AM20100100017_88545772.pdf

Applied Mathematics, 2011, 2, 57-63 doi:10.4236/am.2011.21007 Published Online January 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay

Youquan Luo, Shujing Gao, Shuixian Yan Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques,

Gannan Normal University, Ganzhou, China E-mail: [email protected]

Received August 1, 2010; revised November 10, 2010; accepted November 15, 2010

Abstract In this paper, an impulsive epidemic model with time delay is proposed, which susceptible population is di-vided into two groups: high risk susceptibles and non-high risk susceptibles. We introduce two thresholds R1, R2 and demonstrate that the disease will be extinct if 1 1R and persistent if 2 1R . Our results show that

larger pulse vaccination rates or a shorter the period of pulsing will lead to the eradication of the disease. The conclusions are confirmed by numerical simulations.

Keywords: High Risk Susceptible, Non-High Risk Susceptible, Pulse Vaccination, Extinction

1. Introduction Infectious diseases have tremendous influence on human life. Every year, millions of people die of various infec-tious diseases. Controlling infectious diseases has been an increasingly complex issue in recent years [1]. Over the last fifty years, many scholars have payed great at-tention to construct mathematical models to describe the spread of infectious diseases. See the literatures [2-8], the books [9-11] and the references therein. In the clas-sical epidemiological models, a population of total size N is divided into S (susceptible numbers), I (infective numbers), or S, I and R (recovered numbers) or S, E (exposed numbers), I and R, and corresponding epidemiological models such as SI, SIS, SIR, SIRS, SIER and SEIRS are constructed. All these models are extensions of the SIR model elaborated by Kermack and McKendrick in 1927 [12]. Anderson and May [5,9] discussed the spreading nature of biological viruses, parasites etc. leading to in-fectious diseases in human population through several epidemic models. Cooke and Driessche [8] investigated an SEIRS model with the latent period and the immune period. The consideration of the latent period and the immune period gave rise to models with the incorpora-tion of delays and integral equation formulations.

However, owing to the physical health status, age and other factors, susceptibles population show different in-

fective to a infectious disease. In this paper, we divide the susceptible population into two groups: nonhigh risk susceptibles (S1) and high risk susceptibles (S2), such that individuals in each group have homogeneous susceptibil-ity, but the susceptibilities of individuals from different groups are distinct. In this paper, we propose a new SIR epidemic model, which two noninteracting susceptible subclasses, the nonlinear incidence pS I , time delay and pulse vaccination are considered. The main purpose of this paper is to study the dynamical behavior of the model and establish sufficient conditions that the disease will be extinct or not.

The organization of this paper is as follows. In the next section, we construct a delayed and impulsive SIR epidemic model with two noninteracting susceptible subclasses. In Section 3, using the discrete dynamical sys-tem determined by the stroboscopic map, we establish sufficient conditions for the global attractivity of infec-tion-free periodic solution. And the sufficient conditions for the permanence of the model are obtained in section 4. Finally, we present some numerical simulations to illustrate our results. 2. Model Formulation and Preliminaries Gao etc. [13] proposed a delayed SIR epidemic model with pulse vaccination:

Y. Q. LUO ET AL.


58

( ) ( ) ( ) ( ),

( ) ( ) ( ) ( ) ( ) ( ), ,

( ) ( ) ( ) ( ),

( ) (1 ) ( ),

( ) ( ), ,

( ) ( ) ( ),

S t S t I t S t

I t S t I t e S t I t I t t nT n N

R t e S t I t R t

S t S t

I t I t t nT n N

R t R t S t

(1)

In model (1), the authors assumed that the birth rate ( ) is equal to the death rate, and use a bilinear inci-dence rate. Motivated by [13], in this paper, we assume that there are two cases noninteracting susceptible sub-classes. We denote the density of the susceptible indi-viduals that belong to different subclasses, the infected

individuals, and the recovered individuals in the popula-tion by S1, S2, I and R, respectively, that is, the total va-riable population 1 2 .N S S I R Moreover, if the nonlinear incidence ( 1)pIS p , different constant recruitments and death rates are incorporated into model (1). Then the corresponding model is investigated:

