Global stability for delay SIR epidemic model with vertical
transmission
Junli Liu
School of Science
Xi'an Polytechnic University
Xi'an, China
ztlljl2008@yahoo.com.cn
Tailei Zhang
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an, China
AbstractA SIR epidemic model with delay, saturated contact rate and vertical transmission is considered. The basic
reproduction number
0
R
is calculated. It is shown that this number characterizes the disease transmission dynamics: if
0
1R
, there
only exists the disease-free equilibrium which is globally asymptotically stable; if
0
1R!
, there is a unique endemic equilibrium
and the disease persists, sufficient cond- itions are obtained for the global asymptotic stability of the endemic equilibrium.
Keywords- delay; vertical transmission; saturated contact rate; global asymptotic stability; permanence
1. Introduction
In [1], Cooke formulated an SIR model with time delay
1
2
3
()() ()(),
()() ()() (),
() ()(),
StStIt St
It StItIt
Rt ItRt
EWP
EWPJ
JP
/
° 
®
°
¯
&
&
&
where
()St
denote the number of susceptibles,
()It
the
number of infectives and
()Rt
the number of the removed.
/
is
the recruitment rate. The force of infection at time
t
is given
by
() ()StIt
EW
.
E
is the average number of contacts per
infective per day and
0
W
!
is a fixed time during which the
infectious agents develop in the vector, and it is only after that
time that the infected vector can infect a susceptible human.
123
,,
PPP
are positive constants repre- senting the death rates
of susceptibles, infectives, and recovered, respectively.
J
is
the recovery rate of infectives.
Incidence rate plays an important role in the modelling of
epidemic dynamics. The bilinear incidence rate
SI
E
and the
standard incidence rate
/SI N
E
are mostly used. Different
incidence rate have been used recently, such as
pq
SI
E
[2],
() /NSI N
E
[3],
/(1 )SI I
ED
[4].
Hu et al [5] studied an SIR epidemic model with saturation
incidence rate and vertical transmission as follows
/(1)(1 )(1 ),
/(1)()( ).
SSI SSI
ISI SIt
EDPPG
EDPGJ

°
® 
°
¯
&
&
(1)
In model (1), it is assumed that the total population is unity,
i.e.
1SI R
,
0
P
!
is the birth rate (or death rate),
/(1 )SS
ED
(0,0)
ED
!t
is the saturation inci- dence
rate,
0
J
!
is the recovery rate,
1
G
(0 1)
G
d
is the
proportion of the vertical transmission.
For system (1), they obtained the disease-free equilibrium
0
(1, 0)E
. The basic reproduction for (1) is
0/ [(1)()]R
EDJPG

.
If
0
1R!
, (1) has a unique endemic equilibrium
***
(,)ESI
with *()/[()],S
JPG EDJPG

Also, in [5], it is proved when
0
1,Rd
0
E
is globally asym-
ptotically stable; when
01,
D
dd
*
E
is globally asymptotic-
ally stable.
Motivated by the works [1] and [5], we now consider the
delay effect and formulate the following model
()/(1 )(1)(1),
()/(1 )()().
SSItS SI
IS ItSI t
EW DPPG
EW DPGJ

°
® 
°
¯
&
&
(2)
The initial condition of delay differential equations (2) is
given as
12
()()0,()() 0,[,0],SI
TITTITT W
! !
(3)
*
[(1)( )]/[()(())].I
PEDJPG JPEDJPG

Open Journal of Applied Sciences
Supplement2012 world Congress on Engineering and Technology
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ht © 2012 SciRes.1
where
2
12
((), ())([,0],)CR
IT ITW

, the Banach space
of continuous functions mapping the interval
[,0]
W
into
`
^
2
12
(, ):0,1,2.
i
Rxxxi
t
System (2) has the same basic reproduction number and
equilibria as those of (1).
2. Global Stability of the Equilibria and
the Permanence of the System
Lemma 2.1. (see [6]) Consider the following equation
()( )(),xt axtbxt
W

&
(4)
where
,, 0,ab
W
!
() 0xt!
for
0.t
W
dd
We have
(i) if
,ab
then
lim( )0;
t
xt
of
(ii) if
,ab!
then
lim( ).
t
xt
of
f
Theorem 2.1. The disease-free equilibrium
0
(1, 0)E
is
globa-
lly asymptotically stable for
0
1,R
and unstable for
01.R!
Proof. The characteristic equation of the linearized system of
(2) at
0
(1, 0)E
is
()(/ (1))0.e
OW
OPOJPGE D

(5)
For
0,
W
we have that
()(/ (1))0.
OPOJPGE D

Obviously,
12 0
,( )(1).R
OPOJPG
 
Hence, for
0
1R
the roots of (5) have negative real parts for
0.
W
Note that
0
O
is not a root of (5) while
0
1R
. If (5)
has pure imaginary roots
(0)i
OZZ
r!
for some
0,
W
!
then we have from (5) that
cos() /(1),
sin()/ (1).
JPGE ZWD
ZEZWD
 
® 
¯
This implies that
222
0
()(1)0R
ZJPG

for
0
1R
.
Therefore, any root of (5) must have a negative real part, and
hence the disease-free equilibrium
0
E
is locally asymptotical- ly
stable for any time delay
0.
W
t
Denote
()/(1 ),he
OW
OOJPGE D

then
for
0
1,R!
0
(0) ()(1)0hR
JPG

,
lim( )h
O
O
of
f
,
thus
() 0h
O
has one positive real root. Therefore,
0
E
is
unstable for
01.R!
Also we have
()()/(1)()().It ItIt
EW DJPG
d 
&
If
0
1R
, from Lemma 2.1 and the comparison theorem, we
obtain
() 0It o
as
.tof
Then the limit equation for
()St
is
(1 ),SS
P
&
which implies
lim()1.
t
St
of
This
completes the proof.
Theorem2.2. If
01,R!
the endemic equilibrium
***
(,)ESI
is locally asymptotically stable.
Proof. It is clear that at
***
(,)ESI
the associated transcend-
dental characteristic is
2
()0,ABCDe
OW
OO O

