gaoshujingASB5-8-2009.pdf

8/10/2019 gaoshujingASB5-8-2009.pdf

1/15

Advanced Studies in Biology, Vol. 1, 2009, no. 7, 307 - 321

Pulse Vaccination Strategy in an Epidemic

Model with Time Delays and Nonlinear IncidenceQin Zou, Shujing Gao 0 , Qi Zhong

Key Laboratory of Jiangxi Province for Numerical Simulation and EmulationTechniques, Gannan Normal University, Ganzhou 341000, P.R. China

Abstract

In this paper, we formulate an SEIRS epidemic model with two

time delays and pulse vaccination. By estabishing stroboscopic map,we obtain the exact infection free periodic solution of the impulsive epi-demic system. According to the comparison arguments, we obtain thesufficient conditions of global attractivity of the infection free periodicsolution. Moreover, our results show that a large vaccination or a shortpulse of vaccination or a long latent period or a long immune periodwill lead to the extinction of the disease. The permance of the model isalso discussed.

Keywords: SEIRS epidemic model, Pulse vaccination, Nonlinear inci-dence, Time delay, Permanence

1 Introduction

Controlling infectious diseases has been an increasingly complex issue for everycountries in recent years. Many scholars have investigated and studied lots of epidemic models of ordinary differential equations. The main aim of studyingepidemic models is to help improve our understanding of the global dynamicsof the spread of infectious diseases. By using the compartmental approach, thedynamics of the SIR and SIRS epidemic model have been extensively analyzed,where S (t), I (t) and R(t) denote the number of the susceptible, infective and

recovered at time t, respectively. However, many diseases incubate inside thehosts for a period of time before the hosts become infectious. The resultingmodels are of SEIR or SEIRS types, respectively, depending on whether theacquired immunity is permanent or otherwise, where E (t) denote the numberof exposed at time t. Moreover, considering some factors of the spread of

0 Corresponding author. E-mail address: [email protected] .


2/15


3/15

SEIRS epidemic model, Time delay, Permanence 309

rate kI lS/ (1 + I h ), wherel and h are nonnegative constants.This kinds of incidence rate includes the behavioral change and crowding effect of the infec-tive individuals.It was used by many authors, see, for example, [Alexander etal.,2004; Derrick et al.,1993; Hethcote,2000; Hethcote et al.,1991; Hethcote etal.,1989], etc. In the literature [Ruan et al.,2003], Ruan and Wang studied theglobal dynamics of an epidemic model with nonlinear incidence rate of satu-rated mass action, which is kI 2S/ (1 + I 2). They obtained many dynamicalbehaviors of the model such as a limit cycle, two limit cycles and homoclinicloop, etc. And in the literature [Xiao et al.,2007] and [Cai et al.,2009], Xiaoet al. analysed the epidemic models with nonmonotone incidence rate,whichis IS/ (1 + I 2) and it can be used to interpret the psychological effects .Because in the initial stage the number of the infective is small people mayignore the epidemic, lots of effective contacts between the infective individualsand suspectable. And when the number of the infective is getting larger andlarger, g(I ) is decreasing since many protection measures could be taken by

the susceptible individuals.Generally, death during a latent period and temporary immunity periodshould be considered in modeling, which is called the phenomena of time de-lay . Therefore, many authors in their literatures [Beretta et al.,2001; Berettaet al.,1995; Cooke et al.,1996; Mukherjee et al.,1996; Wang, 2002] have studiedepidemic models with time delay. But epidemic models with time delays andpulse vaccination are seldom studied. In the present paper, we formulate andanalyze an SEIRS disease transmission model with pulse vaccination and twotime delays.

The organization of this paper is following. In the next section, we for-mulate the SEIRS epidemic model with pulse vaccination and time delays. InSection 3, the global attractivity conditions of the disease-free periodic solu-tion is presented. The sufficient conditions for the permanence of the SEIRSmodel is obtained in Section 4. In Section 5, we give some discussion. Themain purpose of this paper is to obtain sufficient conditions that the diseasedies out. The second purpose of this paper is to study the role of incubationin disease transmission.