2

1 2 1

1

1 1 1 1 1 1

2 2 2 2 2

1 2 1 2 2

1 2 2

1 1 1

2 2 2

,

,,

,

,

1 ,

1 ,

,

p

p

p p d p d p

d p d p

S t S t I t S t

S t S t I t S tt nT n N

I t S t I t S t I t e S t I t e S t I t dI t

R t e S t I t e S t I t R t

S t S t

S t S t

I t I t

R t

1 1 2 2

,

,

t nT n N

R t S t S t

(2)

where , 1,2i i are the recruitment rate into the i-th susceptible class, respectively parameters , 1,2i i , d and are the death rates of the susceptible, infected and recovered individuals, and , 1, 2i i are the contact rates, is the infectious period. The term

, 1,2i

d pie S t I t i reflects the fact that

an individual has recovered from the infective compart-ments and are still alive after infectious period .

, 1, 2i i are the vaccination rates, and T is the period of pulsing.

Adding all the equations in model (2), the total varia-ble population size is given by the differential equation

1 2 1 1 2 2 .N t S t S t dI t R t

and we have

1 2 .N t hN t

where 1 2min , , ,h d . It follows that

1 2limsup .t

N th

Note that the first three equations of system (2) do not depend on the forth equation. Thus, we restrict our atten-tion to the following reduced system:

2

1 2 1

1 1 1 1 1 1

2 2 2 2 2

1 2 1 2 2

1 1 1

2 2 2

,

, ,

,

1 ,

1 , ,

,

p

p

p p d p d p

S t S t I t S t

S t S t I t S t t nT n N

I t S t I t S t I t e S t I t e S t I t dI t

S t S t

S t S t t nT n N

I t I t

(3)

The initial conditions of (3) are

1 1 2 2 3, , , ,0 ,S t t S t t I t t for t (4)

where 1 2 3, , ,T

PC and PC is the space of all piecewise functions 3: ,0 R with points of discontinuity at nT n N of the first kind and

Y. Q. LUO ET AL.


59

which are continuous from the left, i.e., 0nT nT , and 3 3

1 2 3, , 0, 1, 2, 3 .iR x x x R x i Define a subset of 3R

31 2

1 21 2

, ,

0

S t S t I t R

S t S t I th

From biological considerations, we discuss system (3) in the closed set. It is easy to verify that is positively invariant with respect to system (3). 3. Global Attractivity of Disease-Free

Periodic Solution To prove our main results, we state some notations and lemmas which will be essential to our proofs.

Lemma 1 (see [6]) Consider the following impulsive differential equation

, ,

1 , ,

u t a bu t t nT

u t u t t nT

where 0, 0,0 1a b . Then above system exists a unique positive periodic solution given by

* , 1 ,b t nTa au t u e for nT t n T

b b

which is globally asymptotically stable, where

1 1.

1 1

bT

bT

eau

b e

Definition 1 (see [14]). Let 3:V R R R , then V is said to belong to class 0V if

i) V is continuous in 3, 1nT n T R and for each

3X R ,

, ,

lim , ,t y nT X

V t y V nT X

exists.

ii) V is locally Lipschitzian in X. Lemma 2 (see [14]). Let 0V V . Assume that

, , , , ,

, , , ,n

D V t x g t V t x t nT

V t x V t x t nTt

where :g R R R is continuous in 1,nT T Rn

and for ,u R n N ,

, ,

lim , ,t y nT u

V t y V nT u

exists, :n R R is non-decreasing. Let r t be the maximal solution of the scalar impulsive differential equation

0

, , ,

, ,

0 ,

n

u g t u t nT

u u

t t

T

u

t t t n

u

existing on 0, . Then 0 00 ,V x u implies that , , 0V t x rt t t , where x t is any solution of

(3). In the following we shall demonstrate that the dis-

ease-free periodic solution * *1 2, ,0S St t is global

attractive. We firstly show the existence of the dis-ease-free periodic solution, in which the infectious indi-viduals are entirely absent from the population perma-nently, i.e. 0I t for all 0t . Under this condition, the growth of the i-th 1,2i susceptible individuals must satisfy

1

, ,

, .

i i i i

i i

S t t

t t

S t nT

S S t nT

According to Lemma 1, we know that the periodic so-lution of the system

* 11 1

for 1 , 1,2

i

i

t nTi i

i Ti i

eS

e

nT t n T i

t

is globally asymptotically stable. Therefore system (3) has a unique disease-free periodic solution

* *1 2, ,0S St t .

