L
EMMA
4.1. B
RAUER
, 2001. Let
f t
be a bounded non-negative function which satisfies an
estimate of the form
t
f t f t
f t a R a d
≤ +
−
∫
a , where
f t
is a non-negative function with and
is a non-negative function with
Then .
lim
t
f t
→∞
= R a
1. R a da
∞
∫
lim
t
f t
→∞
=
P
ROOF
. See [1]. It is also showed in [1] that the lemma is still true if the inequality in the lemma is
replaced by sup
t t a s t
f t f t
f s R a d
− ≤ ≤
≤ +
∫
a . 17
The following lemma is the extension of Brauer’s lemma.
L
EMMA
4.2. Let be bounded non-
negative functions satisfying ,
1, 2
j
f t j
=
1 10
1 1
sup
t t a s t
f t f
t f s R a d
− ≤ ≤
≤ +
∫
a ,
2 20
1 2
sup
t t a s t
f t f
t f s R a d
− ≤ ≤
≤ +
∫
a , where
j
f t is non-negative with
and is non-negative with
Then li
. lim
t j
f t
→∞
=
j
R a 1.
j
R a da
∞
∫
m 0,
1, 2
t j
f t j
→∞
= =
P
ROOF
.
1 10
20 1
2 1
2
sup{ ,
} sup
sup{ , }sup{ ,
}
t t a s t
f t f
t f
t f s f s
R a R a da
− ≤ ≤
≤ +
∫
2 10
20 1
2 1
2
sup{ ,
} sup
sup{ , }sup{ ,
}
t t a s t
f t f
t f
t f s f s
R a R a da
− ≤ ≤
≤ +
∫
and hence,
1 2
10 20
1 2
1 2
sup{ , }
sup{ ,
} sup
sup{ , }sup{ ,
}
t t a s t
f t f
t f
t f
t f s
f s R a R a da
− ≤ ≤
≤ +
∫
From Lemma 4.1 we conclude that , and this is suffice to
show that li
1 2
lim sup{ ,
}
t
f t f t
→∞
=
m 0,
1, 2
t j
f t j
→∞
= =
.
4.1. The stability of the disease-free equilibrium. We investigate the stability of the
disease-free equilibrium for the case of
1 R
. Consider the following inequalities.
1 sup
t H H
t a
I s ds
t H
H t a
H t a s t
H
e I
s a
β
β β
−
− −
− ≤ ≤
∫ −
≤ ∫ ≤
ds I
s
18 Hence we have,
1 sup
t H H
t a H
H
H H
t I
s ds a
H H
H t
a H
H H
t a s t H
I t
I t
B Q a
e e
da I
t B Q
a a I
s e da
β γ
γ
β
−
− −
− − ≤ ≤
= ∫
+ −
≤ +
∫ ∫
19 And
1 1
sup 1
t H H
t a H
H
H H
t I
s ds a
H H
H t
a H
H H
t a s t H
R t
R t
B Q a
e e
da R
t B Q
a a I
s e
da
β γ
γ
β
−
− −
− − ≤ ≤
= ⎡
⎤ ∫
⎡ ⎤
+ −
− ⎢
⎥ ⎣ ⎦
⎣ ⎦
≤ ⎡
⎤ +
− ⎣
⎦
∫ ∫
20 Moreover, since
lim
H
I t
t =
→ ∞
and 1
H
a H
H H
B Q a a
e da
R
γ
β
∞ −
=
∫
then using Lemma 4.1 we conclude that
.
lim
t H
I t
→∞
=
Next, let us see the expression
which, if
, can be written in the form
1
H
a H
H H
B Q a a
e da
γ
β
∞ −
⎡ ⎤
− ⎣
⎦
∫
S H
H H
R B Q
a a β
∞
=
∫
da
0 S
R R
−
. Hence, if then
1
S
R +
R
1 1
H
a H
H H
B Q a a
e da
γ
β
∞ −
⎡ ⎤
− ⎣
⎦
∫
Appendix 1. Furthermore, since
then using Lemma 4.2 we conclude that
lim
H
R t
t =
→ ∞ lim
t H
R t
→∞
=
. Consequently,
This shows that the disease-free equilibrium is globally stable.
lim lim
t H
t H
H H
S t
N t
R t
I t
N
→∞ →∞
= −
−
H
= ,
0,
H H
V H
S I I
N =
4.2. The stability of the endemic equilibrium.
The endemic equilibrium
appears only if . Let us see the
perturbations of
, ,
H H
H
S I
R 1
R
H
I
and
H
R
, respectively, by and
. Define
u t v t
H H
I t
I u t
= +
and substitute this quantity into equation 4 to obtain the
following calculations.
