Modeling and Analysis of an Svirs Epidemic Model Involving External Sources of Disease

8/10/2019 Modeling and Analysis of an Svirs Epidemic Model Involving External Sources of Disease

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INTERNATIONAL JOURNAL OF TECHNOLOGY ENHANCEMENTS AND EMERGING ENGINEERING RESEARCH, VOL 2, ISSUE 10 18ISSN 2347-4289

Copyright 2014 IJTEEE.

Modeling And Analysis Of An SVIRS EpidemicModel Involving External Sources Of DiseaseRaid Kamel Naji, Ahmed Ali Muhseen

Department of mathematics, College of science, University of Baghdad. Baghdad -Iraq. Email: [email protected] Lecturer, Ministry of Education, Rusafa, Baghdad-IraqEmail: [email protected].

Abstract: In this paper a mathematical model that describes the flow of infectious disease in a population is proposed and studied. It is assumed that thedisease divided the population into four classes: susceptible individuals (S), vaccinated individuals (V), infected individuals (I) and recover individuals(R). The impact of immigrants, vaccine and external sources of disease, on the dynamics of SVIRS epidemic model is investigated. The existenceuniqueness and boundedness of the solution of the model are discussed. The local and global stability of the model is studied. The occurrence of locabifurcation as well as Hopf bifurcation in the model is investigated. Finally the global dynamics of the proposed model is studied numerically.

Keywords:Epidemic models, Stability, Vaccinated, Immigrants, external sources, Local and Hopf bifurcation.

1. IntroductionMathematical models have become important tools to studyand analyze the spread and control of infectious disease.

Most the proposed mathematical models those describe thetransmission of infectious disease have been derived fromthe classical susceptible infective recover (SIR) model,which is suggested originally by Kermack and Mckenderick[1]. In that model the susceptible individuals becomeinfective by contact with infected individuals and then theinfected individuals may recover and transfer to removalindividuals at a specific rate. Numbers of mathematicalmodels were developed to study and analyze the spread ofinfectious diseases in order to prevent or minimize thetransmission of them through quarantine and othermeasures see for example [2-5] and the references therein. On the other hand, since the resistance against aninfectious disease represents protection that reduces an

individualsrisk of contracting the disease, therefore manyepidemiological models involving vaccination (V) have beenproposed and studied, see foe example [6-8] and thereference there in. Keeping the above in view, there aremany infectious diseases spread within the population bydirect contact between susceptible and infective individuals,they may spread through external sources in theenvironment such as (air, water, insects, etc). Therefore,recently Das et al. [9] have been proposed and studied amathematical model consisting of eco-epidemiologicalmodel involving external sources of disease. In this paperwe proposed and studied a mathematical model consistingof SVIRS epidemic model involving immigrant individuals,some of them may arrive infected with the disease, and

vaccine in which it is assumed that the disease transmittedby contact as well as external sources in the environment.

2. The mathematical model:Consider a simple epidemiological model in which the totalpopulation (say N(t)) at time t is divided in to three subclasses the susceptible individuals S(t), infected individualsI(t) and recover individuals R(t). Such model can berepresented as follows:

RIdt

dR

IISdt

dI

SISdt

dS

)( (2.1)

Here 0 is the recruitment rate of the population, 0is the natural death rate of the population, 0 is theinfected rate (incidence rate) of the susceptible individuals

due to direct contact with the infected individuals and 0is the natural recovery rate of the infected individuals. Nowsince there are many infectious diseases (Alanfelonzhabirds Anfelonzha and typhoid etc.) spread in theenvironment by different factors including insects, contacor other vectors, therefore, we assumed that the disease in

the above model will transmitted between the populationindividuals by contact as well as external sources odisease in the environment with an external incidence rate

0 . Also it is assumed that the lifetime of removaindividualsimmunity may not continue forever and then theremoval individuals return to be susceptible class with a

constant rate 0 (also known as losing removaindividuals immunity rate). Further, there is a constant flow

say 0A , of a new members arriving into the populationwith the fraction p of A arriving infected ( 10 p )Then the above system (2.1) can be rewritten in the form:

RIdt

dR

ISIpAdt

dI

RSSIApdtdS

)(

)()(

)()1(

(2.2)

Keeping the above in view, in order to study the effect ofvaccination on the system (2.2) let V(t) represented thevaccinated individuals in the population at time t , and thenthe following assumptions are made:


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The susceptible class is vaccinated at per capita

rate 0 . The infection can invade the susceptible class or

vaccinated class depending on vaccine efficiency. The vaccine reduces the possibility of infection by

a factor of , which is known as intensity vaccine

immunity rate, where 10 . The vaccine may not give a permanent immunity

for susceptible individuals, so the vaccine maydisappear and then the individuals loss the

immunity with rate 10 .

