Applied Mathematics, 2011, 2, 57-63
doi:10.4236/am.2011.21007 Published Online January 2011 (http://www.SciRP.org/journal/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
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 11R

and persistent if 21R. 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 num-
bers), 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, SI ER 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 su scep tibles (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 p
SI 
, 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 condition s 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 sub-
classes. 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 vacci nat i on:

Y. Q. LUO ET AL. 58
()()() (),
()() ()() ()(),,
()() ()(),
() (1)(),
() (),,
( )()(),
StStIt St
I
tStIteStIt IttnTnN
Rte StItRt
St St
I
tIt tnTnN
RtRt St


 










 




 







 

(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 12 .NSS IR

Moreover, if the
nonlinear incidence (1)
p
IS p  , different constant
recruitments and death rates are incorporated into model
(1). Then the corresponding model is investigated:
 
 
    
 






2
12 1
1
11 1111
222 22
12122
122
111
222
,
,,
,
,
1,
1,
,
p
p
ppdp dp
dp dp
StStItSt
StS tItSttnTnN
ItStItStIteSt IteSt ItdIt
RteSt IteSt ItRt
St St
St St
It It
Rt


 
 









  
 





 





 
112 2
,
,
tnTnN
RtStSt 
















 


(2)
where

, 1,2
ii
  are the recruitment rate into th e i-th
susceptible class, respectively parameters


, 1,2
ii
 ,
d and

are the death rates of the susceptible, infected
and recovered individuals, and

, 1,2
ii
 are the
contact rates,

is the infectious period. The term
 
, 1,2
i
dp
ieSt Iti


  reflects the fact that
an individual has recovered from the infective compart-
ments and are still alive after infectious period

.

, 1,2
ii
 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






1211 22.NtStSt dItRt

 
and we have


12 .Nt hNt


where


12
min,, ,hd

 . It follows that

12
limsup .
tNt h





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
12 1
11 1111
222 22
12122
111
222
,
,,
,
1,
1, ,
,
p
p
ppdp dp
StStItSt
StS tItStt nTnN
ItStItStIteSt IteSt ItdIt
St St
StStt nTnN
It It

 
 








 


 



  






 
 



(3)
The initial conditions of (3) are














112 23
,,,,0,SttSttItt fort   (4)
where

123
,, ,
T
P
C 

 and PC is the space
of all piecewise functions


3
:,0 R 

 with points
of discontinuity at


nT nN of the first kind and

Y. Q. LUO ET AL. 59
which are continuous from the left, i.e.,

0nT  

nT  , and


33
123
,,0, 1,2,3.
i
RxxxRxi

Define a subset of 3 R
  
 
3
12
12
12
,,
0
StStIt R
St St Ith





 




 

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,,
utabuttnT
utut t nT


  

 


where 0, 0,01ab

. Then above system exists a
unique positive periodic solution given by



*,1,
bt nT
aa
utuefornT tnT
bb


 


which is globally asymptotically stable, where




11 .
11
bT
bT
e
a
ube






Definition 1 (see [14]). Let 3 :VRRR
 
, then V
is said to belong to class 0 V if
i) V is continuous in


3
,1nT nTR 
 and for each
3
X
R ,




,,
lim ,,
tynTXVtyVnT X



 exists.
ii) V is locally Lipschitzian in X.
Lemma 2 (see [14]). Let 0 VV

. Assume that
  
 


,,,,,
,,,,
n
DV txgtVtxtnT
VtxVtxt nTt








where :
g
RR R

 is con tinuous in
 
1,nTTRn





and for ,uR

 nN,






,,
lim ,,
tynT uVtyVnTu




exists, :
nRR


 is non-decreasing. Let


rt be
the maximal solution of the scalar impulsive differential
equation
  

 

0
,, ,
,,
0,
n
ugtutnT
uu
tt
T
u
tttn
u










existing on


0,

. Then

00
0,Vxu
 implies that






,,0Vtx rttt

, where

x
t is any solution of
(3).
In the following we shall demonstrate that the dis-
ease-free periodic solution
 

**
12
,,0SStt 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.


0It

for all 0t. Under this condition,
the growth of the i-th


1, 2i susceptible individuals
must satisfy
 


1
,,
,.
iiii
ii
St t
tt
StnT
SStnT






 




According to Lemma 1, we know that the periodic so-
lution of the system



 
*111
for1 ,1,2
i
i
tnT
ii
iT
ii
e
Se
nT tnTi
t












 
is globally asymptotically stable. Therefore system (3)
has a unique di sease -f ree periodic solut i on




