model Furthermoresome authors [11 13 15 17] show that acute hepatitis B couldbe found today in newborns of infected mothers Pang et al[10] added the vertical transmission term to exposed classfrom chronic carriers class on the basis of the characteristicsof HBV transmission In this paper we improved the modelof [10] with these new features by adding the migrated class119872(119905) and the following parameters

(i) 1205831 the transmission rate from migrated class to

exposed class

(ii) 1205832 the transmission rate frommigrated to acute class

(iii) 120575 the death rate at the migrated class

The paper is organized as follows Section 2 is devoted tothe mathematical formation of the model In Section 3 wefind the Basic Reproduction Number the disease-free equi-librium and endemic equilibrium of the proposed modelLocal asymptotic stability of the disease-free equilibrium andendemic equilibrium is discussed in Section 4 In Section 5we study the global asymptotic stability of disease-freeand endemic equilibria using the Lyapunov function InSection 6 we study the numerical results of the proposedmodel and present the results in the form of plots forillustrations The conclusion and references are presented inSection 7

120575119864

120575119881

119881

120575119862

120575119878119878

119862

120575119872

119872120575119860

(1 minus 119902)1205741119860

119860

1199021205742119860

1205741119864

119864

1205743119862

1205831119872

1205832119872

120575120587120578119862

120573(119860 + 120581119862)119878

120575(1 minus 120587)

1205750119881119901119878

120575120587 minus 120575120587120578119862

Figure 1 The complete flow diagram of hepatitis B virus transmis-sion model

2 Model Formulation

In this section we present the mathematical formulation ofthe compartmental model of hepatitis B which consists of asystem of differential equations The model is based on thecharacteristics of HBV transmissionWe divide the total pop-ulation into six compartments that is Susceptible individuals119878(119905) exposed 119864(119905) Acute 119860(119905) Carrier 119862(119905) Immunity class119881(119905) andMigrated119872(119905)The flow diagram (Figure 1) and thesystem is given in the following

119889119878 (119905)

119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900119881 (119905) minus 119901119878 (119905)

119889119864 (119905)

119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus 1205741119864 (119905) + 120583

1119872(119905)

119889119860 (119905)

119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+ 1205832119872(119905)

119889119862 (119905)

119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)

119889119881 (119905)

119889119905= 1205743119862 (119905) + (1 minus 119902) 120574

1119860 (119905) minus 120575

119900119881 (119905) minus 120575119881 (119905)

+ 120575 (1 minus 120587) + 119901119878 (119905)

119889119872 (119905)

119889119905= minus (120583

1+ 1205832)119872 (119905) minus 120575119872 (119905)

(1)

BioMed Research International 3

Subject to the initial conditions

119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0

119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0

(2)

The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574

1rate the

exposed individuals become infectious andmove to theAcuteclass 120574

2is the rate at which the individuals move to the

carrier class 1205743is the flow of carrier to vaccinated class 120573

shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575

119900represent the loss of

immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed

class and 1205742is the rate of transmission from migrated class

to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again

We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become

119889119878 (119905)

119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)

119889119864 (119905)

119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus 1205741119864 (119905) + 120583

1119872(119905)

119889119860 (119905)

119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+ 1205832119872(119905)

119889119862 (119905)

119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)

119889119872 (119905)

119889119905= minus (120583

1+ 1205832)119872 (119905) minus 120575119872 (119905)

(3)

Let

Γ = (119878 119864 119860 119862119872) isin R5

+ | 119878 (119905) le

120575120587 + 120575119900

120575 + 120575119900+ 119901

(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575

119900

120575 + 120575119900

)

(4)

Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus

3 Basic ReproductionNumberThreshold Quantity

The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of

secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have

F = [

[

0 120573119878119900120573120581119878119900

0 0 0

0 0 0

]

]

119881 = [

[

120575 + 1205741

0 minus120575120587120578

minus1205741

120575 + 1199021205742+ (1 minus 119902) 120574

10

0 minus1199021205742

120575 + 1205743

]

]

119881minus1=

[

(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)

1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574

1

11990212057411205742

1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)

]

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902120574

11205742

(5)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and

119865119881minus1=

[[[[[

[

0120573119878119900120575120587120578119902120574

2

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2)

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

0 0 0

0 0 0

]]]]]

]

(6)

4 BioMed Research International

So the reproduction number given by 120588(119865119881minus1) is

R119900=120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2) + 120575120587120578119902119903

11199032

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) (7)

Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=

(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575

119900)(120575+120575

119900+119901) The endemic

equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system

(3) whose endemic equilibrium is given in the followingsubsection

Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878

lowast 119864 = 119864lowast 119860 = 119860

lowast 119862 = 119862lowast

and 119872 = 119872lowast equating left side of the system (3) equal to

zero we obtained

119878lowast=1198761(120575 + 120574

1) (120575 + 120574

3) (1 minus 119877

0) + 120581120573119878

1199001198761

1205741120573 ((120575 + 120574

3) + 120581119902120574

2)

minus(120575 + 120574

3) [(120575 + 120574

1) + 12058311205741]119872lowast

1205741120573 ((120575 + 120574

3) + 120581119902120574

2) 119860lowast

119864lowast=1198761119860lowastminus 1205832119872lowast

1205741

119862lowast=

1199021205742

(120575 + 1205743)119860lowast

(8)

4 Local Stability Analysis

In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863

119900= (119878119900 0 0 0 0)

in the following theorem

Theorem 1 For 1198770le 1 the disease-free equilibrium of the

system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is

locally asymptotically stable if ((1198761(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) otherwise the disease-free equilibrium of the

system (3) is unstable for 1198770gt 1

Proof To show the local stability of the system (3) about thepoint119863

119900 we set the left-hand side of the system (3) equating

to zero and we obtain the following Jacobian matrix 119869119900(120577)

119869119900(120577) = (

minus120575 minus 120575119900minus 119901 120575

119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878

119900minus 120575119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (9)

By the elementary row operation we get the followingmatrix

119869119900(120577) = (

minus(120575 + 120575119900+ 119901) minus120575

119900minus120573119878119900minus 120575119900

minus120575120587120578 minus 120573120581119878119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 0 minus1198761(120575 + 120574

1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574

1) 1205832+ 12058311205741

0 0 0 1198791

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (10)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and 119879

1= minus(minus119876

1(120575 + 120574

1) +

1205741120573119878119900) minus 119902120574

11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to

the previous Jacobian matrix is given by

(120582+120575 + 120575119900+119901) (120582 + 120583

1+1205832+ 120575) (120582

3+ 11988611205822+1198862120582+1198863) = 0

(11)

The first two eigenvalues minus(120575+120575119900+119901) and minus(120583

1+1205832+120575) have

negative real parts For the rest of the eigenvalues we get

1198861= (120575 + 120575

119900+ 119901) (119876

1(120575 + 120574

3) + 1 + 119876

1) (1 minus 119877

0)

+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574

3) + 119902120574

11205742) minus (119876

1120581 + 1205741))

times (120575 + 120575119900+ 119901)minus1

1198862= (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))

times (120575 + 120575119900+ 119901) [(120575 + 120574

1) + (120575 + 120574

1)2

]

+ (120575 + 1205741) [1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900)]

+ 11990212057411205742(120575 + 120574

1) [120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575 + 119901)]

times (120575 + 120575119900+ 119901)minus2

+ 11990212057411205742(120575 + 120574

1) (120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575

119900+ 119901))

times (120575 + 120575119900+ 119901)minus1

BioMed Research International 5

1198863=

(120575 + 1205741)

(120575 + 120575119900+ 119901)

(1198761(120575 + 120575

119900+ 119901) (120575 + 120574

1) minus 120573120574

1(120575120587 + 120575

119900))

times [11990212057411205742(120573120581 (120575120587 + 120575

119900) + 120575120587120578)

+ (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))]

(12)