Accordingly, the flow of disease in system (2.2) along withthe above assumptions can be representing in the followingblock diagram:

Fig. 1.Block diagram of system (2.3).

Therefore system (2) can be modified to:

RIdt

dR

IVISIpAdt

dI

VVISdt

dV

RVSSIApdt

dS

)(

)()()(

)()(

)()()1(

(2.3)

Clearly for 0 the vaccine is completely affective.While, 1 stand for the situation where the vaccine istotally ineffective. On the other hand, 0 denotes to thecase when immunity is life-long while 1 corresponds tothe case where there is absolutely no vaccine inducedimmunity. Therefore at any point of time t the total numberof population becomes

)()()()( tRtItVtSN .

Obviously, due to the biological meaning of the variables

S(t), V(t), I(t), and R(t), system (2.3) has the domain:0,0,0,0,),,,( 44 RIVSRIVS which is

positively invariant for system (2.3). Clearly, the interactionfunctions on the right hand side of system (2.3) arecontinuously differentiable. In fact they are Liptschizan

function on4 . Therefore the solution of system (2.3)

exists and is unique. Further, all solutions of the system(2.3) with non-negative initial conditions are uniformlybounded as shown in the following theorem.Theorem (2.1): All the solutions of system (2.3), which are

initiate in4 , are uniformly bounded.

Proof: Let ( S(t), V(t), I(t), R(t) ) be any solution of thesystem (2.3) with non-negative initial condition ( S(0), V(0)I(0), R(0) ), since N(t)= S(t)+V(t)+I(t)+R(t), then :

dt

dR

dt

dI

dt

dV

dt

dS

dt

dN

Which gives

ANdt

dN

Now, by solving the above linear differential equation, weget that the total population is asymptotically constant by:

AtN )(

Hence all the solutions of system (2.3) that initiate in4

are confined in the region:

0;:),,,( 4

ANRIVS

which is complete the proof.

3. Existence of Equilibrium points ofsystem(2.3)In this section, the existence of all possible equilibrium

points of system (2.3) is discussed. Clearly, if ,0I thenthe system (2.3) has an equilibrium point called a disease

free equilibrium point and denoted by )0,0,,( VSE

where:

)(

)(

)(

))((

AV

AS

(3.1)

However, if 0I then the system (2.3) has an endemicequilibrium point denoted by ),,,( 11111 RIVSE where

111 ,, IVS and 1R represent the positive solution of the

following set of equations:

0)(

0)()()(

0)()(

0)()()1(

RI

IVISIpA

VVIS

RVSSIAp

(3.2)

Straightforward computation to solve the above system oequations gives that:


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])()[(

)(])()[()(

])1()[(

])1()[()]()([

1

11

11

11

111

I

IIV

IR

V

IApV

V

IApIS

(3.3)

While 1I is a positive root for the following third order

equation:

0413212

311 DIDIDID (3.4)

here:

0)1()())((

))((

)]]()(

))[(()([

)2(

)([)()(

])()[(

)2())((

]2[

)]()[(

0)(

4

3

22

21

Ap

pA

pAD

pA

ApA

pAD

AD

D

Clearly, equation (3.4) has a unique positive root given by

1I and then 1E exists uniquely in Int. 4 if and only if at

least one of the following two conditions hold.

)2())((]2[(

)]()[( 2

A

(3.5a)

)]())([()(

)()]2)(([

pAApA (3.5b)

4. Local stability analysis of system(2.3)In this section, the local stability analysis of the equilibrium

points

E and 1E of the system (2.3) is studied as shown

in the following theorems.