Denote

1 2

1 2

1 21 2

1 2 2

1

1 1

1 1 1 1.

p pT T

T Ti

e e

e eR

d

(5)

Theorem 1 If 1 1R , then the disease-free periodic solution * *

1 2, ,0S St t of system (3) is globally at-tractive.

Proof. Since 1 1R , we can choose 0 suffi-ciently small such that

1

1

2

2

11

1

22

2 2

1

1 1

1.

1 1

pT

Ti

pT

T

e

e

ed

e

(6)

From the first equation and the second equation of system (3), we have , 1, 2i i i iS S it t . Then we consider the following impulsive comparison system

Y. Q. LUO ET AL.


60

, ,

1 , ,

i i i i

i i

u t u t t nT

u t u t t nT

(7)

According to Lemma 1, we obtain the periodic solu-tion of system (7)

* 1 ,1 1

for 1 , 1, 2 ,

i

i

t nTi i

i Ti i

eu

e

nT t n T i

t

which is globally asymptotically stable. By the compari-son theorem [14], we have that there exists 1n Z such that for 11 ,nT t n T n n

* 11 1

, 1, 2 .

i

i

Ti i

i i Ti i

i

eS

et

S i

tu

(8)

Furthermore, from the third equation of system (3), we

get 1 1 2 2p ptI S S I td for t nT and

1n n . From (6) and (8), we have 0I t , then lim 0

ttI

,

i.e., for any sufficiently small 1 0 , there exists an

integer 2 1n n such that 1I t , for all 2t n T .

From the first equation and the second equation of system (3), we have for 2t n T

11 , ( 1, 2).p

i i i i i iS S it S t

Then we consider the following impulsive comparison systems

11 , ,

1 , .

pi i i i i i

i i i

v S v t nTt t

t nTtv vt

(9)

From Lemma 1, we obtain the periodic solution of system (9)

1

1

11

*1

1

1 , 1 , 1, 2 ,1 1

pi ii

pi ii

S t nT

i ii p S T

i i ii

te

v nT t n T iS e

which is globally asymptotically stable. In view of the comparison theorem [14], there exists an integer 3 2n n

such that for 31 , ,nT t n T n n

1

1

11

*1

1

( ) 1 , 1,21 1

pi ii

pi ii

S T

i ii i p S T

i i ii

eS t v i

St

e

(10)

Since and 1 are sufficiently small, from (8) and (10), we know that

*lim , 1,2 .it

S S t it

Hence, disease-free periodic solution * *1 2, ,0S St t

of system (3) is globally attractive. The proof is com-pleted.

Next, we give some accounts of the Theorem 1 for a well biological meaning.

By simple calculation, from (5) we get

1

1

1

2

2

2

1 1 11

1 1 1

1 2 21

2 2 2

10,

1 1

10,

1 1

pTpT

p pT

pTpT

p pT

eRp e

d e

eRp e

d e

(11)

and

1 2

1 2

1 2

1 1

1 1 1 2 21 1 2 21 1

1 21 2

1 10.

1 1 1 1

p pT Tp pT T

p p p pT T

e eRp e p e

T d de e

(12)

Theorem 1 determines the global attravtivity of the disease-free periodic solution of system (3) in for the case 1 1R . Its epidemiology implies that the dis-ease will die out. From (11) and (12), we can see that larger pulse vaccination rates or a shorter period of im-mune vaccination will make for the disease eradication. 4. Permanence In this section, we state the disease is endemic if the in-

fectious population persists above a certain positive level for sufficiently large time. The endemicity of the disease can be well captured and studied through the notation of uniform persistence.

Definition 2. System (3) is said to be uniformly per-sistent if there exist positive constants 0i iM m 1,2,3i (both are independent of the initial values), such that every solution 1 2, ,tS S t I t with posi-tive initial conditions of system (3) satisfies

1 1 1 2 2 2 3 3, , .m S M m S M m It t t M

Y. Q. LUO ET AL.


61

Denote

1 2

1 2

1 21 2

1 21 22

1 11 1

1 1 1 1.

p pT T

d dT T

e ee e

e eR

d

Theorem 2. If 2 1R , then system (3) is uniformly persistent.