By the Routh-Hurwitz criteria 1198861gt 0 119886

3gt 0 and 119886

11198862gt 1198863

Here 1198861gt 0 when 119877

0le 1 and ((119876

1(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) Also 119886

2gt 0 and 119886

3gt 0 and then 119886

11198862gt

1198863 So according to the Routh-Hurwitz criteria the Jacobian

matrix has negative real parts if and only if 1198770le 1 Thus by

Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863

119900= (119878119900 0 0 0 0) The

proof is completed

The stability of the disease-free equilibrium for 1198770le 1

means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877

0gt 1 When the disease-free

equilibrium is locally asymptotically stable for 1198770le 1 then

the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1

41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =

(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of

endemic equilibrium in the following

Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system

(3) is locally asymptotical stable if the following conditionshold

12057411198852120575119900119878lowast2

gt 12057411205732120581

1198661gt 1198662

(13)

otherwise the system is unstable

Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869

lowast(120577) of the system (3) in the following

119869lowast(120577) = (

minus(120575 + 120575119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)) 120575

119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878

lowastminus 120575119900

minus120575119900

120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574

1) 120573119878

lowast120573120581119878lowast+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (120575 + 1205831+ 1205832)

) (14)

By elementary row operation and after simplification we getthe following Jacobian matrix

119869lowast(120577) = (

minus1198851

minus120575119900

minus120573119878lowastminus 120575119900

1198853

minus120575119900

0 minus1198851(120575 + 120574

1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885

11198854+ 11988521198853

11988511205831minus 1198852120575119900

0 0 1198855

1205741(11988511198854+ 11988521198853) 119885

6

0 0 0 1198857

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (15)

where

1198851= 120575 + 120575

119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)

1198852= 120573 (119860

lowast+ 120581119862lowast)

1198853= minus (120575120587120578 + 120573120581119878

lowast+ 120575119900)

1198854= (120573120581119878

lowast+ 120575120587120578)

1198855= minus119876

1(minus1198851(120575 + 120574

1) minus 1198852120575119900)

+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))

1198856= minus1205832(minus1198851(120575 + 120574

1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)

1198857= minus1198855(120575 + 120574

3) minus 11988551205741(11988511198854+ 11988521198853)

(16)

The eigenvalue 1205821= minus(120583

1+ 1205832+ 120575) lt 0 120582

2= minus1198851 and using

the value of 1198851and further 119862lowast we get 120582

2= minus(120575 + 120575

119900+ 119901 +

120573((120575 + 1205743) + 120581119902120574

2)119860lowast) lt 0 120582

3= minus(119885

1(120575 + 120574

1) + 1198852120575119900) lt 0

as 1198851gt 0 and 119885

2gt 0 120582

4= 1198855 1205824lt 0 if and only if 119885

5lt 0

After the simplifications we get

120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2

)

+ 1205741120573120575120587120578120573

lowastlowastlowast119872lowast

minus [(1205741120573120575120587120578120573

lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574

1)

+11987611198852120575119900120573lowast] gt 0

(17)

where 120573lowast = 1205741120573((120575 + 120574

3) + 120581119902120574

2) 120573lowastlowast = 119876

1(120575 + 120574

1)(120575 +

1205743)(1minus119877

0)+120581120573119878

1199001198761 and 120574

1120573120575120587120578120573

lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573

lowastlowast+

120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866

1= 1205741120573120575120587120578120573

lowastlowastlowast119872lowast and

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Signal TransductionJournal of

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BioMed Research International

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International Journal of

Microbiology

2 BioMed Research International

vaccination program in China Wang et al [7] proposed andanalyzed the hepatitis B virus infection in a diffusion modelconfined to a finite domain A hepatitis B virus (HBV) modelwith spatial diffusion and saturation response of the infectionrate is investigated by Xu and Ma [8] Also Zou et al [9]proposed a mathematical model to understand the transmis-sion dynamics and prevalence of HBV in mainland ChinaA model to describe waning of immunity after sometime hasbeen studied by a number of authors [10ndash16] In the context ofrapid global migration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date This isespecially important given that the scope and speed ofmigration is expected to grow in coming years In the contextof rapid globalmigration there is a potential for any disease tobe transferred faster than was previously possible Theseimplications concerning the movement of HBV and HCVmerit far more attention by countries and the internationalcommunity than they have given the problem to date Thisis especially important given that the scope and speed ofmigration is expected to grow in coming years

In this paper we construct the compartmental model ofhepatitis B transmission We have categorized the model intosix compartments Susceptible-119878(119905) Exposed-119864(119905) Acute-119860(119905) Carrier-119862(119905) Vaccinated-119881(119905) and Migrated-119872(119905)individuals The migrated class of individuals comes fromdifferent parts of the world to the host country and theirinteraction occurs in the form of sexual interactions bloodtransportation and transfusion We modify the model fromPang et al [10] by adding some new transmission dynamicsand introduce the migrated class in the model Furthermoresome authors [11 13 15 17] show that acute hepatitis B couldbe found today in newborns of infected mothers Pang et al[10] added the vertical transmission term to exposed classfrom chronic carriers class on the basis of the characteristicsof HBV transmission In this paper we improved the modelof [10] with these new features by adding the migrated class119872(119905) and the following parameters

(i) 1205831 the transmission rate from migrated class to

exposed class

(ii) 1205832 the transmission rate frommigrated to acute class

(iii) 120575 the death rate at the migrated class

The paper is organized as follows Section 2 is devoted tothe mathematical formation of the model In Section 3 wefind the Basic Reproduction Number the disease-free equi-librium and endemic equilibrium of the proposed modelLocal asymptotic stability of the disease-free equilibrium andendemic equilibrium is discussed in Section 4 In Section 5we study the global asymptotic stability of disease-freeand endemic equilibria using the Lyapunov function InSection 6 we study the numerical results of the proposedmodel and present the results in the form of plots forillustrations The conclusion and references are presented inSection 7

120575119864

120575119881

119881

120575119862

120575119878119878

119862

120575119872

119872120575119860

(1 minus 119902)1205741119860

119860

1199021205742119860

1205741119864

119864

1205743119862

1205831119872

1205832119872

120575120587120578119862

120573(119860 + 120581119862)119878

120575(1 minus 120587)

1205750119881119901119878

120575120587 minus 120575120587120578119862

Figure 1 The complete flow diagram of hepatitis B virus transmis-sion model

2 Model Formulation

In this section we present the mathematical formulation ofthe compartmental model of hepatitis B which consists of asystem of differential equations The model is based on thecharacteristics of HBV transmissionWe divide the total pop-ulation into six compartments that is Susceptible individuals119878(119905) exposed 119864(119905) Acute 119860(119905) Carrier 119862(119905) Immunity class119881(119905) andMigrated119872(119905)The flow diagram (Figure 1) and thesystem is given in the following

119889119878 (119905)

119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900119881 (119905) minus 119901119878 (119905)

119889119864 (119905)

119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus 1205741119864 (119905) + 120583

1119872(119905)

119889119860 (119905)

119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+ 1205832119872(119905)

119889119862 (119905)

119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)

119889119881 (119905)

119889119905= 1205743119862 (119905) + (1 minus 119902) 120574

1119860 (119905) minus 120575

119900119881 (119905) minus 120575119881 (119905)

+ 120575 (1 minus 120587) + 119901119878 (119905)

119889119872 (119905)

119889119905= minus (120583

1+ 1205832)119872 (119905) minus 120575119872 (119905)

(1)

BioMed Research International 3

Subject to the initial conditions

119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0

119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0

(2)

The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574

1rate the

exposed individuals become infectious andmove to theAcuteclass 120574

2is the rate at which the individuals move to the

carrier class 1205743is the flow of carrier to vaccinated class 120573

shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575

119900represent the loss of

immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed

class and 1205742is the rate of transmission from migrated class

to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again

We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become

119889119878 (119905)

119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)

119889119864 (119905)

119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus 1205741119864 (119905) + 120583

1119872(119905)

119889119860 (119905)

119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+ 1205832119872(119905)

119889119862 (119905)

119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)

119889119872 (119905)

119889119905= minus (120583

1+ 1205832)119872 (119905) minus 120575119872 (119905)

(3)