Theorem (4.1): The disease free equilibrium point

)0,0,,( VSE of system (2.3) is locally asymptoticallystable if the following sufficient condition is satisfied:

)( VS (4.1)

Proof: The Jacobian matrix of system (2.3) at )( E can

be written as:

44

)(00

0)()(00

0)(

)(

)(

ija

VS

V

S

EJ

Clearly, )( EJ has the following eigenvalues:

)(

)()(

)(4)2(2

1

2

)2( 2,

R

I

VS

VS

here RIVSkk ,,,, represents the eigenvalue in k

direction. Obviously, S and V have negative real parts

while 0R . Therefore E is locally asymptotically stable

if and only if the eigenvalue 0I , which is satisfiedprovided that condition (4.1) holds and hence the proof iscomplete.

Theorem (4.2):Assume that, the endemic equilibrium poin

),,,( 11111 RIVSE of system (2.3) exists in the Int. 4

Then it is locally asymptotically stable if the followingcondition is satisfied:

)(2 11 VS (4.2)

Proof:The Jacobian matrix of system (2.3) at the endemic

equilibrium point 1E that denoted by )( 1EJ can be written:

44

)(00

0)()11()1(1

01)1(

1)()1(

ijb

VSII

VI

SI

Now, according to Gersgorin theorem if the followingcondition holds:

4

1ji

i

ijii bb

Then all eigenvalues of )( 1EJ exists in the region:

4

1

:

jii

ijii bbUCU

Therefore, according to the given condition (4.2) all the

eigenvalues of )( 1EJ exists in the left half plane and

hence, 1E is locally asymptotically stable.


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5. Global stability analysis of system (2.3)In this section, the global dynamics of system (2.3) isstudied with the help of Lyapunov function as shown in thefollowing theorems.

Theorem (5.1):Assume that, the disease free equilibrium

point

E of system (2.3) is locally asymptotically stable.

Then the basin of attraction of E , say4

)( EB , it isglobally asymptotically stable if satisfy the followingconditions:

VSVS

4

2

(5.1a)

IRS

SVSI

S

pAS

))(( (5.1b)

Proof:Consider the following positive definite function:

RIV

VVVV

S

SSSSW

lnln

1

Clearly, RRW 4

1 : is a continuously differentiable

function such that ,0)0,0,,(1 VSW and

)0,0,,(),,,(,0),,,(1 VSRIVSRIVSW . Further wehave:

dt

dR

dt

dI

dt

dV

V

VV

dt

dS

S

SS

dt

dW

1

By simplifying this equation we get:

IRS

S

VSIS

pASVV

V

VVSSVS

SSSdt

dW

))(()(

))(()(

2

21

Therefore, according to condition (5.1a) it is obtain that:

IRS

SVSI

S

pASVV

VSS

Sdt

dW

))((

)()(

2

1

Obviously 01 dt

dW for every initial points satisfying

condition (5.1b) and then 1W is a Lyapunov function

provided that conditions (5.1a)-(5.1b) hold. Thus

E is

globally asymptotically stable in the interior of ),( EB

which means that )( EB is the basin of attraction and that

complete the proof.

Theorem (5.2): Let the endemic equilibrium point 1E o

system (2.3) is locally asymptotically stable. Then it isglobally asymptotically stable provided that:

)( 11 VS (5.2a)

)(

94)(

11

211

VS

IIS

(5.2b)

)(3

22

I

I

(5.2c)

I3

22 (5.2d)

)(

)(3

2))((

11

211

VS

IVI

(5.2e)

)(3

211

2 VS (5.2h)

Proof:Consider the following positive definite function:

2222

21

21

21

21

2RRIIVVSS

W

Clearly, RRW 4

2: is a continuously differentiable

function such that 0),,,( 11112 RIVSW and

),,,(),,,(,0),,,( 11112 RIVSRIVSRIVSW Further, we

have:

dt

dRRR

dt

dIII

dt

dVVV

dt

dSSS

dt

dW)()()()( 1111

2


))(()(2

1

)(3

1))(()(

3

1

)(2

1))((

)(21)(

31

))(()(2

1

)(3

1))((

)(3

1)(

3

1

11342

144

21331123

2133

21221114

21442111

11122

122

21111113

2133

2111

2

RRIIqRRq

IIqIIVVqIIq

VVqRRSSq

RRqSSq

VVSSqVVq

SSqIISSq

IIqSSqdt

dW

With

341123

144422

121133

111311

,)(

,,,)(

,,)(

,,

qVIq

qqIq

qVSq

ISqIq


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Therefore, according to the conditions (5.2a)-(5.2h) weobtain that:

2

144

133

2

133

122

2

144

111

2

122

111

2

133

1112

)(2

)(3

)(3

)(2

)(2

)(3

)(2

)(3

)(3

)(3

RRq

IIq

IIq

VVq

RRq

SSq

VVq

SSq

IIq

SSq

dt

dW

Clearly, 02 dt

dW, and then 2W is a Lyapunov function

provided that the given conditions hold. Therefore, 1E is

globally asymptotically stable.