Proof. Let 1 2, ,S t S t I t be any solution of (3) with initial conditions (4), then it is easy to see that

1 2 1 2 1 21 2, , , for all 0.S S I t

ht t t

h h

We are left to prove there exist positive constants

1 2 3, ,m m m such that 1 1 2 2 3, , ,t tS m S m I t m for all sufficiently large t.

Firstly, from the first and second equations of system (3), we have

1 2 , 1, 2 .p

i i i i iS S ih

t t

Considering the following comparison equations

1 2 ,

,1

,

,

i

i

p

i i i i

i i

uh

u

u t t t nT

u t t t nT

According to Lemma 1 and the comparison theorem, we know that for any sufficiently small 0 , there exists a 0t such that for 0t t ,

1 2

1 2

*

1 2

1 .

1 1

p

i i

p

i i

Th

i ii i i ip

Th

i ii

eS u u m

e

t t

h

t

Now, we shall prove there exist a 3 0m such that 3I t m for all sufficiently large t. For convenience,

we prove it through the following two steps: Step I. Since 2 1R , there exist sufficiently small

* 0Im and 0 such that

1 1 2 21 1 ,p d p de e d (13)

where

*1 2

*1 2*1 2

1 , 1, 2 .

1 1

p

i I i

p

i I i

m Th

i ii p

m Th

i I ii

ei

meh

For *Im , we can claim that there exists a 1 0t such

that *1 II t m . Otherwise, *

II t m for all 0t . It follows from the first and second equations of (3) that

1

*1 2 , 1, 2 .p

i i i I i iS mh

t it S

Considering the following impulsive comparison sys-tems

1*1 2 , ,

1 , ,

p

i i I i i

i i

i

i

tv m vh

t t nT

t t tv nTv

Similarly, we know that there exists 2 0t , such that for 2t t

1*1 2

1*1 2

*1

*1 2

.1

1 1

p

i I i

p

i I i

m Th

i ii i i ip

m Th

i I ii

te

S v v

meh

t t

(14)

Further, the third equations of system (3) can be re-written as

1 1 2 2

1 1

p p

d p

I S I S I

e S t I t

t t t t t

2 2

1 1 2 2

11

1 1

d p

p d p d

ptd

t

e S t I t dI

S I e S I e

ddI e S I d

dt

t

t t t t

t

Y. Q. LUO ET AL.


62

22

ptd

t

de S I d

dt

Define

11

22

.

ptd

t

ptd

t

V I e S I d

e S I d

t t

For 2t t , the derivative of V t along the solution of system (3) is

1 1 2 2

1 1 2 2

1 1

1 1

p d p d

p d p d

V S e S e d I

e e d

t

tI

t t t

From (13), we have 0V t for 2t t , which im-plies that V t as t . This contradicts the boundedness of ( )V t . Hence, there exists a 1 0t such that *

1 II t m . Step II. According to step I, for any positive solution 1 2, , t tS S I t of (3), we are left to consider two

cases. First, If *II t m for all 1t t , then our result is

proved. Second, if *II t m for some 1t t ,we can

choose constants 0 and 0 2 1,T max t t ( 0T is sufficiently large) such that *

II t m , *0 II T m ,

*0 II T m and 1 1 2 2, tS St for

0 0,t T T . Thus, there exists a 0g g such that for 0 0,t T T g

*

.2

II tm

(15)

In this case, we shall discuss three possible cases in term of the sizes of ,g and .

Case 1. If g , then it is obvious that * 2II t m , for 0 0,t T T . Case 2. If g , then from the third equation of

(3), we can deduce

1 1 2 2 ,

t d tp p

tI S S I e dt

(16)

From (14), (15) and (16), we have

0

01 1 2 2 0 0

***

1 1 2 2

, for ,

2

T g d tp p

T

p p dII

I S S I e d t T g T

me

t

g m

Set *

**3 min , 0

2I

I

mm m

, for 0 0,t T T , we

have 3I t m .

Case 3. If g , we will discuss the following two cases, respectively.

Case 3.1. For 0 0,t T T , it is easy to obtain **

II t m . Case 3.2. For 0 0,t T T . We claim that **

II t m . Otherwise, there exists a 0 0,t T T , such that **

II t m for 0 ,t T t , and **II t m .

From (13), (14) and (16), we have

** **1 1 2 2 1 1 2 2 ,

1 dt d tp p p p

I It

eI t S S I e d m m

d

which is contradictory to **II t m . Hence, the claim

holds true. According to the arbitrary of 0T , we can obtain that 3I t m holds for all 0t T . The proof is completed.