Let

Γ = (119878 119864 119860 119862119872) isin R5

+ | 119878 (119905) le

120575120587 + 120575119900

120575 + 120575119900+ 119901

(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575

119900

120575 + 120575119900

)

(4)

Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus

3 Basic ReproductionNumberThreshold Quantity

The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of

secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have

F = [

[

0 120573119878119900120573120581119878119900

0 0 0

0 0 0

]

]

119881 = [

[

120575 + 1205741

0 minus120575120587120578

minus1205741

120575 + 1199021205742+ (1 minus 119902) 120574

10

0 minus1199021205742

120575 + 1205743

]

]

119881minus1=

[

(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)

1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574

1

11990212057411205742

1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)

]

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902120574

11205742

(5)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and

119865119881minus1=

[[[[[

[

0120573119878119900120575120587120578119902120574

2

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2)

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

0 0 0

0 0 0

]]]]]

]

(6)

4 BioMed Research International

So the reproduction number given by 120588(119865119881minus1) is

R119900=120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2) + 120575120587120578119902119903

11199032

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) (7)

Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=

(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575

119900)(120575+120575

119900+119901) The endemic

equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system

(3) whose endemic equilibrium is given in the followingsubsection

Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878

lowast 119864 = 119864lowast 119860 = 119860

lowast 119862 = 119862lowast

and 119872 = 119872lowast equating left side of the system (3) equal to

zero we obtained

119878lowast=1198761(120575 + 120574

1) (120575 + 120574

3) (1 minus 119877

0) + 120581120573119878

1199001198761

1205741120573 ((120575 + 120574

3) + 120581119902120574

2)

minus(120575 + 120574

3) [(120575 + 120574

1) + 12058311205741]119872lowast

1205741120573 ((120575 + 120574

3) + 120581119902120574

2) 119860lowast

119864lowast=1198761119860lowastminus 1205832119872lowast

1205741

119862lowast=

1199021205742

(120575 + 1205743)119860lowast

(8)

4 Local Stability Analysis

In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863

119900= (119878119900 0 0 0 0)

in the following theorem

Theorem 1 For 1198770le 1 the disease-free equilibrium of the

system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is

locally asymptotically stable if ((1198761(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) otherwise the disease-free equilibrium of the

system (3) is unstable for 1198770gt 1

Proof To show the local stability of the system (3) about thepoint119863

119900 we set the left-hand side of the system (3) equating

to zero and we obtain the following Jacobian matrix 119869119900(120577)

119869119900(120577) = (

minus120575 minus 120575119900minus 119901 120575

119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878

119900minus 120575119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (9)

By the elementary row operation we get the followingmatrix

119869119900(120577) = (

minus(120575 + 120575119900+ 119901) minus120575

119900minus120573119878119900minus 120575119900

minus120575120587120578 minus 120573120581119878119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 0 minus1198761(120575 + 120574

1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574

1) 1205832+ 12058311205741

0 0 0 1198791

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (10)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and 119879

1= minus(minus119876

1(120575 + 120574

1) +

1205741120573119878119900) minus 119902120574

11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to

the previous Jacobian matrix is given by

(120582+120575 + 120575119900+119901) (120582 + 120583

1+1205832+ 120575) (120582

3+ 11988611205822+1198862120582+1198863) = 0

(11)

The first two eigenvalues minus(120575+120575119900+119901) and minus(120583

1+1205832+120575) have

negative real parts For the rest of the eigenvalues we get

1198861= (120575 + 120575

119900+ 119901) (119876

1(120575 + 120574

3) + 1 + 119876

1) (1 minus 119877

0)

+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574

3) + 119902120574

11205742) minus (119876

1120581 + 1205741))

times (120575 + 120575119900+ 119901)minus1

1198862= (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))

times (120575 + 120575119900+ 119901) [(120575 + 120574

1) + (120575 + 120574

1)2

]

+ (120575 + 1205741) [1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900)]

+ 11990212057411205742(120575 + 120574

1) [120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575 + 119901)]

times (120575 + 120575119900+ 119901)minus2

+ 11990212057411205742(120575 + 120574

1) (120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575

119900+ 119901))

times (120575 + 120575119900+ 119901)minus1

BioMed Research International 5

1198863=

(120575 + 1205741)

(120575 + 120575119900+ 119901)

(1198761(120575 + 120575

119900+ 119901) (120575 + 120574

1) minus 120573120574

1(120575120587 + 120575

119900))

times [11990212057411205742(120573120581 (120575120587 + 120575

119900) + 120575120587120578)

+ (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))]

(12)

By the Routh-Hurwitz criteria 1198861gt 0 119886

3gt 0 and 119886

11198862gt 1198863

Here 1198861gt 0 when 119877

0le 1 and ((119876

1(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) Also 119886

2gt 0 and 119886

3gt 0 and then 119886

11198862gt

1198863 So according to the Routh-Hurwitz criteria the Jacobian

matrix has negative real parts if and only if 1198770le 1 Thus by

Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863

119900= (119878119900 0 0 0 0) The

proof is completed

The stability of the disease-free equilibrium for 1198770le 1

means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877

0gt 1 When the disease-free

equilibrium is locally asymptotically stable for 1198770le 1 then

the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1

41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =

(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of

endemic equilibrium in the following

Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system

(3) is locally asymptotical stable if the following conditionshold

12057411198852120575119900119878lowast2

gt 12057411205732120581

1198661gt 1198662

(13)

otherwise the system is unstable

Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869

lowast(120577) of the system (3) in the following

119869lowast(120577) = (

minus(120575 + 120575119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)) 120575

119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878

lowastminus 120575119900

minus120575119900

120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574

1) 120573119878

lowast120573120581119878lowast+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (120575 + 1205831+ 1205832)

) (14)

By elementary row operation and after simplification we getthe following Jacobian matrix

119869lowast(120577) = (

minus1198851

minus120575119900

minus120573119878lowastminus 120575119900

1198853

minus120575119900

0 minus1198851(120575 + 120574

1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885

11198854+ 11988521198853

11988511205831minus 1198852120575119900

0 0 1198855

1205741(11988511198854+ 11988521198853) 119885

6

0 0 0 1198857

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (15)

where

1198851= 120575 + 120575

119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)

1198852= 120573 (119860

lowast+ 120581119862lowast)

1198853= minus (120575120587120578 + 120573120581119878

lowast+ 120575119900)

1198854= (120573120581119878

lowast+ 120575120587120578)

1198855= minus119876

1(minus1198851(120575 + 120574

1) minus 1198852120575119900)

+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))

1198856= minus1205832(minus1198851(120575 + 120574

1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)

1198857= minus1198855(120575 + 120574

3) minus 11988551205741(11988511198854+ 11988521198853)

(16)

The eigenvalue 1205821= minus(120583

1+ 1205832+ 120575) lt 0 120582

2= minus1198851 and using

the value of 1198851and further 119862lowast we get 120582

2= minus(120575 + 120575

119900+ 119901 +

120573((120575 + 1205743) + 120581119902120574

2)119860lowast) lt 0 120582

3= minus(119885

1(120575 + 120574

1) + 1198852120575119900) lt 0

as 1198851gt 0 and 119885

2gt 0 120582

4= 1198855 1205824lt 0 if and only if 119885

5lt 0

After the simplifications we get

120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2

)

+ 1205741120573120575120587120578120573

lowastlowastlowast119872lowast

minus [(1205741120573120575120587120578120573

lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574

1)

+11987611198852120575119900120573lowast] gt 0

(17)

where 120573lowast = 1205741120573((120575 + 120574

3) + 120581119902120574

2) 120573lowastlowast = 119876

1(120575 + 120574

1)(120575 +

1205743)(1minus119877

0)+120581120573119878

1199001198761 and 120574

1120573120575120587120578120573

lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573

lowastlowast+

120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866

1= 1205741120573120575120587120578120573

lowastlowastlowast119872lowast and

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Microbiology

BioMed Research International 3

Subject to the initial conditions

119878 (0) ge 0 119864 (0) ge 0 119860 (0) ge 0

119862 (0) ge 0 119881 (0) ge 0 119872 (0) ge 0

(2)