6. The local bifurcation analysis of system(2.3)In this section, the occurrence of local bifurcations (such assaddle-node, transcritical and pitchfork) near the equilibriumpoints of system (2.3) is studied in the following theorem.

Theorem (6.1):System (2.3) has a transcritical bifurcation

near the disease free equilibrium point

E , but neither

saddle-node bifurcation, nor pitchfork bifurcation can accrueat the parameter

)( VS (6.1)

Proof:It is easy to verify that the Jacobian matrix of system

(2.3) at ),( E can be written as:

)(00

0000

0)(

)(

),(

V

S

EDfJ

here )()(),()(),()( 442211 aaa .

Clearly, the third eigenvalue I in I-direction is zero

)0( I , further the eigenvector (sayTkkkkK ),,,( 4321 ) corresponding to 0I satisfy the

following:

KKJ

then 0KJ

From which we get that:

0)( 4321 kkSkk (6.2a)

0)( 321 kVkk (6.2b)

0)( 43 kk (6.2c)

So by solving the above system of equations we get:

343231 ;; zkkykkxkk Where:

)(

)(2

2)()()(

2

)()()(

z

VSVy

SVx

Here 3k be any non zero real number. Thus

3

3

3

3

zkk

yk

xk

K

Similarly the eigenvector TwwwwW 4321 ,,, tha

corresponding to 0I ofTJ

can be written:

0

)(00

0

00)(

00)(

4

3

2

1

w

w

w

w

VSWJ

T

This gives:

0

0

0

3wW

Here 3w is any non-zero real number. Now rewrite system

(2.3) in a vector form as:

)(Xfdt

dX

WhereTRIVSX ),,,( and Tfffff ),,,( 4321 with

4,3,2,1, ifi

are given in system (2.3), and then determine

fd

df we get that:

R

I

V

S

f then

0

0),(

V

S

Ef

Therefore:

0),( EfWT


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Consequently, according to Sotomayor theorem [10] the

system (2.3) has no saddle-node bifurcation near

E at

. Now in order to investigate the accruing of other types

of bifurcation, the derivative of f with respect to vector

X, say ),,( EDf is computed

1000

0100

0010

0001

),( EDf

So

0),( 33 wkKEDfWT

Again, according to Sotomayor theorem, if in addition to theabove, the following holds

0),(),(2 KKEfDWT

here ),( EDf is the Jacobian matrix at E and ,

then the system (2.3) possesses a transcritical bifurcationbut no pitch-fork bifurcation can occur. Now since we havethat:

0

)1)((

2

2

),(),(

33

23

23

2

kkyx

ky

kx

KKEfD

Therefore:

0)1)((),(),( 3332 wkkyxKKEfDWT

Then the system (2.3) has a transcritical bifurcation at

E

when the parameter passes through the bifurcation

value

.

7. SIRS epidemic model without vaccinationConsider the proposed system (2.3) in case of absence of

vaccine, that is when the parameter 0 , then system(2.3) will be reduced to the following subsystem.

),,()(

),,()()(

),,()()1(

3

2

1

RISgRIdt

dR

RISgISIpAdt

dI

RISgRSSIApdt

dS

(7.1)

Clearly, system (7.1) represents a simple SIRS epidemicmodel involving disease, transmitted by contact and byexternal sources in the environment, with partially infectiveimmigrants individuals. Obviously the above subsystem is

uniformly bounded and has two non negative equilibriumpoints: First the disease free equilibrium point that denoted

by ),0,0,( 22 SE which always exists where:

AS

2 (7.2)

Second the endemic equilibrium point of system (7.1) thatdenoted by ),,( 3333 RISE , which exists uniquely in the

Int. 0,0,0,),,( 33 RISRIS ,

where:

))((

)1(

3

33

I

IApS

(7.3a)

33

IR (7.3b)

3122

11

23 4

2

1

2

DDD

DD

DI

(7.3c)

here:

0)1(()()(

)()()(

0)(

3

22

1

AppAD

D

D

The local stability analysis of the above equilibrium points

2E and 3E of system (7.1) is studied as shown in the

following theorems.

