A Threshold Number For Dengue Disease Endemicity In An Age Structured Model.
A.K. SUPRIATNA & E. SOEWONO
A THRESHOLD NUMBER FOR DENGUE DISEASE
ENDEMICITY IN AN AGE STRUCTURED MODEL1
d
S H = BH − β H S H IV − µ H S H ,
dt
d
SV = BV − βV SV I H − µV SV ,
dt
Asep K. Supriatna a & Edy Soewonob
where
a
Department of Mathematics, Universitas Padjadjaran, Indonesia
b Department of Mathematics, ITB, Indonesia
Abstract. In this paper we present a model for dengue disease transmission with
an assumption that individuals in the under-laying populations experience a
monotonically non-increasing survival rate. We show that there is a threshold for
the disease transmission, below which the disease will stop (endemic equilibrium is
not appearing) and above which the disease will stay endemic (endemic equilibrium
is appearing). We also investigate the stability of this endemic equilibrium.
Key-words: Dengue Modeling, Threshold Number, Stability of an Equilibrium
Point.
d
I H = β H S H IV − µ H I H ,
dt
d
IV = βV SV I H − µV IV ,
dt
SH
SV
= the number of susceptibles in the host population
= the number of susceptibles in the vector population
IH
IV
= the number of infectives in the host population
= the number of infectives in the vector population
= host recruitment rate; BV = vector recruitment rate
BH
µH
βH
βV
= host death rate; µV = vector death rate
= the transmission probability from vector to host
= the transmission probability from host to vector
1 Introduction
The model above based on the assumption that the host population
Reducing the number of dengue fever disease prevalence is regarded as an
important public health concern in Indonesia, and in many tropical countries,
since the disease is very dangerous that may lead to fatality. To find a good
management in controlling the disease, ones need to understand the dynamics of
the disease. Many mathematical models have been devoted to address this issue,
examples are [3],[4],[5], and [6]. However, most of the authors have ignored the
presence of age structure in mortality rate of the populations in their models. In
this paper we present a model for dengue disease transmission with the inclusion
that individuals in the under-laying populations experience a monotonically nonincreasing survival rate as their age goes by. We show that there is an endemic
threshold, below which the disease will stop, and above which the disease will stay
endemic.
vector population
2 Host-Vector Model with Monotonic Non-Increasing
Survival Rate
The model discussed here is analogous to the following age-unstructured hostvector SI model:
the host, and
SV
NV
and
each are divided into two compartments,
IV
SH
NH
and
and the
IH
for
for the vector.
An analogous age-structured one for the above model is made by generalizing the
QH (a ) , a function of age describing the
fraction of human population who survives to the age of a or more, such that,
model in [1]. Suppose that there exists
QH (0) = 1 and QH (a ) is a non-negative and monotonically non-increasing
0 ≤ a ≤ ∞ . If it is assumed that human life expectancy is finite, then
∫
Let
∞
0
QH (a)da = L < ∞ and
NH = SH + IH .
I H ( 0) (t )
∞
0
aQH (a )da < ∞
Further, let also assume that
denotes, respectively, the numbers of
S H (0) who survive
t . Then we have
numbers of
at time
∫
at time
t , and
for
1
N H ( 0 ) (t ) , S H ( 0) (t ) ,
and
N H (0) who survive at time t , the
the numbers of
I H (0) who survive
t
N H (t ) = N H ( 0 ) (t ) + ∫ BH QH (a)da .
2
0
Since the per capita rate of infection in human population at time t is
1
Presented in the International Conference on Applied Mathematics (ICAM05) in
Bandung, August 22-26, 2005. Part of the works in this paper is funded by the Indonesian
Government, through the scheme of Penelitian Hibah Bersaing XII (SPK No.
011/P4T/DPPM/PHB/III/2004).
β H I V (t ) ,
the number of susceptibles at time t is given by
t
t
−
β H I V ( s ) ds
S H (t ) = S H ( 0) (t ) + ∫ BH QH (a )e ∫t −a
da .