**
12
,,0SStt.
Denote
 
12
12
12
12
12
2
1
11
1111 .
p
p
TT
TT
i
ee
ee
Rd








 


 
 
 



(5)
Theorem 1 If 11R

, then the disease-free periodic
solution






**
12
,,0SStt of system (3) is globally at-
tractive.
Proof. Since 11R

, we can choose 0

 suffi-
ciently small such that


1
1
2
2
1
11
2
222
1
11
1.
11
p
T
T
i
p
T
T
e
e
ed
e




























(6)
From the first equation and the second equation of
system (3), we have
 
, 1,2
iiii
SSitt   
.
Then we consider the following impulsive comparison
system

Y. Q. LUO ET AL. 60
 


,,
1,,
iiii
ii
ututt nT
ututt nT




 

 


(7)
According to Lemma 1, we obtain the periodic solu-
tion of system (7)



 
*1,
11
for1,1,2 ,
i
i
tnT
ii
iT
ii
e
ue
nT tnTi
t












 
which is globally asymptotically stable. By the compari-
son theorem [14], we have that there exists 1 nZ


such that for

1
1,nTtnT nn
 


*111
,1,2.
i
i
T
ii
ii T
ii
i
e
Se t
Si
tu













 
(8)
Furthermore, from the third equation of system (3), we
get





112 2
pp
t
I
SSItd 
 for tnT and
1
nn.
From (6) and (8), we have


0
I
t
, then


lim 0
ttI 

,
i.e., for any sufficiently small 10

, there exists an
integer 21
nn such that

1
It

, for all 2 tnT.
From the first equation and the second equation of
system (3), we have for 2 tnT




11,( 1,2). p
iiii ii
SS itSt  


Then we consider the following impulsive comparison
systems







11,,
1,.
p
iiiiii
iii
vSvtnTtt
tnTtvvt
 



 

 


(9)
From Lemma 1, we obtain the periodic solution of
system (9)






 
11
11
*
11
1,1,1,2,
11
p
ii
i
p
ii
i
StnT
ii
ipST
ii ii
te vnTtnTi
Se










 




which is globally asymptotically stable. In view of the
comparison theorem [14], there exists an integer 32 nn such that for


3
1, ,nTtnTn n 






11
11
*
11
()1, 1,2
11
p
ii i
p
ii i
ST
ii
ii pST
ii ii
e
St vi
S
t
e











 




(10)
Since

and 1

are sufficiently small, from (8) and
(10), we know that

*
lim,1,2 .
i
tSStit  
Hence, disease-free periodic solution




**
12
,,0SStt
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
111 1
111
122 1
222
10,
11
10,
11
p
T
pT
pp
T
p
T
pT
pp
T
e
Rpe
de
e
Rpe
de



















 






(11)
and







1 2
1 2
1 2
11
111 22
112 2
11
12
12
11
0.
11 11
p
p
TT
pp
TT
ppp p
TT
ee
Rpep e
Td d
ee



 
 









 
 (12)
Theorem 1 determines the global attravtivity of the
disease-free periodic solution of system (3) in

for
the case 11R. 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 p ersists above a certain positive lev el
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 0
ii
Mm


1,2,3i (both are independent of the initial values),
such that every solution
 


12
,,tSStIt
with posi-
tive initial condition s of system (3) satisfies


1112 2233
,,.mS MmSMmIttt
M
 

Y. Q. LUO ET AL. 61
Denote




12
12
12
12
12 12
2
11
11
1111 .
pp
TT
dd
TT
ee
ee
ee
Rd












 


 


Theorem 2. If 21R, then system (3) is uniformly
persistent. Proof. Let






12
,,StStIt be any solution of (3)
with initial conditions (4), then it is easy to see that
  
1212 12
12
,,,forall0.SSIt
h
ttt
hh
  
 
We are left to prove there exist positive constants
123
,,mmm such that



112 23
,,,ttSmSmItm
for all sufficiently large t.
Firstly, from the first and second equations of system
(3), we have

12 ,1,2.
p
iiiii
SSi
h
tt

 



 





Considering the following comparison equations
 

 
12 ,
,1
,
,
i
i
p
ii ii
ii
u
h
u
uttt nT
utt t nT




















According to Lemma 1 and the comparison theorem,
we know that for any sufficiently small 0