Theorem (7.1): The disease free equilibrium poin)0,0,( 22 SE of system (7.1) is locally asymptotically

stable provided that:

2S (7.4a)

While it is a saddle point provided that:

2S (7.4b)

Proof: The Jacobian matrix of system (7.1) at 2E can be

written as:

)(0

0)(0)( 2

2

2

S

S

EJ

Clearly, )( 2EJ has the following eigenvalues:

0)(

;)(;0 2

R

IS S (7.5)


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here RISkk ,,, represents the eigenvalue in k-direction. Obviously, S and R are negative and then

2E is locally asymptotically stable if and only if the

eigenvalue 0I , which is satisfied provided thatcondition (7.4a) holds, while it is saddle point provided thatcondition (7.4b) holds and hence the proof is complete.

Theorem (7.2): The endemic equilibrium point

),,( 3333 RISE of system (7.1) is locally asymptoticallystable provided that:

3S (7.6a)

)(3 S (7.6b)

Proof: The Jacobian matrix of system (7.1) at the

endemic equilibrium point 3E can be written:

33

33

33

3

)(0

0)()(

)(

ijd

SISI

EJ

Then the characteristic equation of )( 3EJ is given by:

0322

13 (7.7)

here:

)(33

3322111

SI

ddd

33223311211222112 dddddddd

)(]))[(()(

)(

33

33

33123213213322113

SI

SI

dddddddd

Further:

332211

2211211233223322

3311331122112211

321

2

)()(

)()(

ddd

dddddddd

dddddddd

)()(2)(

)()()()(

)())(

)((

)(

33

3333

33

33

33

33

SI

SIIS

SS

II

SI

SI

Now according to (Routh-Hurtiz) criterion 3E will be locally

asymptotically stable provided that 01 ; 03 and

0321 . Clearly: 01 ; 03 and

0321 provided that conditions (7.6a)-(7.6b)

are hold. Hence the proof is completeThe global dynamics of system (7.1) is studied with the helpof Lyapunov function as shown in the following theorems.

Theorem (7.3):Assume that, the disease free equilibrium

point 2E of system (7.1) is locally asymptotically stable

Then the basin of attraction of 2E , say3

2)( EB

satisfy the following condition:

RSIpA )( (7.8)

Proof: Consider the following positive definite function:

RIdt

dF

F

SFL

S

S

2

21

Clearly, RRL 3

1 : is a continuously differentiable

function such that 0)0,0,( 21 SL , and

)0,0,(),,(,0),,( 21 SRISRISL .

Further, we have:

dt

dR

dt

dI

dt

dS

S

SS

dt

dL

21


22

21 )()()( S

S

RI

S

pARISS

Sdt

dL

Obviously, 0

1

dt

dL

for every initial point satisfying

condition (7.8) and then 1L is a Lyapunov function provided

that condition (7.8) holds. Thus 2E is globally

asymptotically stable in the interior of )( 2EB , which means

that )( 2EB is the basin of attraction and this complete the

proof.

Theorem (7.4): Let the endemic equilibrium point 3E o

system (7.1) is locally asymptotically stable. Then it isglobally asymptotically stable provided that the followingconditions are satisfied:

)()( 2 II (7.9a)

))((2 I (7.9b)

))((2 (7.9c)

Proof: Consider the following positive definite function:

222

23

23

23

2RRIISS

L


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Clearly, RRL 3

2: is a continuously differentiable

function such that 0),,( 3332 RISL and

),,(),,(,0),,( 3332 RISRISRISL . Further, wehave:

dt

dRRR

dt

dIII

dt

dSSS

dt

dL333

2 )(


))(()(2

1)(

2

1

))(()(2

1)(

2

1

))(()(2

1)(

2

1

33232

3332

322

33132

3332

311

33122

3222

3112

RRIIqRRqIIq

RRSSqRRqSSq

IISSqIIqSSqdt

dL

with

23331322

1211

;;;

;;

qqqq

IqIq

Therefore, according to the conditions (7.9a)-(7.9c), weobtain that:

2

333

322

2

333

311

2

322

3112

)(2

)(2

)(2

)(2

)(2

)(2

RRq

IIq

RRq

SSq

IIq

SSq

dt

dL

Clearly, 02 dt

dL, and then 2L is a Lyapunov function

provided that the given conditions (7.9a)-(7.9c) hold.