The proportion of failure immunization is shown by 120587120575 represent both the death and birth rate At the 120574

1rate the

exposed individuals become infectious andmove to theAcuteclass 120574

2is the rate at which the individuals move to the

carrier class 1205743is the flow of carrier to vaccinated class 120573

shows the transmission coefficient 120581 represents the carrierinfectiousness to acute infection 119902 is the proportion of acuteindividuals that become carrier 120575

119900represent the loss of

immunity rate and the individual become the susceptibleagain 119901 represents the vaccination of susceptible individuals1205831represents the rate of flow from migrated class to exposed

class and 1205742is the rate of transmission from migrated class

to acute class 120578 is the unimmunized children born to carriermothers 120575(1 minus 120587) measures the successful immunization ofnewborn babies and the term 120575120587(1 minus 120578119862(119905)) shows that thenewborns are unimmunized and become susceptible again

We assume that the total population size is equal to 1and just for simplifications 119878(119905) is the susceptible 119864(119905) theexposed 119860(119905) the acute 119862(119905) the carrier 119881(119905) the immunityand119872(119905) is themigrated class representing the state variablesin our proposed population model The sum of the totalpopulation is 119878(119905) + 119864(119905) + 119860(119905) + 119862(119905) + 119881(119905) + 119872(119905) = 1holds We just add (1) and we can easily get We ignore thefifth equation in system (1) so the new models become

119889119878 (119905)

119889119905= 120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878(119905)+119864(119905)+119860 (119905)+119862(119905)+119872(119905)))minus119901119878(119905)

119889119864 (119905)

119889119905= 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus 1205741119864 (119905) + 120583

1119872(119905)

119889119860 (119905)

119889119905= 1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+ 1205832119872(119905)

119889119862 (119905)

119889119905= 1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)

119889119872 (119905)

119889119905= minus (120583

1+ 1205832)119872 (119905) minus 120575119872 (119905)

(3)

Let

Γ = (119878 119864 119860 119862119872) isin R5

+ | 119878 (119905) le

120575120587 + 120575119900

120575 + 120575119900+ 119901

(119878 + 119864 + 119860 + 119862 +119872 le120575120587 + 120575

119900

120575 + 120575119900

)

(4)

Here Γ is a positively invariant set All the solutions lie insideΓ which is our main focus

3 Basic ReproductionNumberThreshold Quantity

The basic reproduction number or the threshold quantityR0for the proposed model gives an average number of

secondary infection when an infection is introduced in apurely susceptible population We use the idea developed by[18] and also for detail see [19] We have

F = [

[

0 120573119878119900120573120581119878119900

0 0 0

0 0 0

]

]

119881 = [

[

120575 + 1205741

0 minus120575120587120578

minus1205741

120575 + 1199021205742+ (1 minus 119902) 120574

10

0 minus1199021205742

120575 + 1205743

]

]

119881minus1=

[

(120575+1199021205742+(1minus119902)1205741)(120575+1205743) 1199021205742120575120587120578 120575120587120578(120575+1199021205742+(1minus119902)1205741)

1205741(120575+1205743) (120575+1205741)(120575+1205743) 120575120587120578120574

1

11990212057411205742

1199021205742(120575+1205741) (120575+1205741)(120575+1199021205742+(1minus119902)1205741)

]

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902120574

11205742

(5)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and

119865119881minus1=

[[[[[

[

0120573119878119900120575120587120578119902120574

2

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2)

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) minus 120575120587120578119902119903

11199032

0 0 0

0 0 0

]]]]]

]

(6)

4 BioMed Research International

So the reproduction number given by 120588(119865119881minus1) is

R119900=120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2) + 120575120587120578119902119903

11199032

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) (7)

Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=

(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575

119900)(120575+120575

119900+119901) The endemic

equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system

(3) whose endemic equilibrium is given in the followingsubsection

Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878

lowast 119864 = 119864lowast 119860 = 119860

lowast 119862 = 119862lowast

and 119872 = 119872lowast equating left side of the system (3) equal to

zero we obtained

119878lowast=1198761(120575 + 120574

1) (120575 + 120574

3) (1 minus 119877

0) + 120581120573119878

1199001198761

1205741120573 ((120575 + 120574

3) + 120581119902120574

2)

minus(120575 + 120574

3) [(120575 + 120574

1) + 12058311205741]119872lowast

1205741120573 ((120575 + 120574

3) + 120581119902120574

2) 119860lowast

119864lowast=1198761119860lowastminus 1205832119872lowast

1205741

119862lowast=

1199021205742

(120575 + 1205743)119860lowast

(8)

4 Local Stability Analysis

In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863

119900= (119878119900 0 0 0 0)

in the following theorem

Theorem 1 For 1198770le 1 the disease-free equilibrium of the

system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is

locally asymptotically stable if ((1198761(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) otherwise the disease-free equilibrium of the

system (3) is unstable for 1198770gt 1

Proof To show the local stability of the system (3) about thepoint119863

119900 we set the left-hand side of the system (3) equating

to zero and we obtain the following Jacobian matrix 119869119900(120577)

119869119900(120577) = (

minus120575 minus 120575119900minus 119901 120575

119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878

119900minus 120575119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (9)

By the elementary row operation we get the followingmatrix

119869119900(120577) = (

minus(120575 + 120575119900+ 119901) minus120575

119900minus120573119878119900minus 120575119900

minus120575120587120578 minus 120573120581119878119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 0 minus1198761(120575 + 120574

1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574

1) 1205832+ 12058311205741

0 0 0 1198791

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (10)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and 119879

1= minus(minus119876

1(120575 + 120574

1) +

1205741120573119878119900) minus 119902120574

11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to

the previous Jacobian matrix is given by

(120582+120575 + 120575119900+119901) (120582 + 120583

1+1205832+ 120575) (120582

3+ 11988611205822+1198862120582+1198863) = 0

(11)

The first two eigenvalues minus(120575+120575119900+119901) and minus(120583

1+1205832+120575) have

negative real parts For the rest of the eigenvalues we get

1198861= (120575 + 120575

119900+ 119901) (119876

1(120575 + 120574

3) + 1 + 119876

1) (1 minus 119877

0)

+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574

3) + 119902120574

11205742) minus (119876

1120581 + 1205741))

times (120575 + 120575119900+ 119901)minus1

1198862= (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))

times (120575 + 120575119900+ 119901) [(120575 + 120574

1) + (120575 + 120574

1)2

]

+ (120575 + 1205741) [1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900)]

+ 11990212057411205742(120575 + 120574

1) [120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575 + 119901)]

times (120575 + 120575119900+ 119901)minus2

+ 11990212057411205742(120575 + 120574

1) (120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575

119900+ 119901))

times (120575 + 120575119900+ 119901)minus1

BioMed Research International 5

1198863=

(120575 + 1205741)

(120575 + 120575119900+ 119901)

(1198761(120575 + 120575

119900+ 119901) (120575 + 120574

1) minus 120573120574

1(120575120587 + 120575

119900))

times [11990212057411205742(120573120581 (120575120587 + 120575

119900) + 120575120587120578)

+ (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))]

(12)

By the Routh-Hurwitz criteria 1198861gt 0 119886

3gt 0 and 119886

11198862gt 1198863

Here 1198861gt 0 when 119877

0le 1 and ((119876

1(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) Also 119886

2gt 0 and 119886

3gt 0 and then 119886

11198862gt

1198863 So according to the Routh-Hurwitz criteria the Jacobian

matrix has negative real parts if and only if 1198770le 1 Thus by

Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863

119900= (119878119900 0 0 0 0) The

proof is completed

The stability of the disease-free equilibrium for 1198770le 1

means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877

0gt 1 When the disease-free

equilibrium is locally asymptotically stable for 1198770le 1 then

the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1

41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =

(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of

endemic equilibrium in the following

Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system

(3) is locally asymptotical stable if the following conditionshold

12057411198852120575119900119878lowast2

gt 12057411205732120581

1198661gt 1198662

(13)

otherwise the system is unstable

Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869

lowast(120577) of the system (3) in the following

119869lowast(120577) = (

minus(120575 + 120575119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)) 120575