0
3
A.K. SUPRIATNA & E. SOEWONO
Threshold Number of a Dengue Model
See also [2]. The number of human infectives is
I H (t ) = N H (t ) − S H (t ) , given by
t
−
β H I V ( s ) ds
I H (t ) = I H ( 0) (t ) + ∫ BH Q(a ) 1 − e ∫t −a
da .
0
The Equilibrium of the system is given by
∞
t
t→∞
N H ( 0) (t ) = 0 ,
lim
t→∞
S H ( 0) (t ) = 0 , and
lim
0
I =∫
*
V
t→∞
I H ( 0 ) (t ) = 0 .
5
Analogously, we can derive similar equations for the mosquitoes, which are
t
NV (t ) = NV ( 0) (t ) + ∫ BV QV (a )da ,
6
0
t
t
−
β V I H ( s ) ds
SV (t ) = SV ( 0) (t ) + ∫ BV QV (a)e ∫t −a
da ,
7
t
−
β V I H ( s ) ds
IV (t ) = IV ( 0 ) (t ) + ∫ BV QV (a) 1 − e ∫t −a
da .
0
8
0
lim
t→∞
∞
0
V
lim
t →∞
SV ( 0) (t ) = 0 , and
lim
t→∞
IV ( 0) (t ) = 0 .
2
*
H
14
).
15
*
− β H ∫0∞ BV QV ( a )(1− e − βV I H a ) da a
∞
16
I H* = F1 ( F2 ( I H* )) = ∫ BH QH (a )1 − e
da ,
0
*
*
Note that F1 ! F2 is bounded. It is easy to see that ( I H , IV ) = (0,0) is the disease-
free equilibrium. To find a non-trivial equilibrium (an endemic equilibrium), we could
observe the following
dF1 ( F2 ( I H ))
>0
dI H
d 2 F1 ( F2 ( I H ))
< 0.
dI H2
and
17
I H* occurs if and only if
∞
∞
dF1 ! F2 (0)
= BH BV β H βV ∫ aQH (a) ∫ aQV (a)da da > 1 .
0
0
dI H
9
Hence, equations (3), (4), (7), and (8) constitute an age-structured of a host-vector
SI model.
*
a
− βV I H
V
Therefore, a unique non-trivial value of
NV ( 0) (t ) = 0 ,
*
The last equations can be reduced as
t
It is also clear that
]
]da = F (I
I H* = ∫ BH QH (a ) 1 − e − β H I V a da = F1 ( IV* ) ,
4
It is clear that
lim
[
B Q (a )[1 − e
( I H* , IV* ) satisfying
The existence of the corresponding non-trivial value of
IV* follows
18
immediately. The
LHS of (18) will be refereed as a threshold number R0 of the model. We conclude
that an endemic equilibrium
3 The existence of a threshold number
In this section we will show that there is a threshold number for the model
discussed above. Let us consider the limit values of equations (2) and (4). Whenever
t → ∞ , and by considering (5) holds, the equations (2) and (4) can be written as
∞
N H (t ) = ∫ BH QH (a)da ,
10
∞
β H I V ( s ) ds
−
I H (t ) = ∫ BH QH (a ) 1 − e ∫t −a
da .
0
11
( I H* , IV* ) ≠ (0,0) occurs if and only if R0 > 1 .
4 The Stability of the Equilibria
To investigate the stability of the equilibria we use the method in [1] and use the
lemma therein.
0
t
Lemma 1 (Brauer, 2001). Let
t
0
∞
0
∞
−
β V I H ( s ) ds
IV (t ) = ∫ BV QV (a) 1 − e ∫t −a
da .
0
and (12) show that the value of N H (t ) and NV (t )
12
where
f 0 (t ) is
a non-negative function with
negative function with
t
Equations (10)
a bounded non-negative function which
f (t ) ≤ f 0 (t ) + ∫ f (t − a) R(a)da ,
Similarly, equations (6) and (8) can be written as
NV (t ) = ∫ BV QV (a)da .
f (t ) be
satisfies an estimate of the form
13
are constants,
hence the equations for the age-structured host-vector SI model reduce to two
equations, (11) and (13).
∫
∞
0
R(a)da < 1.
Then
limt → ∞ f 0 (t ) = 0 and R(a ) is
a non-
limt →∞ f (t ) = 0 .