Therefore, 3E is globally asymptotically stable and hence

the proof is complete. The occurrence of local bifurcations(such as saddle-nod, transcritical and pitchfork) near theequilibrium point of system (7.1) is studied in the followingtheorem.

Theorem (7.5):System (7.1) has a transcritical bifurcation

near the disease free equilibrium point 2E , but neither

saddle-node bifurcation, nor pitchfork bifurcation can accrueat the parameter

2S

(7.10)

Proof: It is easy to check that the Jacobian matrix of

system (7.1) at ),( 2E can be written as:

)(0

000),(

2

2

S

EJJ

Clearly, the second eigenvalue I in I-direction is zero

)0( I , while S and R those are given in equation

(7.7) are negative. Further, the eigenvector (sayTkkkK ),,( 321 ) corresponding to 0I can be written

as:

2

2

2

yk

k

zk

K

here 2321 ; ykkzkk , 2k be any non zero rea

number with

z and

y . Similarly the

eigenvector TwwwW ),,( 321 corresponding to 0I oTJ can be written:

0

0

2wW

Here 2w is any non-zero real number. Now rewrite system

(7.1) in a vector form as:

)(Xgdt

dX

WhereTRISX ),,( and Tgggg ),,( 321 with

3,2,1, igi are given in system (7.1), and then determine

gd

dg we get that:

0),( 2 EgW

T

Consequently, according to Sotomayor theorem [10] the

system has no saddle-node bifurcation near 2E at

Now in order to investigate the accruing of other types of

bifurcation, the derivative of g with respect to vector X

say ),,( 2 EDg is computed and then we obtain that:

0),( 2222 wkSKEDgWT

Moreover, since

02),(),( 22222 kzwKKEgDWT


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Then the system (7.1) has a trnscritical bifurcation but not

pitch-fork bifurcation at 2E when the parameter passes

through the bifurcation value . The occurrence of Hopf-

bifurcation near the endemic equilibrium point of system(7.1) is also studied. Not that, it is well known that thenecessary conditions of the three dimensional dynamical

system (7.1) to have a Hopf bifurcation around 3E at a

specific parameter value ( sayq ) are given by

0)(1 q , 0)(2

q and

0)()()()( 321 qqqq , where 1 and 2

represent the coefficients of the characteristic equation ofthe dynamical system (7.1). Now since the conditions that

guarantee the positivity of 1 and 2 are the same

conditions that guarantee the positivity of 321 .Hence there is no possibility of occurrence of Hopfbifurcation.

8. Numerical analysis of systems (2.3) and(7.1):In this section, the global dynamics of systems (2.3) and(7.1) is studied numerically. The objectives of this study areconfirming our obtained analytical results and understandthe effects of immigration, existence of vaccine andexistence of the external sources for disease on thedynamics of SVIRS and SIRS epidemic models.Consequently, first system (2.3) is solved numerically fordifferent sets of initial conditions and for different sets ofparameters. It is observed that, for the following set ofhypothetical parameters that satisfies stability condition(4.1) of disease free equilibrium point, system (2.3) has a

globally asymptotically stable disease free equilibrium pointas shown in following figure.

5.0,8.0

,01.0,05.0,5.0,1.0

,0,0005.0,0,100,400

pA

(7.11)

Fig. 2, Time series of the solution of system (2.3). (a)trajectories of S, (b) trajectories of V, (c) trajectories of I

and (d) trajectories of R. The solid line refers to thetrajectory started at (1500,1200,1500,1500) while dotted

line refers to trajectory started at (500,400,500,900).

Fig. 3,Time series of the solution of system (2.3). (a)trajectories of S , (b) trajectories of V, (c) trajectories of I

and (d) trajectories of R. the solid line refers to thetrajectory started at (1500,1200,1000,900) while the dotted

line refers to the trajectory started at (700,800,500,100).

Fig. 4,Time series of the solution of system (2.3). (a) for

1.00 , (b) for 5.00 , (c) for 10 .


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Fig. 5, Time series of the solution of system (2.3). (a) for

01.0 , (b) for 2.0 , (c) for 9.0 .


1.0 , (b) for 2.0 , (c) for 1 .