119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878

lowastminus 120575119900

minus120575119900

120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574

1) 120573119878

lowast120573120581119878lowast+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (120575 + 1205831+ 1205832)

) (14)

By elementary row operation and after simplification we getthe following Jacobian matrix

119869lowast(120577) = (

minus1198851

minus120575119900

minus120573119878lowastminus 120575119900

1198853

minus120575119900

0 minus1198851(120575 + 120574

1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885

11198854+ 11988521198853

11988511205831minus 1198852120575119900

0 0 1198855

1205741(11988511198854+ 11988521198853) 119885

6

0 0 0 1198857

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (15)

where

1198851= 120575 + 120575

119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)

1198852= 120573 (119860

lowast+ 120581119862lowast)

1198853= minus (120575120587120578 + 120573120581119878

lowast+ 120575119900)

1198854= (120573120581119878

lowast+ 120575120587120578)

1198855= minus119876

1(minus1198851(120575 + 120574

1) minus 1198852120575119900)

+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))

1198856= minus1205832(minus1198851(120575 + 120574

1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)

1198857= minus1198855(120575 + 120574

3) minus 11988551205741(11988511198854+ 11988521198853)

(16)

The eigenvalue 1205821= minus(120583

1+ 1205832+ 120575) lt 0 120582

2= minus1198851 and using

the value of 1198851and further 119862lowast we get 120582

2= minus(120575 + 120575

119900+ 119901 +

120573((120575 + 1205743) + 120581119902120574

2)119860lowast) lt 0 120582

3= minus(119885

1(120575 + 120574

1) + 1198852120575119900) lt 0

as 1198851gt 0 and 119885

2gt 0 120582

4= 1198855 1205824lt 0 if and only if 119885

5lt 0

After the simplifications we get

120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2

)

+ 1205741120573120575120587120578120573

lowastlowastlowast119872lowast

minus [(1205741120573120575120587120578120573

lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574

1)

+11987611198852120575119900120573lowast] gt 0

(17)

where 120573lowast = 1205741120573((120575 + 120574

3) + 120581119902120574

2) 120573lowastlowast = 119876

1(120575 + 120574

1)(120575 +

1205743)(1minus119877

0)+120581120573119878

1199001198761 and 120574

1120573120575120587120578120573

lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573

lowastlowast+

120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866

1= 1205741120573120575120587120578120573

lowastlowastlowast119872lowast and

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Signal TransductionJournal of

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Virolog y

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Volume 2014

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Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

4 BioMed Research International

So the reproduction number given by 120588(119865119881minus1) is

R119900=120581120573119878119900(120575 + 120574

1(1 minus 119902) + 119902120574

2) + 120575120587120578119902119903

11199032

(120575 + 1205741) (120575 + 119902120574

2+ (1 minus 119902) 120574

1) (120575 + 120574

3) (7)

Here 119878119900 shows the disease-free equilibrium (DFE) and119863119900=

(119878119900 0 0 0 0) giving 119878119900 = (120575120587+120575

119900)(120575+120575

119900+119901) The endemic

equilibrium point 119879lowast = (119878lowast 119864lowast 119860lowast 119862lowast119872lowast) for system

(3) whose endemic equilibrium is given in the followingsubsection

Endemic Equilibria To find the endemic equilibria of thesystem (3) by setting 119878 = 119878

lowast 119864 = 119864lowast 119860 = 119860

lowast 119862 = 119862lowast

and 119872 = 119872lowast equating left side of the system (3) equal to

zero we obtained

119878lowast=1198761(120575 + 120574

1) (120575 + 120574

3) (1 minus 119877

0) + 120581120573119878

1199001198761

1205741120573 ((120575 + 120574

3) + 120581119902120574

2)

minus(120575 + 120574

3) [(120575 + 120574

1) + 12058311205741]119872lowast

1205741120573 ((120575 + 120574

3) + 120581119902120574

2) 119860lowast

119864lowast=1198761119860lowastminus 1205832119872lowast

1205741

119862lowast=

1199021205742

(120575 + 1205743)119860lowast

(8)

4 Local Stability Analysis

In this section we find the local stability of disease-free andendemic equilibria First we show the local asymptoticalstability of DFE equilibrium and then we find the localasymptotical stability of endemic equilibrium Now we showthe local stability of DFE about the point 119863

119900= (119878119900 0 0 0 0)

in the following theorem

Theorem 1 For 1198770le 1 the disease-free equilibrium of the

system (3) about an equilibrium point 119863119900= (119878119900 0 0 0 0) is

locally asymptotically stable if ((1198761(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) otherwise the disease-free equilibrium of the

system (3) is unstable for 1198770gt 1

Proof To show the local stability of the system (3) about thepoint119863

119900 we set the left-hand side of the system (3) equating

to zero and we obtain the following Jacobian matrix 119869119900(120577)

119869119900(120577) = (

minus120575 minus 120575119900minus 119901 120575

119900minus120573119878119900minus 120575119900minus120575120587120578 minus 120573120581119878

119900minus 120575119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (9)

By the elementary row operation we get the followingmatrix

119869119900(120577) = (

minus(120575 + 120575119900+ 119901) minus120575

119900minus120573119878119900minus 120575119900

minus120575120587120578 minus 120573120581119878119900

0

0 minus (120575 + 1205741) 120573119878

119900120573120581119878119900+ 120575120587120578 120583

1

0 0 minus1198761(120575 + 120574

1) + 12057411205731198781199001205741(120573120581119878119900+ 120575120587120578) (120575 + 120574

1) 1205832+ 12058311205741

0 0 0 1198791

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (10)

where 1198761= (120575 + 119902120574

2+ (1 minus 119902)120574

1) and 119879

1= minus(minus119876

1(120575 + 120574

1) +

1205741120573119878119900) minus 119902120574

11205742(120573120581119878119900+ 120575120587120578) The characteristic equation to

the previous Jacobian matrix is given by

(120582+120575 + 120575119900+119901) (120582 + 120583

1+1205832+ 120575) (120582

3+ 11988611205822+1198862120582+1198863) = 0

(11)

The first two eigenvalues minus(120575+120575119900+119901) and minus(120583

1+1205832+120575) have

negative real parts For the rest of the eigenvalues we get

1198861= (120575 + 120575

119900+ 119901) (119876

1(120575 + 120574

3) + 1 + 119876

1) (1 minus 119877

0)

+ 120573 (120575120587 + 120575119900) ((1205741(120575 + 120574

3) + 119902120574

11205742) minus (119876

1120581 + 1205741))

times (120575 + 120575119900+ 119901)minus1

1198862= (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))

times (120575 + 120575119900+ 119901) [(120575 + 120574

1) + (120575 + 120574

1)2

]

+ (120575 + 1205741) [1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900)]

+ 11990212057411205742(120575 + 120574

1) [120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575 + 119901)]

times (120575 + 120575119900+ 119901)minus2

+ 11990212057411205742(120575 + 120574

1) (120573120581 (120575120587 + 120575

119900) + 120575120587120578 (120575 + 120575

119900+ 119901))

times (120575 + 120575119900+ 119901)minus1

BioMed Research International 5

1198863=

(120575 + 1205741)

(120575 + 120575119900+ 119901)

(1198761(120575 + 120575

119900+ 119901) (120575 + 120574

1) minus 120573120574

1(120575120587 + 120575

119900))

times [11990212057411205742(120573120581 (120575120587 + 120575

119900) + 120575120587120578)

+ (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))]

(12)

By the Routh-Hurwitz criteria 1198861gt 0 119886

3gt 0 and 119886

11198862gt 1198863

Here 1198861gt 0 when 119877

0le 1 and ((119876

1(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) Also 119886

2gt 0 and 119886

3gt 0 and then 119886

11198862gt

1198863 So according to the Routh-Hurwitz criteria the Jacobian

matrix has negative real parts if and only if 1198770le 1 Thus by

Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863

119900= (119878119900 0 0 0 0) The

proof is completed

The stability of the disease-free equilibrium for 1198770le 1

means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877

0gt 1 When the disease-free

equilibrium is locally asymptotically stable for 1198770le 1 then

the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1

41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =

(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of

endemic equilibrium in the following

Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system

(3) is locally asymptotical stable if the following conditionshold

12057411198852120575119900119878lowast2

gt 12057411205732120581

1198661gt 1198662

(13)

otherwise the system is unstable

Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869

lowast(120577) of the system (3) in the following

119869lowast(120577) = (

minus(120575 + 120575119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)) 120575