Proof. See [1]. It is also showed in [1] that the lemma is still true if the inequality in
the lemma is replaced by
f (t ) ≤ f 0 (t ) + ∫ sup t − a ≤ s ≤ t f ( s ) R(a )da .
t
0
19
A.K. SUPRIATNA & E. SOEWONO
Threshold Number of a Dengue Model
I H (t ) = I H* + v(t )
Further, we generalize Lemma 1 using a similar argument as in [1] as follows.
Lemma 2. Let
f j (t ), j = 1, 2 be bounded non-negative functions satisfying
t
t
0
t
with
∫
0
non-negative with
R j (a )da < 1.
Then
t
−
β H I V* ds − ∫ β H u ( s ) ds
da
v(t ) = − I H* + I H ( 0) (t ) + ∫ BH QH (a)1 − e ∫t −a
e t −a
0
t
0
f j 0 (t ) is
*
0
f 2 (t ) ≤ f 20 (t ) + ∫ sup t − a ≤ s ≤ t f1 ( s ) R2 (a)da
∞
and substitute these quantities into
t
−
β H [ I V + u ( s )] ds
I H* + v(t ) = I H ( 0 ) (t ) + ∫ BH QH (a )(1 − e ∫t −a
)da
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤ t f 2 ( s ) R1 (a)da ,
where
IV (t ) = IV* + u (t ) ,
and
equation (4) to obtain the following calculations:
limt→∞ f j 0 (t ) = 0 and R j (a) is
non-negative
∞
t
= − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
limt → ∞ f j (t ) = 0, j = 1, 2 .
*
0
t
*
−
β H u ( s ) ds
da
+ ∫ BH QH (a )1 − e − β H I V a e ∫t −a
0
t
4.1
The stability of the disease-free equilibrium
We investigate the stability of the disease-free equilibrium for the case of
R0 < 1 .
Consider the following inequalities.
*
t
t
t
*
*
β H u ( s ) ds
−
− ∫ BH QH (a )(1 − e − β H I V a )da + ∫ BH QH (a)1 − e − β H I V a e ∫t −a
da
0
0
t
t
β H I V ( s ) ds
t
1 − e ∫t −a
≤ ∫t − a β H IV ( s )ds ≤ aβ H sup t − a ≤ s ≤ t IV ( s ) .
−
∞
v(t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
20
t
β V I H ( s ) ds
−
t
1 − e ∫t −a
≤ ∫t − a βV I H ( s )ds ≤ aβV supt − a ≤ s ≤ t I H ( s ) .
21
∞
v(t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
*
t
t
*
β H u ( s ) ds
−
+ ∫ BH QH (a)e − β H IV a 1 − e ∫t −a
da
0
t
Hence we have,
t
t
−
β H I V ( s ) ds
I H (t ) = I H ( 0) (t ) + ∫ BH QH (a )(1 − e ∫t −a
)da
∞
0
≤ − ∫ BH QH (a)(1 − e − β H IV a )da + I H ( 0 ) (t )
t
≤ I H ( 0 ) (t ) + ∫ BH QH (a )(aβ H sup t − a ≤ s ≤ t IV ( s ))da
22
0
IV (t ) = IV ( 0 ) (t ) + ∫
t
0
0
∞
0
aBH β H QH (a)da < 1
then using Lemma 2 we conclude that
and
limt→∞ I H (t ) = 0
This shows that the disease-free equilibrium
∫
∞
0
23
aBV βV QV (a)da < 1 ,
and
( I H* , IV* ) = (0,0)
*
Hence, we have
t
∫
t
0
≤ IV ( 0) (t ) + ∫ BV QV (a)(aβV supt − a ≤ s ≤ t I H ( s ))da
If further we assume that
+ ∫ BH QH (a)e − β H IV a β H a sup t − a ≤ s ≤ t u ( s )da
t
β V I H ( s ) ds
BV QV (a)(1 − e ∫t −a
)da
−
*
t
lim t →∞ I V (t ) = 0 .