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1.0 , (b) for 3.0 , (c) for 6.0 .

Fig. 8, Time series of the solution of system (2.3) for the

data given by (7.11) with varying . (a) for 05.0 , (b)for 01.0 .

Fig. 9, Time series of the solution of system (2.3) for the

data given by (7.11) with 001.0 and varying . (a)for 15.0 , (b) for 2.0 .

In the following the global dynamics of system (7.1) iscarried out. System (7.1) is solved numerically for thefollowing set of parameters, which satisfies condition (7.4a)and then the trajectories are drawn in Fig. 10.

5.0,8.0,1.0,0

,00015.0,0,100,400

pA

(7.12)

Fig. 10, Time series of the solution of system (7.1). (a)trajectories of S, (b) trajectories of I and (c) trajectories of

R , the solid starting at (1500,1200,1200) and dottedstarting at (500,500, 300).


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Fig. 11,Phase plot of system (7.1) starting from threedifferent initial points.

Fig. 12, Phase plot of system (7.1). (a) The solution

approaches to (5000, 0, 0) for 0 , (b) The solution

approaches to (3210.12,767.02,1022.79) for 1.0 , (c)The solution approaches to (1763.19,1387.2,1849.6) for

5.0 .

Fig. 13 Phase plot of system (7.1). (a) The solution

approaches to (800,4064.52,135.48) for 02.0 , (b) Thesolution approaches to (2666.67,1555.56,777.77

for 3.0 , (c) The solution approaches to (5000,0,0) fo9.0 .

In Fig. 2 shows that the solution of system (2.3)approaches asymptotically to the disease free equilibriumpoint has a globally asymptotically stable disease free

)0,0,15.3846,85.1153(E starting from two different initiapoints and this is confirming our obtained analytical resultsFig. 3shows clearly the convergence of system (2.3) to theendemic equilibrium poin

)61.758,96.568,43.2800,996.871(1E asymptotically fromtwo different initial points. This is indicates to occurrence oa transcritical bifurcation near the disease free equilibrium

point at a specific value of )001.0,0005.0( , so 0Ebecame unstable and the solution of system (2.3)


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approaches to 1E . In addition to that, the above two figures

refer to that increasing the contact rate between Sand I causes destabilizing to disease free equilibrium point andthe system approaches instead to the endemic point. Notethat, in the above figures (4-9), we will use the followingrepresentations: Solid line for describing trajectory of S;dashed line for describing trajectory of V; dash dot line for

describing trajectory of I; dotted line for describingtrajectory of Rand starting at (2000, 1500, 1000, 1250). InFig. 4 as the incidence rate of disease resulting from

external sources increases (through increasing 0 ), the

disease free equilibrium point of system (2.3) becomesunstable point and the trajectory of system (2.3)approaches asymptotically to the endemic equilibrium point.

In fact as 0 increases it is observed that the number of

susceptible and vaccinated individuals decrease and thenumber of recover and infected individuals increases. Fig. 5it is clear that as the rate of vaccine coverage increases theendemic equilibrium point of system (2.3) becomesunstable point and the trajectory of the system approachesasymptotically to the disease free equilibrium pointattendant that increasing in vaccined individuals anddecreasing in susceptible individuals. Fig. 6the increasing

(that is decreasing the lifetime of vaccine immunity)

destabilizes the disease free equilibrium point and then thesolution of system (2.3) approaches to endemic equilibriumpoint attended that increasing in the susceptible, infectedand recover individuals while the number of vaccinatedindividuals decreases. From Fig. 7 that, as the recoveryrate increases from 0.1 to 0.6 the endemic equilibrium pointof system (2.3) becomes unstable point and the trajectoryof system (2.3) approaches asymptotically to the diseasefree equilibrium point. But the number of susceptible andvaccinated individuals increases while the number of the

infected and recover individuals decreases. In Fig. 8however, , increases the parameter more than 0.1

keeping other parameters fixed as in (7.11) with 001.0 causes transferring in the stability of system (2.3) fromendemic equilibrium point to disease free equilibrium pointas shown in Fig. 9. Therefore, the death rate due to thedisease plays a vital role as bifurcation parameter of system(2.3). Fig. 10 Shows that the solution of system (7.1)approaches asymptotically to the disease free equilibrium