119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878

lowastminus 120575119900

minus120575119900

120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574

1) 120573119878

lowast120573120581119878lowast+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (120575 + 1205831+ 1205832)

) (14)

By elementary row operation and after simplification we getthe following Jacobian matrix

119869lowast(120577) = (

minus1198851

minus120575119900

minus120573119878lowastminus 120575119900

1198853

minus120575119900

0 minus1198851(120575 + 120574

1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885

11198854+ 11988521198853

11988511205831minus 1198852120575119900

0 0 1198855

1205741(11988511198854+ 11988521198853) 119885

6

0 0 0 1198857

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (15)

where

1198851= 120575 + 120575

119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)

1198852= 120573 (119860

lowast+ 120581119862lowast)

1198853= minus (120575120587120578 + 120573120581119878

lowast+ 120575119900)

1198854= (120573120581119878

lowast+ 120575120587120578)

1198855= minus119876

1(minus1198851(120575 + 120574

1) minus 1198852120575119900)

+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))

1198856= minus1205832(minus1198851(120575 + 120574

1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)

1198857= minus1198855(120575 + 120574

3) minus 11988551205741(11988511198854+ 11988521198853)

(16)

The eigenvalue 1205821= minus(120583

1+ 1205832+ 120575) lt 0 120582

2= minus1198851 and using

the value of 1198851and further 119862lowast we get 120582

2= minus(120575 + 120575

119900+ 119901 +

120573((120575 + 1205743) + 120581119902120574

2)119860lowast) lt 0 120582

3= minus(119885

1(120575 + 120574

1) + 1198852120575119900) lt 0

as 1198851gt 0 and 119885

2gt 0 120582

4= 1198855 1205824lt 0 if and only if 119885

5lt 0

After the simplifications we get

120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2

)

+ 1205741120573120575120587120578120573

lowastlowastlowast119872lowast

minus [(1205741120573120575120587120578120573

lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574

1)

+11987611198852120575119900120573lowast] gt 0

(17)

where 120573lowast = 1205741120573((120575 + 120574

3) + 120581119902120574

2) 120573lowastlowast = 119876

1(120575 + 120574

1)(120575 +

1205743)(1minus119877

0)+120581120573119878

1199001198761 and 120574

1120573120575120587120578120573

lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573

lowastlowast+

120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866

1= 1205741120573120575120587120578120573

lowastlowastlowast119872lowast and

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Microbiology

BioMed Research International 5

1198863=

(120575 + 1205741)

(120575 + 120575119900+ 119901)

(1198761(120575 + 120575

119900+ 119901) (120575 + 120574

1) minus 120573120574

1(120575120587 + 120575

119900))

times [11990212057411205742(120573120581 (120575120587 + 120575

119900) + 120575120587120578)

+ (1198761(120575 + 120574

1) (120575 + 120575

119900+ 119901) minus 120573120574

1(120575120587 + 120575

119900))]

(12)

By the Routh-Hurwitz criteria 1198861gt 0 119886

3gt 0 and 119886

11198862gt 1198863

Here 1198861gt 0 when 119877

0le 1 and ((119876

1(120575 + 120575

119900+ 119901)(120575 + 120574

1) gt

1205731205741(120575120587 + 120575

119900)) Also 119886

2gt 0 and 119886

3gt 0 and then 119886

11198862gt

1198863 So according to the Routh-Hurwitz criteria the Jacobian

matrix has negative real parts if and only if 1198770le 1 Thus by

Routh-Hurwitz criteria the DFE of the system (3) is locallyasymptotically stable about the point119863

119900= (119878119900 0 0 0 0) The

proof is completed

The stability of the disease-free equilibrium for 1198770le 1

means that the disease dies out from the population Next weshow that the endemic equilibrium of the system (3) is locallyasymptotically stable for 119877

0gt 1 When the disease-free

equilibrium is locally asymptotically stable for 1198770le 1 then

the endemic equilibrium does not exist but we are interestedto know about the properties of an endemic equilibrium for1198770gt 1

41 Stability of Endemic Equilibrium (EE) In this subsectionwe find the local asymptotic stability of EE about 119863lowast =

(119878lowast 119864lowast 119860lowast 119862lowast119872lowast) and we prove the local stability of

endemic equilibrium in the following

Theorem2 For1198770gt 1 the endemic equilibrium119863lowast of system

(3) is locally asymptotical stable if the following conditionshold

12057411198852120575119900119878lowast2

gt 12057411205732120581

1198661gt 1198662

(13)

otherwise the system is unstable

Proof Here we prove the that the system (3) about theequilibrium point119863lowast is locally stable and for this we obtainthe Jacobian matrix 119869

lowast(120577) of the system (3) in the following

119869lowast(120577) = (

minus(120575 + 120575119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)) 120575

119900minus120573119878lowastminus 120575119900minus120575120587120578 minus 120573120581119878

lowastminus 120575119900

minus120575119900

120573 (119860lowast+ 120581119862lowast) minus (120575 + 120574

1) 120573119878

lowast120573120581119878lowast+ 120575120587120578 120583

1

0 1205741

minus1198761

0 1205832

0 0 1199021205742

minus (120575 + 1205743) 0

0 0 0 0 minus (120575 + 1205831+ 1205832)

) (14)

By elementary row operation and after simplification we getthe following Jacobian matrix

119869lowast(120577) = (

minus1198851

minus120575119900

minus120573119878lowastminus 120575119900

1198853

minus120575119900

0 minus1198851(120575 + 120574

1) minus 11988521205751199001198851120573119878lowastminus 1198852(120573119878lowast+ 120575119900) 119885

11198854+ 11988521198853

11988511205831minus 1198852120575119900

0 0 1198855

1205741(11988511198854+ 11988521198853) 119885

6

0 0 0 1198857

0

0 0 0 0 minus (1205831+ 1205832+ 120575)

) (15)

where

1198851= 120575 + 120575

119900+ 119901 + 120573 (119860

lowast+ 120581119862lowast)

1198852= 120573 (119860

lowast+ 120581119862lowast)

1198853= minus (120575120587120578 + 120573120581119878

lowast+ 120575119900)

1198854= (120573120581119878

lowast+ 120575120587120578)

1198855= minus119876

1(minus1198851(120575 + 120574

1) minus 1198852120575119900)

+ 1205741(1198854120573119878lowastminus 1198852(120573119878lowast+ 120575119900))

1198856= minus1205832(minus1198851(120575 + 120574

1) minus 1198852120575119900) + 1205741(11988511205831minus 1198852120575119900)

1198857= minus1198855(120575 + 120574

3) minus 11988551205741(11988511198854+ 11988521198853)

(16)

The eigenvalue 1205821= minus(120583

1+ 1205832+ 120575) lt 0 120582

2= minus1198851 and using

the value of 1198851and further 119862lowast we get 120582

2= minus(120575 + 120575

119900+ 119901 +

120573((120575 + 1205743) + 120581119902120574

2)119860lowast) lt 0 120582

3= minus(119885

1(120575 + 120574

1) + 1198852120575119900) lt 0

as 1198851gt 0 and 119885

2gt 0 120582

4= 1198855 1205824lt 0 if and only if 119885

5lt 0

After the simplifications we get

120573lowast119860lowast(12057411198852120575119900minus 12057411205732120581119878lowast2

)

+ 1205741120573120575120587120578120573

lowastlowastlowast119872lowast

minus [(1205741120573120575120587120578120573

lowastlowast+ 120573lowast1198761) 1198851(120575 + 120574

1)