5. Numerical Simulations In this section, we give some numerical simulations to illustrate the effects of different probability on popula-tion. In system (3), 1 0.25, 2 0.2, 1 0.03,

2 0.1, 1 0.03, 2 0.05, 2,p 0.1,d 2, 3T . Time series are drawn in Figure 1(a) and Figure

1(b) with initial values 1 2 0.5sin ,t t 2 t 2 0.8cos ,t 3 1 0.5sin , 2,0t tt for 30 puls-ing cycles. If we take 1 0.45, 2 0.90 , then 1R 0.9956 . By Theorem 1, we know that the disease will disappear (see Figure 1(a)). If we let 1 0.10, 2

0.20 , then 2 1.4685R . According to Theorem 2, we know that the disease will be permanent (see Figure 1(b)).

(a)

Y. Q. LUO ET AL.


63

(b)

Figure 1. Two figures show that movement paths of S1, S2 and I as functions of time t. (a) Disease will be extinct with R1 = 0.9956 and θ1 = 0.45, θ2 = 0.9; (b) Disease will be per-sistent with R2 = 1.4685 and θ1 = 0.1, θ2 = 0.2. 6. Acknowledgements The research of Shujing Gao has been supported by The Natural Science Foundation of China (10971037) and The National Key Technologies R & D Program of Chi-na (2008BAI68B01). 7. References [1] X. B. Zhang, H. F. Huo, X. K. Sun and Q. Fu, “The Dif-

ferential Susceptibility SIR Epidemic Model with Time Delay and Pulse Vaccination,” Journal of Applied Mathe- matics and Computing, Vol. 34, No. 1-2, 2009, pp. 287- 298.

[2] Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, “Pulse Mass Measles Vaccination across Age Cohorts,” Pro-ceedings of the National Academy of Sciences of the United States of America, Vol. 90, No. 24, 1993, pp. 11698-11702. doi:10.1073/pnas.90.24.11698

[3] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—II: The Problem

of Endemicity,” Proceedings of the Royal Society Series A, Vol. 138, No. 834, 1932, pp. 55-83. doi:10.1098/rspa. 1932.0171

[4] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—III: Further Stu-dies of the Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 141, No. 843, 1933, pp. 94-122. doi:10.1098/rspa.1933.0106

[5] R. M. Anderson and R. M. May, “Population Biology of Infectious Disease: Part I,” Nature, Vol. 280, 1979, pp. 361-367. doi:10.1038/280361a0

[6] S. Gao, L. Chen and J. J. Nieto, “Angela Torres, Analysis of a Delayed Epidemic Model with Pulse Vaccination and Saturation Incidence,” Vaccine, Vol. 24, No. 35-36, 2006, pp. 6037-6045. doi:10.1016/j.vaccine.2006.05.018

[7] C. McCluskey, “Global Stability for a Class of Mass Action Systems Allowing for Latency in Tuberculosis,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 518-535. doi:10.1016/j.jmaa.2007. 05.012

[8] K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS Epidemic Model with Two Delays,” Journal of Mathematical Biology, Vol. 35, No. 2, 1996, pp. 240-260. doi:10.1007/s002850050051

[9] R. M. Anderson and R. M. May, “Infectious Diseases of Humans, Dynamics and Control,” Oxford University Press, Oxford, 1992.

[10] O. Diekmann and J. A. P. Heesterbeek, “Mathematical Epidemiology of Infectious Diseases,” John Wiley & Sons, Chisteter, 2000.

[11] F. Brauer and C. C. Castillo, “Mathematical Models in Population Biology and Epidemiology,” Springer, New York, 2000.

[12] W. O. Kermack and A. G. McKendrick, “A Contribution to the Mathematical Theory of Epidemics I,” Proceedings of the Royal Society Series A, Vol. 115, No. 772, 1927, pp. 700-721. doi:10.1098/rspa.1927.0118

[13] S. Gao, Z. Teng and D. Xie, “Analysis of a Delayed SIR Epidemic Model with Pulse Vaccination,” Chaos, Soli-tons & Fractals, Vol. 40, No. 2, 2009, pp. 1004-1011. doi:10.1016/j.chaos.2007.08.056

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

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AM20100100017_88545772.pdf