∞
t
v(t ) ≤ − ∫ BH QH (a )(1 − e − β H IV a )da + I H ( 0) (t ) + ∫ sup t −a≤s≤t u ( s) BH QH (a )e − β H IV a β H ada
*
t
*
0
∞
Next define
f (t ) = v(t ) , f 0 (t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
*
t
R(a) = BH QH (a)e − β H I V a β H a .
*
is globally stable.
It
can
be
shown
that
∫
∞
0
, and
R(a)da < 1 .
If
v(t ) = u (t ) , that is, the perturbation is symmetrical, then by Lemma 1 we conclude
4.2
The stability of the endemic equilibrium
The endemic equilibrium
perturbations
of
I H*
*
H
*
V
( I , I ) appears
and
IV* ,
only if
respectively,
by
that
R0 > 1 .
v(t )
Let us
and
u (t ) .
see the
Define
limt → ∞ v(t ) = 0 .
lim t→∞ I V (t ) = IV* can
equilibrium
This
shows
that
lim t →∞ I H (t ) = I H* .
The
fact
that
be shown analogously. Hence, we conclude that the endemic
( I H* , IV* ) ≠ (0,0)
is globally stable if
R0 > 1 .
Threshold Number of a Dengue Model
5
Concluding Remarks
We found a threshold value determining the appearance of the endemic equilibrium,
in which this equilibrium is occurring only if this threshold value is greater than
one. The global stability of this equilibrium is confirmed as long as the perturbation
of the equilibrium is symmetrical.
References
[1] Brauer, F. (2002). A Model for an SI Disease in an Age-Structured Population.
Discrete and Continuous Dynamical Systems – Series B . 2 , 257-264.
[2] Diekmann, O. & J.A.P. Heesterbeek (2000). Mathematical Epidemiology of
Infectious Diseases . John Wiley & Son. New York.
[3] Esteva, L. & C. Vargas (1998). Analysis of a Dengue Disease Transmission
Model, Math. Biosci. 150 , 131-151.
[4] Supriatna, A.K. & E. Soewono (2003). Critical Vaccination Level for Dengue
Fever Disease Transmission. SEAMS-GMU Proceedings of International
Conference 2003 on Mathematics and Its Applications , pages 208-217.
[5] Soewono, E.
& A.K. Supriatna (2001). A Two-dimensional Model for
Transmission of Dengue Fever Disease. Bull. Malay. Math. Sci. Soc. 24, 49-57.
[6] Soewono, E. & A.K. Supriatna. A Paradox of Vaccination Predicted by a Simple
Host-Vector Epidemic Model (to appear in an Indian Journal of Mathematics).
ASEP K. S UPRIATNA: Department of Mathematics, Universitas Padjadjaran, Jl. Raya
Bandung-Sumedang km 21, Sumedang 45363, Indonesia. Phone/Fax: +62 +22
7794696
E DY S OEWONO: Department of Mathematics & Center of Mathematics, Institut
Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia. Phone/Fax: +62
+22 250 8126
A THRESHOLD NUMBER FOR DENGUE DISEASE
ENDEMICITY IN AN AGE STRUCTURED MODEL1
d
S H = BH − β H S H IV − µ H S H ,
dt
d
SV = BV − βV SV I H − µV SV ,
dt
Asep K. Supriatna a & Edy Soewonob
where
a
Department of Mathematics, Universitas Padjadjaran, Indonesia
b Department of Mathematics, ITB, Indonesia
Abstract. In this paper we present a model for dengue disease transmission with
an assumption that individuals in the under-laying populations experience a
monotonically non-increasing survival rate. We show that there is a threshold for
the disease transmission, below which the disease will stop (endemic equilibrium is
not appearing) and above which the disease will stay endemic (endemic equilibrium
is appearing). We also investigate the stability of this endemic equilibrium.