+11987611198852120575119900120573lowast] gt 0

(17)

where 120573lowast = 1205741120573((120575 + 120574

3) + 120581119902120574

2) 120573lowastlowast = 119876

1(120575 + 120574

1)(120575 +

1205743)(1minus119877

0)+120581120573119878

1199001198761 and 120574

1120573120575120587120578120573

lowastlowastlowast119872lowastgt [(1205741120573120575120587120578120573

lowastlowast+

120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast] say119866

1= 1205741120573120575120587120578120573

lowastlowastlowast119872lowast and

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

6 BioMed Research International

1198662= [(1205741120573120575120587120578120573

lowastlowast+120573lowast1198761)1198851(120575+1205741)+11987611198852120575119900120573lowast]120573lowastlowastlowast

= (120575+

1205743)[(120575 + 120574

1) + 12057411205831] So 120582

4has negative real part if (120574

11198852120575119900gt

12057411205732120581119878lowast2) For 120582

5= 1198857 we obtained negative real parts

and by using the 1198855which is positive under the conditions

described in 1205824 and 119885

1gt 0 119885

2gt 0 we get the negative

real partsThus all the eigenvalues have negative real parts soby the Routh-Hurwitz criteria the endemic equilibrium point119863lowast is locally asymptotically stable when 119877

0gt 1 The proof is

completed

5 Global Stability of DFE

In this section we present the global stability of disease-freeequilibrium DFE of the system (3) For different biologicalmodel the Lyapunov function was used by [20 21] for theglobal stability For our model we define and constructLyapunov function in the following for the global stability ofDFE Further the global stability of endemic equilibrium weuse the Lyapunov function and find its global asymptoticalstability

Theorem 3 For 1198770le 1 the disease-free equilibrium of the

system (3) is stable globally asymptotically if 119878 = 119878119900 and

unstable for 1198770gt 1

Proof Here we define the Lyapunov function for the globalstability of disease-free equilibrium given by

119881 (119905) = [1198891(119878 minus 119878

119900) + 1198892119864 + 1198893119860 + 119889

4119862 + 119889

5119872] (18)

Differentiating the previous function with respect to 119905 andusing the system (3)

1198811015840(119905) = 119889

11198781015840+ 11988921198641015840+ 11988931198601015840+ 11988941198621015840+ 11988951198721015840

1198811015840(119905)

= 1198891[120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905) minus 120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905)

+ 120575119900(1minus(119878 (119905) + 119864 (119905) +119860 (119905) + 119862 (119905) +119872 (119905)))minus119901119878 (119905)]

+ 1198892[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905) + 120575120587120578119862 (119905)

minus1205741119864 (119905) + 120583

1119872(119905)]

+ 1198893[1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905) minus (1 minus 119902) 120574

1119860 (119905)

+1205832119872(119905)]

+ 1198894[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1198895[minus (1205831+ 1205832)119872 (119905) minus 120575119872 (119905)]

(19)

where 119889119894 119894 = 1 2 5 are some positive constants to be

chosen later

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 2 The plot shows the HBV transmission model of hepatitisB with 120583

1= 090 and 120583

2= 090

Time (year)0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 3 The plot shows the HBV transmission model of hepatitisB with 120583

1= 080 and 120583

2= 090

After the arrangement we obtain

1198811015840(119905) = [119889

2minus 1198891] (120573 (119860 + 120581119862)) 119878 + [119889

2minus 1198891] 120575120587120578119862

+ [11988931205741minus 1198892(120575 + 120574

1)] 119864 + [119889

21205831minus 1198895120575]119872

+ [11988941199021205742minus 11988931198761] 119860 + 119889

1120575120587 minus 119889

1120575119878 + 119889

1120575119900

minus 1198891120575119900(119878 + 119864 + 119860 + 119862 +119872) minus 119889

1119901119878

minus 1198894(120575 + 120574

3) 119862

(20)

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

BioMed Research International 7

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 4 The plot shows the HBV transmission model of hepatitisB with 120583

1= 070 and 120583

2= 080

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 5 The plot shows the HBV transmission model of hepatitisB with 120583

1= 050 and 120583

2= 060

Choosing the constants 1198891= 1198892= 1205741 1198893= (120575 + 120574

1) 1198894=

1198761(120575 + 120574

1)1199021205742 and 119889

5= 12057411205831120575

After the simplification we get

1198811015840(119905) = minus (119878 minus 119878

119900) minus 1205741120575119900119864 minus 1205741120575119900119860

minus (1205741120575119900+(120575 + 120574

3) (120575 + 120574

1) 1198761

1199021205742

)119862

minus (1205741120575119900+ (120575119900+ 1205741) 1205832)119872

(21)

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 6 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 030

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 7 The plot shows the HBV transmission model of hepatitisB with 120583

1= 020 and 120583

2= 040

where 119878119900 = (120575120587 + 120575119900)(120575 + 120575

119900+ 119901) 1198811015840(119905) = 0 if and only if

119878 = 119878119900 and 119864 = 119860 = 119862 = 119872 = 0 Also 1198811015840(119905) is negative for

(119878 gt 119878119900) So by [22] the DFE is globally asymptotically stable

in Γ The proof is completed

51 Global Stability of Endemic Equilibrium In this subsec-tion we show the global asymptotical stability of the system(3) To do this we state and prove the following theorem forthe global stability of endemic equilibrium

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

8 BioMed Research International

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

Time (year)

Com

part

men

tal p

opul

atio

ns

Population behavior of individuals

SEA

CM

Figure 8 The plot shows the HBV transmission model of hepatitisB with 120583

1= 010 and 120583

2= 020

Theorem 4 For 1198770gt 1 system (3) is globally asymptotically

stable if 119878 = 119878lowast and 120575119900gt 1205831 and unstable for 119877

0le 1

Proof To prove that system (3) is globally asymptoticallystable we define the Lyapunov in the following

119871 (119905) =(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

(119878 minus 119878lowast) +

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

119864

+ (1205831+ 1205832+ 1205833) 119860 +

(1205831+ 1205832+ 1205833) 1198761

1199021205742

119862 + 1205832119872

(22)

Taking the derivative with respect to time 119905 using the system(3)

1198711015840(119905) =

(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

times [120575120587 (1 minus 120578119862 (119905)) minus 120575119878 (119905)minus120573 (119860 (119905)+120581119862 (119905)) 119878 (119905)

+ 120575119900(1 minus (119878 (119905) + 119864 (119905) + 119860 (119905) + 119862 (119905) + 119872 (119905)))

minus119901119878 (119905)]

+(1205831+ 1205832+ 1205833) 1205741

1205741+ 120575

[120573 (119860 (119905) + 120581119862 (119905)) 119878 (119905) minus 120575119864 (119905)

+120575120587120578119862 (119905) minus 1205741119864 (119905) + 120583

1119872(119905)]

+ (1205831+ 1205832+ 1205833) [1205741119864 (119905) minus 120575119860 (119905) minus 119902120574

2119860 (119905)

minus (1 minus 119902) 1205741119860 (119905) + 120583

2119872(119905)]

+(1205831+ 1205832+ 1205833) 1198761

1199021205742

[1199021205742119860 (119905) minus 120575119862 (119905) minus 120574

3119862 (119905)]

+ 1205832[minus (1205831+ 1205832+ 120575)119872]

(23)Simplifying we obtained

1198711015840(119905) = minus

(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575) (120575 + 120575

119900+ 119901)

(119878 minus 119878lowast)

minus(1205831+ 1205832+ 1205833) 1205741120575119900

(1205741+ 120575)

[119864 + 119860 + 119862]

minus(1205831+ 1205832+ 1205833) 1205741

(1205741+ 120575)

[120575119900minus 1205831]119872

(24)

The endemic equilibrium of the system (3) is globally asymp-totically stable for 119877

0gt 1 if 119878 = 119878

lowast and 120575119900gt 1205831

So the endemic equilibrium of the system (3) is globallyasymptotically stable The proof is completed

6 Numerical Simulations

In this section we present the numerical simulation of theproposed model (3) by using the Runge-Kutta order fourscheme For different values of the parameters the numericalresults are presented in Figures 2 3 4 5 6 7 and 8 Thevariation of migration parameters 120583