2 Model Formulation and Preliminary

In this section, we rstly formulate an SEIRS epidemic model with pulsevaccination and time delays. Then we introduce some lemmas which are usefulin proving the global attractivity of infection free periodic solution and thepermanence of the system.

Comparing with bilinear and standard incidence, saturation incidence maybe more suitable for our real world. Zhang and Teng [Zhang et al.,2008]introduced and studied the following pulse vaccination delayed SEIRS epidemic


4/15

310 Qin Zou, Shujing Gao and Qi Zhong

model with saturation incidence

S (t) = b bS (t) S (t)I (t)1 + mS (t)

+ R(t)

E (t) = S (t)I (t)

1 + mS (t)

S (t )I (t )

1 + mS (t ) e b bE (t)

I (t) = S (t )I (t )

1 + mS (t ) e b (b + + )I (t)

R (t) = I (t) bR(t) R(t)

t = n, n N,

S (t+ ) = (1 )S (t)E (t+ ) = E (t)I (t+ ) = I (t)R(t+ ) = R(t) + S (t)

t = n.

(1)They investigated the existence of the disease-free periodic solution and ob-

tained its exact expression. Further, they established the sufficient conditionsof global attractivity of the disease-free periodic solution and the permanenceof disease.

In the model (1), they only considered the latent period of the disease(). The form of incidence is S (t)I (t)/ (1 + mS (t)), which is saturated withthe susceptible. However, many epidemic diseases have the latent period. Inthis paper, we formulate an pulse vaccination SEIRS epidemic model with twotime delays which includes the latent period and the latent period. And theincidence rate in our model is kI lS/ (1 + I h ). We study the following model:

S (t) = b S (t) kS (t)I l(t)

1 + I h

(t) + I (t )e

E (t) = kS (t)I l(t)1 + I h (t)

kS (t )I l(t )

1 + I h (t ) e E (t)

I (t) = kS (t )I l(t )

1 + I h (t ) e ( + )I (t)

R (t) = I (t) I (t )e R(t)

t = nT, n N,

S (t+ ) = (1 )S (t)E (t+ ) = E (t)I (t+ ) = I (t)R(t+ ) = R(t) + S (t)

t = nT, n N,

(2)where parameters b, , , , , l,hand are positive constants, b is the re-cruitment rate of susceptible individuals, represents the natural death rateof the population, is the recovery rate from the infected individuals, is theimmune period of the population, is the latent period of the disease.

In this paper, we will focus on the case when l = 1. Since the rst andthird equations of system (2) do not include the variables E (t) and R(t) the


5/15


dynamics of system (2) is determined by the following subsystem:

S (t) = b S (t) kS (t)I (t)1 + I h (t)

+ I (t )e

I (t) = kS (t )I (t )

1 + I h (t ) e ( + )I (t)

t = nT, n N,

S (t+ ) = (1 )S (t)I (t+ ) = I (t) t = nT, n N,

(3)

Set j = max {, }. The initial conditions of (3) is given as

S ( ) = 1( ), I ( ) = 2( ), [ j, 0], i (0) > 0, i = 1, 2 (4)

where = ( 1, 2) C and C is the space of all piecewise continuous func-tions : [ j, 0] R+2 , R+2 = {X R2 : X 0}. has the rst kind of discontinuity points at nT (n N ) which are continuous from the left, i.e.,

( nT

) = ( nT ). C is the Banach space with uniform norm

= sup[ j, 0]

{|1|, |2|} .

The solution of system (3) is a piecewise continuous function Y : R+ R+2 ,Y (t) is continuous on (nT, (n + 1) T ] and Y (nT + ) = lim

t nT +Y (t) exists.