(6)
where
**2
/(1 )AIS
JPGPE D

,
**2
()()/(1)BIS
PJPGPJ ED

,
C
JPG
,
()D
PJ PG
.
For
0,
W
we have from (6) that
2() 0ACB D
OO

. (7)
Since
0AC!
,
0BD!
, then from Routh-Hurwitz
criterion it follows that for
0
W
both roots of (6) have
negative real parts.
Since
BD!
, then
0
O
is not a root of (6). If (6)
has pure imaginary roots
(0)i
OZZ
r!
for some
0,
W
!
then we have from (6) that
2
sin( )cos( ),
cos( )sin( ).
BC D
AC D
ZZ ZWZW
ZZ ZWZW
 
®
¯
We have that
22222
()(2)0gzzAB CzBD 
,
2
z
Z
.
And
22
2ABC
2**2 *2*4
2/ (1)()/(1)0,ISIS
PPGED ED
!
since
0BD!!
, then we have
22
0BD!
. Hence, we
have that
() 0gz!
for any
0z!
, this is a contradicts to
() 0.gz
This shows that all roots of the characteristic (6)
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have negative real parts for any time delay
0.
W
t
This
completes the proof.
Theorem 2.3. If
01R!
, the unique endemic equilibrium
*
E
is globally asymptotically stable whenever
*,IId
*
SSd
or
**
,.IISStt
Proof. Denote
(,)/(1 )fSI SIS
ED
. Consider the
following Lyapunov functional
*
()
*** *
** * **
**
()()(, )1/(,)
ln( /)(,)
[( )/1ln(( )/)].
St
S
t
t
VtStSfSIfId
II IIIfSI
IIIId
W
TT
TTT

 
u
³
³
Differentiating this function with respect to time yields
*****
****
** ***
*** **
***
*
()(1()/)[1( , )/((),)]
(1)(())[1(,)/(( ),)]
(,)[1 (,)/((),)
ln ((,)/((),)](, )
[1(/( ))((( ),())/(,))
ln((/( ))((
VtSStSf SIfStI
IItfSI fStI
fS IfS IfStI
fS IfStIfS I
IItfStItfSI
IItf
P
PG
W

 


u 
&
**
(),()) /(,)))],StItf SI
W
in the above equation, we have used the fact that
ln(() /())It It
W
** *
***
**
ln((,)/((),))
ln((( ),())/(( )(,)))
ln(()(( ),)/((( ),())))
fSIfSt I
I f StItIt f SI
Itf StII f StIt
W
WW

 
and
**
((),)/((),( ))/( )fSt IfStItIIt
WW
 
.
From the monotonicity of the function
(, )fSI
with respect
to
S
, we have
**** *
(1() /)[1(,) /((),)]0SStSfSIfStI
P
 d
.
The conditions in this theorem implies that
****
(1)(())[1(,)/(( ),)]IItfSI fStI
PG
 
***
(1)(())(( ))/(()(1))0.IItStSStS
PG D
d
Since the function
()1 ln()0Hxxx  d
for any
0x!
,
and
() 0Hx
if and only if
1x
. Therefore,
() 0Vt d
&
and the equality holds only at
**
,.SSII
Hence, from Kuang [7] (1993, Corollary 5.2, p. 30), we have
that
*
E
is globally asymptotically stable under the condition
0
1R!
. This completes the proof of Theorem 2.3.
From the above proof, we obtain the following result.
Corollary 2.1. If
0
1R!
,
1
G
, then the unique endemic
equilibrium
*
E
is globally asymptotically stable.
Corollary 2.1 shows that if there is no vertical transmission,
when the endemic equilibrium exists, it is globally asymptotic-
ally stable.
We now consider the permanence of system (2). Using the
same method as that in Theorem 3.2 of [8], we get the
following result.
Theorem 2.4. If
01R!
, then there exists an
0
H
!
such that
every solution
((),())St It
of system (2) with initial cond-
itions (3) satisfies
lim inf().
t
It
H
of
t
It is easy to obtain that
lim inf()/()
t
St
PGP E
of
t
,
and also
() 1,() 1StItdd
, from Theorem 2.4, we have
Theorem 2.5. If
01R!
, then system (2) is permanent.
3. Discussion
In this paper, a SIR epidemic model with delay, saturated
contact rate and vertical transmission is investigated. We
obtained the basic reproduction number
0
.R
It is shown that if
0
1,R
the disease-free equilibrium is globally asymptotical-
ly stable; if
01R!
, there is a endemic equilibrium and the
system is permanent. When
01R!
, in Theorem 2.3, by cons-
tructing a Lyapunov functional, sufficient conditions are obta-
ined for the global asymptotic stability of the endemic
equilibrium. We conjecture that for any
01
G
d
, when the
endemic equilibrium exists, it is globally asymptotically stable.
New technique is required to prove this.
4. Acknowledgement
This research was partially supported by the National
Natural Science Foundation of China (Nos. 11001215,
11101323), and Scientific Research Program Funded by
Shaanxi Provincial Education Department (No. 12JK0859).
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