1and 120583

2 with different

values is presented In the numerical solution of the modelthe parameters and their values are presented in Table 1In our simulation the susceptible individuals are shown bydashed line the exposed individuals by dotted dashed theacute individuals by bold line the carrier by dashed lineand the migrated individuals by red bold line Figures 2to 8 represent the compartmental population of hepatitis Bindividuals with migration effect The values presented inTable 1 are fixed except for120583

1and1205832 In Figure 2 by the values

for 1205831= 09 and 120583

2= 09 we see that the population of

exposed and acute individuals is increasing In Figure 3 weset 1205831= 08 and 120583

2= 09 and the population of exposed

and acute individuals decreased Decreasing the values of 1205831

and 1205832 we obtain different results see Figures 2 to 8 When

we decrease the population of immigrants who have the HBVvirus we see the decrease in the population of exposed andacute individuals The parameters presented in Table 1 wereused by different authors for example the natural deathrate equally birth rate by [4] the rate at which the latentindividuals become infectious by [3] the rate at which theindividuals move to the carrier class by [3] the rate at whichthe individuals move to the carrier class [3] the transmissioncoefficient 120573 by [24] 120575

119900the loss of immunity rate by [13] the

value of 120578 by [13] the value of 120581 by [10] and120587 and 119902 from [23]We assume the values for the parameters 120574

3 119901 1205831 and 120583

2in

our simulation

7 Conclusion

A compartmental model of HBV transmission virus hasbeen presented A mathematical model has been obtained

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

BioMed Research International 9

Table 1 Parameter values used in numerical simulations

Notation Parameter description Range Source120575 Natural death rate equally birth rate 00143 [4]120587 The failure immunization 0-1 [23]1205741

The rate at which the latent individuals become infectious 6 [3]1205742

The rate at which the individuals move to the carrier class 4 [3]1205743

The rate of flow from carrier to the vaccinated class 034 Assumed120573 The transmission coefficient 08 [24]119902 The proportion of individuals become carrier 0005 [23]120575119900

Represent the loss of immunity 006ndash003 [13]119901 Represent the vaccination of susceptible 03 Assumed1205831

The rate of flow fromMigrated class to exposed class 023 Assumed1205832

The rate of flow fromMigrated class to acute class 056 Assumed120578 Unimmunized children born to carrier mothers 07 [13]120581 The infectiousness of carriers related to acute infection 0-1 [10]

by adding (1) the migrated class (2) HBV transmission ratebetween migrated class and exposed class (3) transmissionrate between migrated class and acute class and (4) deathrate of individuals in the migrated class By adding thesenew features we have obtained a compartmental model ofHBVwithmigration effect First we obtained the basic repro-duction number for the proposed model The disease-freeequilibrium is locally as well as globally asymptotically stablefor 1198770le 1 We obtained the local and global asymptotical

stability for the endemic equilibrium For the reproductionnumber 119877

0gt 1 the endemic equilibrium is locally as

well as globally asymptotically stable Furthermore we havesolved the compartment model numerically and the resultsare presented in Figures 2 to 8 By changing the values of 120583

1

and 1205832 different results have been obtained It is concluded

that when the value of 1205831and 120583

2decreases the population

of (exposed acute and carrier) individuals also decreaseThe proportion of infected individuals decreases when theproportion ofmigrated individuals (who have theHBVvirus)decreases So the number of infected individuals is directlyproportional to the number of migrated individuals

Last yet not the least the authors of this work have agreedto devise in a course of time a more advanced model onrestraining HBV transmission through migration

Acknowledgment

Theauthors would like to thank the Kanury V S Rao for theircareful reading of the original manuscripts and their manyvaluable comments and suggestions that greatly improve thepresentation of this work

References

[1] F Ahmed and G R Foster ldquoGlobal hepatitis migration and itsimpact onWestern healthcarerdquoGut vol 59 no 8 pp 1009ndash10112010

[2] ldquoChinese center for disease control and prevention (CCDC)rdquo 2010httpwwwchinacdccnn272442n272530n3479265n347930337095html

[3] G F Medley N A Lindop W J Edmunds and D JNokes ldquoHepatitis-B virus endemicity heterogeneity catas-trophic dynamics and controlrdquo Nature Medicine vol 7 no 5pp 619ndash624 2001

[4] S Thornley C Bullen and M Roberts ldquoHepatitis B in ahigh prevalence New Zealand population a mathematicalmodel applied to infection control policyrdquo Journal ofTheoreticalBiology vol 254 no 3 pp 599ndash603 2008

[5] R M Anderson and R M May Infectious Disease of HumansDynamics and Control Oxford University Press Oxford UK1991

[6] S Zhao Z Xu and Y Lu ldquoA mathematical model of hepatitis Bvirus transmission and its application for vaccination strategyin Chinardquo International Journal of Epidemiology vol 29 no 4pp 744ndash752 2000

[7] K Wang W Wang and S Song ldquoDynamics of an HBV modelwith diffusion and delayrdquo Journal ofTheoretical Biology vol 253no 1 pp 36ndash44 2008

[8] R Xu andZMa ldquoAnHBVmodelwith diffusion and time delayrdquoJournal of Theoretical Biology vol 257 no 3 pp 499ndash509 2009

[9] L Zou W Zhang and S Ruan ldquoModeling the transmissiondynamics and control of hepatitis B virus in Chinardquo Journal ofTheoretical Biology vol 262 no 2 pp 330ndash338 2010

[10] J Pang J A Cui and X Zhou ldquoDynamical behavior of ahepatitis B virus transmission model with vaccinationrdquo Journalof Theoretical Biology vol 265 no 4 pp 572ndash578 2010

[11] K P Maier Hepatitis-Hepatitisfolgen Georg Thieme VerlagStuttgart Germany 2000

[12] G L Mandell R G Douglas and J E Bennett Principles andPractice of Infectious Diseases A Wiley Medical PublicationJohn Wiley and Sons New York NY USA 1979

[13] CW Shepard E P Simard L Finelli A E Fiore and B P BellldquoHepatitis B virus infection epidemiology and vaccinationrdquoEpidemiologic Reviews vol 28 no 1 pp 112ndash125 2006

[14] X Weng and Y Zhang Infectious Diseases Fudan UniversityPress Shanghai China 2003

[15] J Wu and Y Luo Infectious Diseases Central South UniversityPress Changsha China 2004

[16] J BWyngaarden L H Smith and J C BennettCecil Text Bookof Medicine WB Saunders Philadelphia Pa USA 19th edition1992

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

10 BioMed Research International

[17] B JMcMahonW LM Alward andD BHall ldquoAcute hepatitisB virus infection relation of age to the clinical expressionof disease and subsequent development of the carrier staterdquoJournal of Infectious Diseases vol 151 no 4 pp 599ndash603 1985

[18] P Van Den Driessche and J Watmough ldquoReproduction num-bers and sub-threshold endemic equilibria for compartmentalmodels of disease transmissionrdquoMathematical Biosciences vol180 pp 29ndash48 2002

[19] O Diekmann and J A P HeesterbeekMathematical Epideniol-ogy of Infectious Diseases 2000

[20] A Mwasa and J M Tchuenche ldquoMathematical analysis of acholera model with public health interventionsrdquo BioSystemsvol 105 no 3 pp 190ndash200 2011

[21] J Li and Y Yang ldquoSIR-SVS epidemic models with continuousand impulsive vaccination strategiesrdquo Journal of TheoreticalBiology vol 280 no 1 pp 108ndash116 2011

[22] J P LaSalle ldquoStability of nonautonomous systemsrdquo NonlinearAnalysis vol 1 no 1 pp 83ndash90 1976

[23] World Health Organization ldquoHepatitis B WHOCDSCSRLYO20022Hepatitis Brdquo httpwwwwhointcsrdiseasehepa-titiswhocdscsrlyo20022en

[24] W J Edmunds G FMedley andD J Nokes ldquoThe transmissiondynamics and control of hepatitis B virus in the GambiardquoStatistics in Medicine vol 15 pp 2215ndash2233 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

GenomicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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