Let N (t) = S (t) + E (t) + I (t) + R(t). By adding the equations in model(2), we obtain

N (t) = b N (t). (5)

So the total population size may vary in time and N (t) is continuous on

t [0, + ). From(5), we have limsupt +

N (t) b

. Since S (t)|S =0 > 0 and

I (t)| I =0 = 0, for t = nT, n N. Moreover, S (nT + ) = (1 )S (nT ), I (nT + ) =I (nT ) for n N. Hence, we obtain the following lemma.

Lemma 2.1 is a positive invariant set of system (3), where = {(S, I )

R+2 : 0 S + I } and = b

+ , > 0 is sufficiently small.Lemma 2.2 [Gao et al., 2006] Consider the following impulsive differential

equation u (t) = a bu(t), t = nT,u(t+ ) = (1 )u(t), t = nT, (6)

where a > 0, b > 0, 0 < < 1. Then there exists a unique positive periodicsolution of system (6)

ue(t) = ab + ( u

ab

)e b(t nT ) , nT < t (n + 1) T


6/15


which is globally asymptotically stable, where u = ab

(1 )(1 e bT )1 (1 )e bT

.

Lemma 2.3 [Kuang, 1993; Xiao et al.,2001] Consider the following delaydifferential equation:

x (t) = a1x(t ) a2x(t)where a1, a2 and are all positive constants and x(t) > 0 for t [ , 0]. Wehave:

(i) If a1 < a 2, then limt x(t) = 0;

(ii) If a1 > a 2, then limt x(t) = + .

3 Global attractivity

In this section, we discuss the global attractivity of infection free periodic so-lution. The technique of the proofs is to use comparison arguments. Firstly,we analyze the existence of the infection free periodic solution of system (3),in which infective individuals are entirely absent from the population perma-nently, i.e., I (t) = 0 for all t 0. So we know that the growth of the susceptiblein the time-interval nT < t (n + 1) T must satisfy

S (t) = b S (t), t = nT,S (t+ ) = (1 )S (t), t = nT. (7)

By Lemma 2.2, we derive the periodic solution of system (7)

S e(t) = b 1

1 (1 )e T e (t nT ) , nT < t (n + 1) T

which is globally asymptotically stable. Therefore, system (3) has a uniqueinfection free periodic solution (S e(t), 0).

Now we give the conditions which assure the global attractivity of theinfection free periodic solution (S e(t), 0).

Theorem 3.1 If R < 1, then the infection free periodic solution ( S e(t), 0) of system (3) is global attractive on , where

R = ke

(b + e

)(1 e T

) ( + )(1 (1 )e T )

.

Proof. Since R < 1, we can choose 0 > 0 sufficiently small such that

ke < + , (8)


7/15


where = (b + e )(1 e T )

(1 (1 )e T ) + 0.

We derive from the rst equation of system (3) that

S (t) < (b + e ) S (t).

So we estabish the following comparison impulsive system

u (t) = ( b + e ) u(t), t = nT,u(t+ ) = (1 )u(t), t = nT. (9)

In view of Lemma 2.2, we obtain the unique periodic solution of system (9)

ue(t) = b + e

1

1 (1 )e T

e (t nT ) , nT < t (n + 1) T

which is globally asymptotically stable.Let (S (t), I (t)) be the solution of system (3) and S ( ) = 1( ), [ j, 0],

u(t) be the solution of system (9) with initial condition u( ) = 1( ), [ j, 0]. By the comparison theorem of impulsive differential equations [Bainovet al.,1993; Lakshmikantham et al.,1989], there exists an integer n1 > 0 suchthat

S (t) u(t) < ue(t) + 0, nT < t (n + 1) T, n > n 1,which yieldsS (t) n 1.

It follows from the second equation of system (3) that

I (t) < ke I (t ) ( + )I (t) t > nT + , n > n 1.

Then we consider the following auxiliary system

v (t) = ke v(t ) ( + )v(t), (10)

By Lemma 2.3,we havelim

t v(t) = 0 .

According to the comparison theorem and nonnegative of I (t), we obtain that

limt

I (t) = 0 . (11)

Hence, for 0 > 0 sufficiently small there is an integer n2 > n 1 (where n2T >n1T + ) such that if t > n 2T, I (t) < 0.