, there
exists a 0 t such that for 0 tt,
 

12
12
*
12
1.
11
p
ii
p
ii
T
h
ii
iii i
pT
h
ii
i
e
Suu m
e
tt
h
t






























 











Now, we shall prove there exist a 30msuch that

3
It m for all sufficiently large t. For convenience,
we prove it through the following two steps:
Step I. Since 21R, there exist sufficiently small
*0
I
m and 0

 such that


112 2
11,
pd pd
eed

 

  (13)
where


*
12
* 12
*
12
1, 1,2.
11
p
iIi
p
iIi
mT
h
ii
ipmT
h
iIi
i
ei
me
h






























 


 

 

 
For *
I
m, we can claim that there exists a 10t such
that

*
1
I
I
tm. Otherwise,


*
I
It m for all 0t. It
follows from the first and second equations of (3) that
 
1*
12 ,1,2.
p
iiiIii
Sm
htitS

 




 





Considering the following impulsive comparison sys-
tems
 

 
1*
12 ,,
1,,
p
ii Iii
ii
i
i
tvmv
htt nT
tttvnTv

 

















Similarly, we know that there exists 20t, such that
for 2 tt


1*
12
1*
12
*
1*
12
.1
11
p
iIi
p
iIi
mT
h
ii
iii i
pmT
h
iIi
i
te Svv
me
h
tt













 







 









 


 

 

 
 (14)
Further, the third equations of system (3) can be re-
written as
  

112 2
11
pp
dp
ISISI
eSt It
ttttt










 

 

 
22
112 2
11
11
dp
pdpd
p
t
d
t
eSt ItdI
SI eSI e
d
dIeSId
dt
t
tt tt
t














 

Y. Q. LUO ET AL. 62

22
p
t
d
t
d
eSId
dt







Define
 

11
22
.
p
t
d
t
p
t
d
t
VIe SId
eSId
tt















For 2 tt, the derivative of

Vt along the solution
of system (3) is
 






112 2
112 2
11
11
pdpd
pd pd
VSeSedI
eed
t
tI
tt t



 








From (13), we have

0Vt
 for 2
tt, which im-
plies that

Vt as t. This contradicts the
boundedness of ()Vt. Hence, there exists a 10t such
that

*
1
I
I
tm.
Step II. According to step I, for any positive solution
 
12
, , ttSSIt of (3), we are left to consider two
cases. First, If


*
I
It m for all 1 tt, then our result is
proved. Second, if


*
I
It m for some 1 tt,we can
choose constants 0

 and

021
,Tmaxtt

 (0 T
is sufficiently large) such that

*
I
It m,


*
0
I
I
Tm,


*
0
I
I
Tm 

 and
 
112 2
, tSSt

  for


00
,tTT


.
Thus, there exists a

0gg

 such that for


00
,tTT g



*.
2
I
It m  (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


*2
I
It m, for


00
,tTT


.
Case 2. If g



, then from the third equation of
(3), we can deduce
 




112 2,
tdt
pp
t
I
SSIedt



 



 (16)
From (14), (15) and (16), we have
 





0
0112 200
***
112 2
,for ,
2
Tg dt
pp
T
pp d
II
ISSIedtTgT
me
t
gm



 
 


 
 

Set ***
3min ,0
2II
m
mm




, for


00
,tTT

, we
have

3
It m.
Case 3. If g

  , we will discuss the following
two cases, respectively.
Case 3.1. For


00
,tTT


, it is easy to obtain


**
I
It m.
Case 3.2. For


00
,tT T



. We claim that


**
I
It m. Otherwise, there exists a


00
,tT T

  ,
such that


**
I
It m for


0,tT t
  , and


**
I
I
tm.
From (13), (14) and (16), we have
 




** **
112 21122,
1d
tdt
pp pp
I
I
t
e
I
tSSIedm m
d


 




 

which is contradictory to

**
I
I
tm. Hence, the claim
holds true.
According to the arbitrary of 0 T, we can obtain that

3
It m holds for all 0 tT. 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), 10.25,

 20.2,

 10.03,


20.1,

10.03,

20.05,

2,p0.1,d2,


3T. Time series are drawn in Figure 1(a) and Figure
1(b) with initial values


120.5sin,tt
 


2t


20.8cos,t
 

310.5sin, 2,0ttt
  for 30 puls-
ing cycles. If we take 10.45,

20.90

, then 1 R

0.9956 . By Theorem 1, we know that the disease will
disappear (see Figure 1(a)). If we let 10.10,

 2


0.20 , then 21.4685R

. 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.
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