point )0,0,5000(2E from two different initial data. Fig. 11shows the existence of a unique endemic equilibrium pointof system (7.1), which is globally asymptotically stable. Fig.12 the external incidence rate increases the disease free

equilibrium point of system (7.1) becomes unstable pointand the trajectory of system (7.1) approachesasymptotically to the endemic equilibrium point, and thenthe number of susceptible individuals decrease while thenumber of the infected and recover individuals increases.Fig. 13the recovery rate increases the endemic equilibriumpoint of system (7.1) becomes unstable point and thetrajectory of system (7.1) approaches asymptotically to thedisease free equilibrium point attendant that increasing thenumber of susceptible individuals and decreasing in thenumbers of the infected and recover individuals.

9. Conclusion and discussion:In this paper, two mathematical models have beenproposed and analyzed. The objective is to study the effecof immigrants, existence and nonexistence vaccine, andthen existence of external sources of the disease in theenvironment on the dynamical behavior of SVIRS and SIRSepidemic models. The existence and the stability analysis oall possible equilibrium points are studied analytically as

well as numerically. It is observed that system (2.3) andsystem (7.1) have transcritical bifurcation near the diseasefree equilibrium point, but neither saddle node nor pitchforkbifurcation can accrue. Further both the systems (2.3) and(7.1) do not have Hopf bifurcation near the endemicequilibrium point. Finally according to the numericallysimulation the following results are obtained:

1. Both the systems (2.3) and (7.1) do not haveperiodic dynamic, instead it they approacheither to the disease free equilibrium point oelse to endemic equilibrium point.

2. As the number of the infected immigranindividuals and the incidence rate of disease(external incidence rate or contact incidence

rate) increase, the asymptotic behavior of thesystems (2.3) and (7.1) transfer fromapproaching to disease free equilibrium point tothe endemic equilibrium point.

3. As the lifetime of vaccine immunity decreases

(the losing vaccine immunity rate )(

increases), then the disease free equilibriumpoint of system (2.3) becomes unstable and thesolution will approaches to the endemicequilibrium point. Further, similar result isobtained in systems (2.3) and (7.1) when thenatural death rate decreases.

4. As the recovery rates in the systems (2.3) and(7.1) increase then the solution in both thesystems will be transfer from stability aendemic equilibrium point to stability at diseasefree equilibrium point. Further, similar result isobtained in case of system (2.3) when thevaccine coverage rate increases.

5. Finally, changing the lifetime of removaindividual's immunity in both the system (2.3)and (7.1) do not have vital effect on thedynamical behavior of each of them.

References:[1]W.O. Kermack, A.G. Mckendrick, A contribution to the

mathematical theory of epidemics, Proc. R. Soc

London a 115 (1927) 700-721.

[2]R.M. Anderson, R. M. May, Population Biology oInfectious Disease, Springer Verlag, New York(1982).

[3]F. Brauer, C. Castillo-Chavez, Mathematical Models inPopulation Biology and Epidemiology, Springer, NewYork, (2001).

[4]O. Diekmann, J. A. P. Heesterbeek, MathematicaEpidemiology of Infectious Disease: Model BuildingAnalysis and Interpretation, Wiley, New York, (2000).


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[5]H.W. Hethcote, The mathematics of infectious disease,SIAM Rev. 42 (2000) 599-653.

[6]Kribs-Zaleta, C.M., Velasco-Hernandez, J.X.,(2000). Asimple vaccination model with multiple endemic states.Math. Biosci. 164: 183-201.

[7]Alexander, M.E., Bowman, C., Moghadas, S.M.,

Summers, R., Gumel, A.B., Sahai, B.M., A vaccinationmodel for transmission dynamics of influenza. SIAM J.Appl. Dyn. Syst. (2004), 3:503-524.

[8]Shurowq k. Shafeeq, The effect of treatment,immigrants and vaccinated on the dynamic of SISepidemic model. M.Sc. thesis. Department ofMathematics, College of Science, University ofBaghdad. Baghdad, Iraq (2011).

[9]Krishna pada Das, Shovonlal Roy and J.Chattopadhyay, Biosystem. 95 (2009) 188-199.

[10]Sotomayor, J., Generic bifurcations of dynamical

systems, in dynamical systems, M. M. Peixoto, NewYork, academic press (1973).

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Modeling and Analysis of an Svirs Epidemic Model Involving External Sources of Disease

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