8/15


From the rst equation of system (3), we have that for t > n 2T +

S (t) = b S (t) kS (t)I (t)1 + I h (t)

+ I (t )e

> b ( + k 0)S (t)

andS (t) < (b + e 0) S (t).

Then, we consider the following two comparison impulsive differential equa-tions for t > n 2T + and n > n 2,

u1(t) = b ( + k 0)u1(t), t = nT,u1(t+ ) = (1 )u1(t), t = nT,

(12)

and

u2(t) = ( b + e 0) u2(t), t = nT,

u2(t+ ) = (1 )u2(t). t = nT. (13)

In view of Lemma 2.2, we obtain that the unique periodic solutions of system(12)

u1e (t) = b

k 0 + 1

1 (1 )e (k 0 + )T

e (k 0 + )( t nT ) , nT < t (n+1) T,

and the unique periodic solutions of system (13)

u2e (t) = b + e 0

1

1 (1 )e T e (t nT )

, nT < t (n + 1) T,

respectively. And they are globally asymptotically stable.According to the comparison theorem of impulsive differential equations,

there exists an integer n3 > n 2 such that n3 > n 2 + and

u1e (t) 0 < S (t) < u2e (t) + 0, nT < t (n + 1) T, n > n 3. (14)Because 0 can be arbitrarily small, from (14) we obtain thatlim

t S (t) =

S e(t) (15)

It follows from (11) and (15) that the infection free periodic solution ( S e(t), 0)of system (3) is globally attractive. The completes the proof.

According to Theorem 3.1, we can easily obtain the following results.

Corollary 3.1 If ke (b + e )

+ , then the infection free periodic


9/15


solution (S e(t), 0) of system (2.2) is globally attractive.

Corollary 3.2 Assume that ke (b + e )

> + . Then the infection

free periodic solution (S e(t), 0) of system (2.2) is globally attractive provided

that > or T < T or > where

=ke (b + e )

( + ) 1 (eT 1),

T = 1 ln 1 +

( + )ke (b + e ) ( + )

,

= 1 ln

k(b + e )(1 e T ) (1 (1 )e T )( + )

.

Corollary 3.3 If > , then the infection free periodic solution ( S e(t), 0)of system (2.2) is globally attractive, where

= 1 ln

ke (1 e T ) ( + )(1 (1 )e T ) kbe (1 e T )

.

4 Permanence

In this section we say the disease is endemic if the infectious population persistsabove a certain threshold level for sufficiently large time. Before starting ourtheorem, we give the following denitions and lemma.

Denition 4.1 System (3) is said to be uniformly persistent if there is an > 0(independent of the initial conditions) such that every solution S (t), I (t)with initial conditions (4) of system (3) satises

lim inf t

S (t) , liminf t

I (t) .

Denition 4.2 System (3) is said to be permanent if there exists a compactregion 0 such that every solution with initial conditions (4) of system(3) will eventually enter and remain the region 0.

Denote two quantities

R = kbe (1 )(1 e T )

( + )(1 + h )(1 (1 )e T ) ,

and I = k

(R 1) = k

kbe (1 )(1 e T ) ( + )(1 + h )(1 (1 )e T )

1 .

Lemma 4.1 If R > 1, then for any t0 > 0, it is impossible that I (t) < I for


10/15


all t t0.Proof. Suppose the contrary. Then there is a t0 > 0 such that I (t) < I for all t t0. According to any positive solution ( S (t), I (t)) of system (3), wedene

V (t) = I (t) + ke

t

t

I (u)S (u)

1 + I h (u)du (16)

The second equation of system (3) can be rewritten as

I (t) =ke S (t)1 + I h (t)

( + ) I (t) ke ddt tt I (u)S (u)1 + I h (u) du.

We get the derivative of V along the solution of system (3)

V (t) = ( + )I (t) ke S (t)

(1 + I h (t))( + ) 1 (17)

From the rst equation of system (3) we have that

S (t) > b S (t) kI S (t) = b ( + kI )S (t), t t0.

