Let . Further let also assume that
, , and
denotes, respectively, the numbers of
0 who survive at time
t
, the numbers of 0 who survive at time
t
, and the numbers of 0 who survive at time
t
, then we have
H H
H
I S
N +
=
t N
H
t S
H
t I
H
H
N
H
S
H
I
∫
+ =
t H
H H
H
da a
Q B
t N
t N
. 3
Since the per capita rate of infection in human population at time t is
t I
V H
β
, the number of susceptibles at time t is given by
∫
∫ +
=
−
−
t ds
s I
H H
H H
da e
a Q
B t
S t
S
t a
t V
H
β
. 4
The number of human infectives is
t S
t N
t I
H H
H
− =
, and given by
∫
∫ −
+ =
−
−
t ds
s I
H H
H
da e
a Q
B t
I t
I
t a
t V
H
1
β
. 5
It is clear that
lim =
∞ →
t N
t
H
, , and
.
lim =
∞ →
t S
t
H
lim =
∞ →
t I
t
H
6 Analogously, we can derive similar equations for the mosquitoes, which are
∫
+ =
t V
V V
V
da a
Q B
t N
t N
. 7
∫
∫ +
=
−
−
t ds
s I
V V
V V
da e
a Q
B t
S t
S
t a
t H
V
β
. 8
∫
∫ −
+ =
−
−
t ds
s I
V V
V V
da e
a Q
B t
I t
I
t a
t H
V
1
β
. 9
It is also clear that
lim =
∞ →
t N
t
V
, , and
.
lim =
∞ →
t S
t
V
lim =
∞ →
t I
t
V
10
3. The existence of a threshold number
Next we will show that there is a threshold number for the model discussed above. Let us consider the limit values of equations 3 and 5. Whenever
, and by considering 6 holds, the equations 3 and 5 can be written as
∞ →
t
,
∫
∞
= da
a Q
B t
N
H H
H
11
∫
∞ −
∫ −
=
−
1 da
e a
Q B
t I
t a
t V
H
ds s
I H
H H
β
. 12
Similarly equations 7 and 9 can be written as
∫
∞
= da
a Q
B t
N
V V
V
. 13
33
∫
∞ −
∫ −
=
−
1 da
e a
Q B
t I
t a
t H
V
ds s
I V
V V
β
. 14
The value of is constant, that is,
. Similarly the value of is also constant,
that is, . Hence the system reduces to two equations, namely equations 12 and 14.
t N
H
∫
∞
= da
a Q
B K
H H
H
t N
V
∫
∞
= da
a Q
B K
V V
V
The equilibrium of the system are given by and
satisfying
H
I
V
I
∫
∞ −
− =
1 da
e a
Q B
I
a I
H H
H
V H
β
, 15
∫
∞ −
− =
1 da
e a
Q B
I
a I
V V
V
H V
β
. 16
Or
∫
∞ ⎥⎦
⎤ ⎢⎣
⎡ ∫
− −
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
− =
∞ −
1
1 da
e a
Q B
I
a da
e a
Q B
H H
H
a H
I V
V V
H
β
β
, 17
∫
∞ ⎥⎦
⎤ ⎢⎣
⎡ ∫
− −
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎝ ⎛
− =
∞ −
1
1 da
e a
Q B
I
a da
e a
Q B
V V
V
a V
I H
H H
V
β
β
. 18
It is easy to see that is a disease-free equilibrium. To find other equilibrium, without loss of
generality, let us consider only equation 17. An endemic equilibrium satisfies
, ,
=
V H
I I
∫
∞ ⎥⎦
⎤ ⎢⎣
⎡ ∫
− −
⎟ ⎟
⎟ ⎠
⎞ ⎜
⎜ ⎜
⎝ ⎛
− =
∞ −
1
1 1
da I
e a
Q B
H a
da e
a Q
B H
H
a H
I V
V V
H
β
β
. 19
The RHS of equation 20 is a function of
I
, say
∫
∞ ⎥⎦
⎤ ⎢⎣
⎡ ∫
− −
⎟ ⎟
⎟ ⎠
⎞ ⎜
⎜ ⎜
⎝ ⎛
− =
∞ −
1
1 da
I e
a Q
B I
f
H a
da e
a Q
B H
H H
H
a H
I V
V V
H β
β
20
satisfying
34
da a
da a
e a
Q B
e a
Q B
I da
dI e
d a
Q B
I da
I e
a Q
B I
I f
I
V a
I V
V H
I u
H H
H H
a da
e a
Q B
H H
H H
a da
e a
Q B
H H
H H
H H
H V
H a
H I
V V
V H
a H
I V
V V
H
∫ ∫
∫ ∫
∞ ∞
− −
+ ∞
⎥⎦ ⎤
⎢⎣ ⎡
∫ −
− +
∞ ⎥⎦
⎤ ⎢⎣
⎡ ∫
− −
+ +
⎟ ⎠
⎞ ⎜
⎝ ⎛
⎥⎦ ⎤
⎢⎣ ⎡
→ =
⎟ ⎟
⎟ ⎟
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎜ ⎜
⎜ ⎜
⎝ ⎛
− →
= ⎟
⎟ ⎟
⎠ ⎞
⎜ ⎜
⎜ ⎝
⎛ −
→ =
→
∞ −
∞ −
1 1
lim 1
1 lim
1 lim
lim
β β
β β
β
β β
21 with
1 lim
lim
1
= →
= →
⎥⎦ ⎤
⎢⎣ ⎡
∫ −
− +
− +
∞ −
a da
e a
Q B
H I
u H
a H
I V
V V
H H
e I
e I
β
β
; .
1 lim
= →
− +
a I
H
H V
e I
β
22 Hence we have
lim ⎟
⎠ ⎞
⎜ ⎝
⎛ =
→
∫ ∫
∞ ∞
+
da da
a aQ
a aQ
B B
I f
I
V H
V H
V H
H H
H
β β
. 23
Similarly, we obtain
lim =
∞ →
H H
H
I f
I
. 24
This will ensure that a host endemic equilibrium exists if and only if
H
I 1
⎟ ⎠
⎞ ⎜
⎝ ⎛
∫ ∫
∞ ∞
da da
a aQ
a aQ
B B
V H
V H
V H
β β
. 25
The LHS of 25 will be refereed as a threshold number R of the model.
4. Estimating the threshold number for exponential survival rate