Key-words: Dengue Modeling, Threshold Number, Stability of an Equilibrium
Point.
d
I H = β H S H IV − µ H I H ,
dt
d
IV = βV SV I H − µV IV ,
dt
SH
SV
= the number of susceptibles in the host population
= the number of susceptibles in the vector population
IH
IV
= the number of infectives in the host population
= the number of infectives in the vector population
= host recruitment rate; BV = vector recruitment rate
BH
µH
βH
βV
= host death rate; µV = vector death rate
= the transmission probability from vector to host
= the transmission probability from host to vector
1 Introduction
The model above based on the assumption that the host population
Reducing the number of dengue fever disease prevalence is regarded as an
important public health concern in Indonesia, and in many tropical countries,
since the disease is very dangerous that may lead to fatality. To find a good
management in controlling the disease, ones need to understand the dynamics of
the disease. Many mathematical models have been devoted to address this issue,
examples are [3],[4],[5], and [6]. However, most of the authors have ignored the
presence of age structure in mortality rate of the populations in their models. In
this paper we present a model for dengue disease transmission with the inclusion
that individuals in the under-laying populations experience a monotonically nonincreasing survival rate as their age goes by. We show that there is an endemic
threshold, below which the disease will stop, and above which the disease will stay
endemic.
vector population
2 Host-Vector Model with Monotonic Non-Increasing
Survival Rate
The model discussed here is analogous to the following age-unstructured hostvector SI model:
the host, and
SV
NV
and
each are divided into two compartments,
IV
SH
NH
and
and the
IH
for
for the vector.
An analogous age-structured one for the above model is made by generalizing the
QH (a ) , a function of age describing the
fraction of human population who survives to the age of a or more, such that,
model in [1]. Suppose that there exists
QH (0) = 1 and QH (a ) is a non-negative and monotonically non-increasing
0 ≤ a ≤ ∞ . If it is assumed that human life expectancy is finite, then
∫
Let
∞
0
QH (a)da = L < ∞ and
NH = SH + IH .
I H ( 0) (t )
∞
0
aQH (a )da < ∞
Further, let also assume that
denotes, respectively, the numbers of
S H (0) who survive
t . Then we have
numbers of
at time
∫
at time
t , and
for
1
N H ( 0 ) (t ) , S H ( 0) (t ) ,
and
N H (0) who survive at time t , the
the numbers of
I H (0) who survive
t
N H (t ) = N H ( 0 ) (t ) + ∫ BH QH (a)da .
2
0
Since the per capita rate of infection in human population at time t is
1
Presented in the International Conference on Applied Mathematics (ICAM05) in
Bandung, August 22-26, 2005. Part of the works in this paper is funded by the Indonesian
Government, through the scheme of Penelitian Hibah Bersaing XII (SPK No.
011/P4T/DPPM/PHB/III/2004).
β H I V (t ) ,
the number of susceptibles at time t is given by
t
t
−
β H I V ( s ) ds
S H (t ) = S H ( 0) (t ) + ∫ BH QH (a )e ∫t −a
da .
0
3
A.K. SUPRIATNA & E. SOEWONO
Threshold Number of a Dengue Model
See also [2]. The number of human infectives is
I H (t ) = N H (t ) − S H (t ) , given by
t
−
β H I V ( s ) ds
I H (t ) = I H ( 0) (t ) + ∫ BH Q(a ) 1 − e ∫t −a
da .
0
The Equilibrium of the system is given by
∞
t
t→∞
N H ( 0) (t ) = 0 ,
lim
t→∞
S H ( 0) (t ) = 0 , and
lim
0
I =∫
*
V
t→∞
I H ( 0 ) (t ) = 0 .
5
Analogously, we can derive similar equations for the mosquitoes, which are
t
NV (t ) = NV ( 0) (t ) + ∫ BV QV (a )da ,
6
0
t
t
−
β V I H ( s ) ds
SV (t ) = SV ( 0) (t ) + ∫ BV QV (a)e ∫t −a
da ,
7
t
−
β V I H ( s ) ds
IV (t ) = IV ( 0 ) (t ) + ∫ BV QV (a) 1 − e ∫t −a
da .
0
8
0
lim
t→∞
∞
0
V
lim
t →∞
SV ( 0) (t ) = 0 , and
lim
t→∞
IV ( 0) (t ) = 0 .
2
*
H
14
).