Then we estabish the following auxiliary system for t t0 :

w (t) = b ( + kI )w(t), t = nT,w(t+ ) = (1 )w(t), t = nT. (18)

By Lemma 2.2, we derive that for nT < t (n + 1) T ,

we(t) = b

+ kI +

b + kI

(1 )(1 e (+ kI )T )

1 (1 )e (+ kI )T

b + kI

e (+ kI )( t nT )

is globally asymptotically stable. In view of the comparison theorem of impul-sive differential equation, we obtain that there exists an integer t1(t1 > t 0 + )and sufficiently small > 0 such that

S (t) > we(t) , t t1.Sincewe(t) >

b + kI

(1 )(1 e (+ kI )T )

1 (1 )e (+ kI )T ,

then we get

S (t) > b + kI

(1 )(1 e (+ kI )T

)1 (1 )e (+ kI )T

= , t t1. (19)

From(17) and (19), we have

V (t) > ( + )I (t) ke

(1 + h )( + ) 1), t t1. (20)


11/15


Since R < 1, it is easy to get I > 0. Moreover, there exists sufficiently small > 0 such that

ke (1 + h )( + )

> 1. (21)

From the above results, we obtain that

V (t) > ( + )I (t)( ke

(1 + h )( + ) 1) > 0, t t1. (22)

Dene m = mint [t 1 ,t 1 + ]

I (t). Next we want to prove that I (t) m for all

t t1. Suppose that this claim is not valid. Then there exists T such thatI (t) m for t1 t t1 + + T , I (t1 + + T ) = m and I (t1 + + T ) 0.We can derive from the second equation of system (3) and (19) that

I (t1 + + T ) = ke I (t1 + + T )S (t1 + + T )

1 + I h

(t1 + + T ) ( + )I (t1 + + T )

> ke

(1 + h )( + ) 1 ( + )m.

From (21), we get I (t1 + + T ) > 0. This contradicts I (t1 + + T ) 0.Therefore, I (t) m for all t t1. From (22), we have

V (t) > ( + ) ke

(1 + h )( + ) 1 m > 0, t t1,

which yields that as t , V (t) . However, from (16) we obtain thatV (t) b + ke

2. This is a contradiction. Therefore, for any t0 > 0, it isimpossible that I (t) < I for all t t0. The proof of Lemma 4.1 is complete.

Theorem 4.1 If R > 1, there exists a positive constant p such that anypositive solution of system (3) satises I (t) p, for t large enough.Proof. Based on the result of Lemma 4.1, we will discuss the following twocases:

1. I (t) I , for t large enough; 2. I (t) oscillates about I , for t largeenough.

Obviously, we only need to consider the second case. We assume that thereexists sufficiently large t1 and t2 such that

I (t1) = I (t2) = I , I (t) I , t [t1, t2], S (t) > , t [t1, t2].

Since the positive solutions of (3) are ultimately bounded and I (t) is noteffected by impulses, I (t) is uniformly continuous. Therefore there exists a


12/15


positive constant such that I (t) > I 2 fort [t1, t1 + ], where satises < and is independent of the choice of t1 and t2. Set p = min { I 2 , p },where p = I e (+ ) . Now we discuss the following three subcases.

Case I. If t2 t1 , then it is evident that I (t) p for t large enough.Case II. If < t 2 t1 , then from the second equation of system (2.2)

we have I (t) > ( + )I (t). Since I (t1) = I , by the comparison theorem weobtain that

I (t) > I e (+ )( t t ) I e (+ ) = p , t [t1, t2].

Considering I (t) > I 2 , for t [t1, t1 + ], we obtain I (t) p, for t [t1, t2].Case III. If t2 t1 > , then by the same analysis of subcase (2), we have

I (t) p, for t [t1, t1 + ]. Next, we will prove that I (t) p is still validfor t [t1 + , t2]. Suppose the contrary. Then there exists T 0 such thatI (t) p for t [t1, t1 + +

T ], I (t1 + +

T ) = p and I (t1 + +

T ) 0. We

can derive from the second equation of system (3) that

I (t1 + + T ) = ke I (t1 + + T )S (t1 + + T )1 + I h (t1 + + T ) ( + )I (t1 + + T )> ( ke

(1 + ( b )h )( + ) 1)( + ) p > 0.