[ ]
1
t H
H t a
H
H H
t I
u s ds a
H H
I u t
I t
B Q a
e e
da
β γ
−
− +
−
+ =
⎛ ⎞
∫ +
− ⎜
⎟ ⎝
⎠
∫
1 1
1
t t
H H H
t a t a
H H H
H t
H t a
H H H
H H
t I ds
u s ds a
H H
I a a
H H
H t
u s ds I a
a H
H
u t I
I t
B Q a
e e
e da
B Q a
e e
da I
t B Q
a e
e e
da
β β
γ β
γ β
β γ
− −
−
− −
− ∞
− −
− −
−
= − + ⎛
⎞ ∫
∫ +
− ⎜
⎟ ⎝
⎠ = −
− +
⎛ ⎞
∫ +
− ⎜
⎟ ⎝
⎠
∫ ∫
∫
1 1
1
H H H
H H t
H t a
H H H
I a a
H H
H t
t I a
H H
t u s ds
I a a
H H
u t B Q
a e
e da
I t
B Q a
e da
B Q a
e e
e da
β γ
β β
β γ
−
∞ −
− −
− −
−
= − −
+ −
− ⎛
⎞ ∫
+ −
⎜ ⎟
⎝ ⎠
∫ ∫
∫
1 1
1 sup
H H H
t H
t a H H
H H H
H H H
H
I a a
H H
H t
t u s ds
I a a
H H
I a a
H H
H t
I a a
H H
H t a s t
u t B Q
a e
e da
I t
B Q a e
e e
da B Q
a e
e da
I t
B Q a e
ae u s
β γ
β β
γ β
γ β
γ
β
−
∞ −
− −
− −
∞ −
− −
− − ≤ ≤
= − −
+ ⎛
⎞ ∫
+ −
⎜ ⎟
⎝ ⎠
≤ − −
+ +
∫ ∫
∫
t
da
∫
Hence, we have
1 sup
H H H
H H H
I a a
H H
H t
t I a
a t a s t
H H
H
u t B Q
a e
e da
I t
u s B Q a e
ae da
β γ
β γ
β
∞ −
− −
− − ≤ ≤
≤ − −
+ +
∫ ∫
By defining
1
f t u t
=
,
1
H H
I a a
H H
H
R a B Q
a e ae
H
β γ
β
−
=
−
, and
10
1
H H H
I a a
H H
H t
f t
B Q a
e e
da I
β γ
∞ −
−
= − −
+
∫
t , then we have
1 10
1 1
sup
t t a s t
f t f
t f s R a d
− ≤ ≤
≤ +
∫
a . We see that
and it can be shown that
is non-negative with
10
lim
t
f t
→∞
=
1
R a
1
1 R a da
∞
∫
see Appendix 2. Then by Lemma 4.1 we have , means that
1
lim
t
f t
→∞
= lim
t H
H
I t
I
→∞
=
. Next, define
and substitute these quantities into equation 5 to
obtain the following calculations:
H H
R t
R v t
= +
[ ]
1 1
t H
H t a
H
H H
t I
u s ds a
H H
R v t
R t
B Q a
e e
da
β γ
−
− +
−
+ =
∫ ⎡
⎤ +
− −
⎣ ⎦
∫
1 1
1 1
1 1
t t
H H H
t a t a
H H H
H t
H t a
H H H
H H
t I ds
u s ds a
H H
I a a
H H
H t
u s ds I a
a H
H
v t R
R t
B Q a
e e
e da
B Q a
e e
da R
t B Q
a e
e e
da
β β
γ β
γ β
β γ
− −
−
− −
− ∞
− −
− −
−
= − + ⎛
⎞ ∫
∫ ⎡
⎤ +
− −
⎜ ⎟⎣
⎦ ⎝
⎠ ⎡
⎤ = −
− −
+ ⎣
⎦ ⎛
⎞ ∫
⎡ ⎤
+ −
− ⎜
⎟⎣ ⎦
⎝ ⎠
∫ ∫
∫
1 1
1 1
1 1
H H H
H H H
t H
t a H H
H
I a a
H H
H t
t I a
a H
H t
u s ds I a
a H
H
v t B Q
a e
e da
R t
B Q a
e e
da B Q
a e
e e
da
β γ
β γ
β β
γ
−
∞ −
− −
− −
− −
⎡ ⎤
= − −
− +
⎣ ⎦
⎡ ⎤
− −
− ⎣
⎦ ⎛
⎞ ∫
⎡ ⎤
+ −
− ⎜
⎟⎣ ⎦
⎝ ⎠
∫ ∫
∫
1 1
1 1
1 1
1
H H H
t H
t a H H
H H H
H H H
I a a
H H
H t
t u s ds
I a a
H H
I a a
H H
H t
I a H
H H
v t B Q
a e
e da
R t
B Q a e
e e
da B Q
a e
e da
R t
B Q a e
a
β γ
β β
γ β
γ β
β
−
∞ −
− −
− −
∞ −
− −
⎡ ⎤
= − −
− +
⎣ ⎦
⎛ ⎞
∫ ⎡
⎤ +
− −
⎜ ⎟⎣
⎦ ⎝
⎠ ⎡
⎤ ≤ −
− −
+ ⎣
⎦ +
∫ ∫
∫
sup
H
t a
t a s t
e u
γ −
− ≤ ≤
⎡ ⎤
− ⎣
⎦
∫
s da
Hence, we have
1 1
sup 1
H H H
H H H
I a a
H H
H t
t I a
a t a s t
H H
H
v t B Q
a e
e da
I t
u s B Q a e
a e
da
β γ
β γ
β
∞ −
− −
− − ≤ ≤
⎡ ⎤
≤ −
− −
+ ⎣
⎦ ⎡
⎤ +
− ⎣
⎦
∫ ∫
By defining
1
f t u t
=
,
2
f t v t
=
, and
2
1
H H H
I a a
H H
H
R a B Q
a e a
e
β
β
−
⎡ =
⎣
γ
−
⎤ −
⎦
20
1 1
H H H
I a a
H H
H t
f t
B Q a
e e
da I
β γ
∞ −
−
⎡ ⎤
= − −
− +
⎣ ⎦
∫
t
, then we have
2 20
1 2
sup
t t a s t
f t f
t f s R a d
− ≤ ≤
≤ +
∫
a . We can show that in Appendix 3 that
20
lim
t
f t
→∞
=
and is non-negative with
2
R a
2
1 R a da
∞
∫
. Then by Lemma 4.2 we have
2
lim
t
f t
→∞
=
, means that .
lim
t H
R t
R
→∞
=
H
Finally, since
lim
t H
N t
→∞
is a constant, , and
lim
t H
R t
R
→∞
=
H H
lim
t H
I t
I
→∞
=
then is globally stable.
, ,
H H
H
S I
R
V. Concluding Remarks