15
*
− β H ∫0∞ BV QV ( a )(1− e − βV I H a ) da a
∞
16
I H* = F1 ( F2 ( I H* )) = ∫ BH QH (a )1 − e
da ,
0
*
*
Note that F1 ! F2 is bounded. It is easy to see that ( I H , IV ) = (0,0) is the disease-
free equilibrium. To find a non-trivial equilibrium (an endemic equilibrium), we could
observe the following
dF1 ( F2 ( I H ))
>0
dI H
d 2 F1 ( F2 ( I H ))
< 0.
dI H2
and
17
I H* occurs if and only if
∞
∞
dF1 ! F2 (0)
= BH BV β H βV ∫ aQH (a) ∫ aQV (a)da da > 1 .
0
0
dI H
9
Hence, equations (3), (4), (7), and (8) constitute an age-structured of a host-vector
SI model.
*
a
− βV I H
V
Therefore, a unique non-trivial value of
NV ( 0) (t ) = 0 ,
*
The last equations can be reduced as
t
It is also clear that
]
]da = F (I
I H* = ∫ BH QH (a ) 1 − e − β H I V a da = F1 ( IV* ) ,
4
It is clear that
lim
[
B Q (a )[1 − e
( I H* , IV* ) satisfying
The existence of the corresponding non-trivial value of
IV* follows
18
immediately. The
LHS of (18) will be refereed as a threshold number R0 of the model. We conclude
that an endemic equilibrium
3 The existence of a threshold number
In this section we will show that there is a threshold number for the model
discussed above. Let us consider the limit values of equations (2) and (4). Whenever
t → ∞ , and by considering (5) holds, the equations (2) and (4) can be written as
∞
N H (t ) = ∫ BH QH (a)da ,
10
∞
β H I V ( s ) ds
−
I H (t ) = ∫ BH QH (a ) 1 − e ∫t −a
da .
0
11
( I H* , IV* ) ≠ (0,0) occurs if and only if R0 > 1 .
4 The Stability of the Equilibria
To investigate the stability of the equilibria we use the method in [1] and use the
lemma therein.
0
t
Lemma 1 (Brauer, 2001). Let
t
0
∞
0
∞
−
β V I H ( s ) ds
IV (t ) = ∫ BV QV (a) 1 − e ∫t −a
da .
0
and (12) show that the value of N H (t ) and NV (t )
12
where
f 0 (t ) is
a non-negative function with
negative function with
t
Equations (10)
a bounded non-negative function which
f (t ) ≤ f 0 (t ) + ∫ f (t − a) R(a)da ,
Similarly, equations (6) and (8) can be written as
NV (t ) = ∫ BV QV (a)da .
f (t ) be
satisfies an estimate of the form
13
are constants,
hence the equations for the age-structured host-vector SI model reduce to two
equations, (11) and (13).
∫
∞
0
R(a)da < 1.
Then
limt → ∞ f 0 (t ) = 0 and R(a ) is
a non-
limt →∞ f (t ) = 0 .
Proof. See [1]. It is also showed in [1] that the lemma is still true if the inequality in
the lemma is replaced by
f (t ) ≤ f 0 (t ) + ∫ sup t − a ≤ s ≤ t f ( s ) R(a )da .
t
0
19
A.K. SUPRIATNA & E. SOEWONO
Threshold Number of a Dengue Model
I H (t ) = I H* + v(t )
Further, we generalize Lemma 1 using a similar argument as in [1] as follows.
Lemma 2. Let
f j (t ), j = 1, 2 be bounded non-negative functions satisfying
t
t
0
t
with
∫
0
non-negative with
R j (a )da < 1.
Then
t
−
β H I V* ds − ∫ β H u ( s ) ds
da
v(t ) = − I H* + I H ( 0) (t ) + ∫ BH QH (a)1 − e ∫t −a
e t −a
0
t
0
f j 0 (t ) is
*
0
f 2 (t ) ≤ f 20 (t ) + ∫ sup t − a ≤ s ≤ t f1 ( s ) R2 (a)da
∞
and substitute these quantities into
t
−
β H [ I V + u ( s )] ds
I H* + v(t ) = I H ( 0 ) (t ) + ∫ BH QH (a )(1 − e ∫t −a
)da
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤ t f 2 ( s ) R1 (a)da ,
where
IV (t ) = IV* + u (t ) ,
and
equation (4) to obtain the following calculations:
limt→∞ f j 0 (t ) = 0 and R j (a) is
non-negative
∞
t
= − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
limt → ∞ f j (t ) = 0, j = 1, 2 .