This contradicts I (t1 + + T ) 0. Therefore, I (t) p for t [t1, t2].Moreover, in view of Lemma 4.1 we obtain that for any t0 > 0, it is impossiblethat I (t) < I for all t t0. Hence, if R > 1, any positive solution of system(3) satises I (t) p, for t large enough. This completes the proof.

Theorem 4.2 If R > 1, system (3) is permanent.Proof. Let (S (t), I (t)) be any solution of system (3). We derive from the rstequation of system (3) that

S (t) b ( + k)S (t).

Then we estabish the following comparison system

x (t) = b ( + k)x(t) t = nT,x(t+ ) = (1 )x(t) t = nT. (23)

By the similar analysis in the proof of Theorem 3.1, we obtain that lim t S (t) M, where

M = b + k

(1 )(1 e (+ k )T )1 (1 )e (+ k )T

1

( 1 is sufficiently small.)


13/15


Dene 0 = {(S, I )|S M, I p, S + I }. From the above discuss andTheorem 4.1, we know that 0 is global attractor in and 0 is a positivelyinvariant set. Therefore,system (3) is permanent.

Corollary 4.1 Assume that < or < , then system (3) is permanent,where

= 1 ( ( + )(1 + h )

kbe (1 e T ) + ( + )(1 + ( b )h )e T ,

= 1 ln

( + )(1 + h )(1 (1 )e T )kb(1 )(1 e T )

.

5 Discussion

In this paper, we have studied the delayed SEIRS epidemic model with

pulse vaccination and nonlinear incidence. We obtained two thresholds R andR (see Theorem 3.1 and 4.1) and further discussed if R < 1 then the diseasewill be extinct, if R > 1 then the disease will be permanent which means thatafter some period of time the disease will become endemic. Corollary 3.2 im-plies that the disease will disappear if the pulse vaccination rate is larger than or the length of latent period exceeds or the length of pulse vaccinationperiod is smaller than T . From Corollary 3.3, we obtain that if the lengthof immune period is larger than , the disease will will lead to the extinc-tion. And Corollary 4.1 show that the disease will be uniformly persistent if the pulse vaccination rate is smaller than or the length of latent period issmaller than .

We have discussed two cases: (1)R < 1, (2)R > 1. When R 1 R ,the dynamical behavior of system (3) has not been clear. These works will beleft as our future consideration.

ACKNOWLEDGEMENTS. The research has been partially supportedby The Natural Science Foundation of China (10971037), The National KeyTechnologies R & D Program of China(2008BAI68B01), The Natural ScienceFoundation of Jiangxi Province (2008GZS0027) and The Foundation of Edu-cation Committee of Jiangxi (GJJ0931).

References

[1] Z. Agur, L. Cojocaru, G. Mazor, et al, Pulse mass measles vaccinationacross age cohorts, Proc Nat Acad Sci USA, 90 (1993), 11698 - 11702.


14/15


[2] M.E. Alexander, S.M. Moghadas, Periodicity in an epidemic model witha generalized non-linear incidence, Math. Biosci 189 (2004), 75.

[3] D.D. Bainov, P.S. Simeonov, Impulsive differential equations: periodicsolutions and applications, New York: Longman Scientic and Technical,

1993.[4] E. Beretta, T. Hara, W. Ma, Y. Takeuchi, Global asymptotic stability of

an SIR epidemic model with distributed time delay, Nonlinear Anal, 47(2001), 4107 - 4115.

[5] E. Beretta, Y. Takeuchi, Global stability of an SIR epidemic model withtime delays, J. Math. Biol, 33 (1995), 250 - 260.

[6] L. Cai, J. Li, X.Li, Analysis of an SIS epidemic model with nonmonotoneincidence rate, Journal of Biomathematics, 24 (2009), 193 - 200.

[7] V. Capasso, G. Serio, A generalization of the Kermack-Mckendrick deter-ministic epidemic model, Math. Biosci, 42 (1978), 43 - 61.

[8] K.L. Cooke, P. Van Den Driessche, Analysis of an SEIRS epidemic modelwith two delays, J. Math. Biol, 35 (1996), 240 - 260.

[9] W.R. Derrick, P. van den Driessche, A disease transmission model in anonconstant population, J. Math. Biol, 31 (1993), 495.

[10] S. Gao, L. Chen, The effect of seasonal harvesting on a single-speciesdiscrete model with stage structure and birth pulses, Chaos, Solitons and Fractals, 24 (2005), 1013 - 1023.

[11] S. Gao, L. Chen, J.J. Nieto, A. Torres, Analysis of a delayed epi-demic model with pulse vaccination and saturation incidence, Vaccine,24 (2006), 6037 - 6045.

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

[13] H.W. Hethcote, P. van den Driessche, Some epidemiological models withnonlinear incidence, J. Math. Biol, 29 (1991), 271.

[14] H.W. Hethcote, S.A. Levin, Periodicity in epidemiological models, in: L.Gross, T.G. Hallam, S.A. Levin (Eds.), Applied Mathematical Ecology,Springer-Verlag, Berlin, (1989), 193.

[15] Y. Jiang, H. Wei, X. Song, L. Mei, G. Su, S. Qiu. Global attractivity andpermanence of a delayed SVEIR epidemic model with pulse vaccinationand saturation incidence, Applied Mathematics and Computation, 213(2009), 312 - 321.


15/15


[16] Y. Kuang, Delay Differential Equation with Application in PopulationDynamics, Academic Press, New York, 1993.

[17] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsivedifferential equations. Singapore, World Scientic, 1989.

[18] W. Liu, S.A. Levin, Y. Iwasa, Inuence of nonlinear incidence rates uponthe behavior of SIRS epidemiological models, J. Math. Biol, 23 (1986),187 - 204.

[19] D. Mukherjee. Stability analysis of an S-I epidemic model with time delay,Mathl. Comput. Modelling, 24 (1996), 63 - 68.

[20] M. Ramsay, N. Gay, E. Miller, The epidemiology of measles in Englandand Wales: Rationale for 1994 national vaccination, campaign, Commun Dis Rep, 4 (1994), 141 - 146.

[21] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with non-linear incidence rate, J. Differen. Equat , 188 (2003), 135 - 163.

[22] B. Shulgin, L. Stone, Z. Agur, Pulse vaccination strategy in the SIR epi-demic model, Bull Math Biol, 60 (1998), 1123 - 1148.

[23] X. Song, Y. Jiang, H. Wei. Analysis of a saturation incidence SVEIRSepidemic model with pulse and two time delays, Applied Mathematics and Computation, 214 (2009), 381 - 390.

[24] L. Stone, B. Shulgin, Z. Agur, Theoretical examination of the pulse vacci-

nation policy in the SIR epidemic model, Math Comput Modell, 31 (2000),207 - 215.

[25] W. Wang. Global behavior of an SEIRS epidemic model with time delays,Appl. Math. Lett 15 (2002), 423 - 428.

[26] D. Xiao, S. Ruan, Global Analysis of an Epidemic Model with Nonmono-tone Incidence Rate, Mathematical Biosciences, 188 (2007), 419 - 429.

[27] Y. Xiao, L. Chen, Modelling and analysis of a predator-prey model withdisease in the prey, Math. Biosci, 171 (2001), 59 - 82.

[28] T. Zhang, Z. Teng, Pulse vaccination delayed SEIRS epidemic model withsaturation incidence, Applied Mathematical Modelling, 32 (2008), 1403 -1416.

Received: September, 2009

Documents

gaoshujingASB5-8-2009.pdf