*
0
t
*
−
β H u ( s ) ds
da
+ ∫ BH QH (a )1 − e − β H I V a e ∫t −a
0
t
4.1
The stability of the disease-free equilibrium
We investigate the stability of the disease-free equilibrium for the case of
R0 < 1 .
Consider the following inequalities.
*
t
t
t
*
*
β H u ( s ) ds
−
− ∫ BH QH (a )(1 − e − β H I V a )da + ∫ BH QH (a)1 − e − β H I V a e ∫t −a
da
0
0
t
t
β H I V ( s ) ds
t
1 − e ∫t −a
≤ ∫t − a β H IV ( s )ds ≤ aβ H sup t − a ≤ s ≤ t IV ( s ) .
−
∞
v(t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
20
t
β V I H ( s ) ds
−
t
1 − e ∫t −a
≤ ∫t − a βV I H ( s )ds ≤ aβV supt − a ≤ s ≤ t I H ( s ) .
21
∞
v(t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
*
t
t
*
β H u ( s ) ds
−
+ ∫ BH QH (a)e − β H IV a 1 − e ∫t −a
da
0
t
Hence we have,
t
t
−
β H I V ( s ) ds
I H (t ) = I H ( 0) (t ) + ∫ BH QH (a )(1 − e ∫t −a
)da
∞
0
≤ − ∫ BH QH (a)(1 − e − β H IV a )da + I H ( 0 ) (t )
t
≤ I H ( 0 ) (t ) + ∫ BH QH (a )(aβ H sup t − a ≤ s ≤ t IV ( s ))da
22
0
IV (t ) = IV ( 0 ) (t ) + ∫
t
0
0
∞
0
aBH β H QH (a)da < 1
then using Lemma 2 we conclude that
and
limt→∞ I H (t ) = 0
This shows that the disease-free equilibrium
∫
∞
0
23
aBV βV QV (a)da < 1 ,
and
( I H* , IV* ) = (0,0)
*
Hence, we have
t
∫
t
0
≤ IV ( 0) (t ) + ∫ BV QV (a)(aβV supt − a ≤ s ≤ t I H ( s ))da
If further we assume that
+ ∫ BH QH (a)e − β H IV a β H a sup t − a ≤ s ≤ t u ( s )da
t
β V I H ( s ) ds
BV QV (a)(1 − e ∫t −a
)da
−
*
t
lim t →∞ I V (t ) = 0 .
∞
t
v(t ) ≤ − ∫ BH QH (a )(1 − e − β H IV a )da + I H ( 0) (t ) + ∫ sup t −a≤s≤t u ( s) BH QH (a )e − β H IV a β H ada
*
t
*
0
∞
Next define
f (t ) = v(t ) , f 0 (t ) = − ∫ BH QH (a)(1 − e − β H I V a )da + I H ( 0) (t )
*
t
R(a) = BH QH (a)e − β H I V a β H a .
*
is globally stable.
It
can
be
shown
that
∫
∞
0
, and
R(a)da < 1 .
If
v(t ) = u (t ) , that is, the perturbation is symmetrical, then by Lemma 1 we conclude
4.2
The stability of the endemic equilibrium
The endemic equilibrium
perturbations
of
I H*
*
H
*
V
( I , I ) appears
and
IV* ,
only if
respectively,
by
that
R0 > 1 .
v(t )
Let us
and
u (t ) .
see the
Define
limt → ∞ v(t ) = 0 .
lim t→∞ I V (t ) = IV* can
equilibrium
This
shows
that
lim t →∞ I H (t ) = I H* .
The
fact
that
be shown analogously. Hence, we conclude that the endemic
( I H* , IV* ) ≠ (0,0)
is globally stable if
R0 > 1 .
Threshold Number of a Dengue Model
5
Concluding Remarks
We found a threshold value determining the appearance of the endemic equilibrium,
in which this equilibrium is occurring only if this threshold value is greater than
one. The global stability of this equilibrium is confirmed as long as the perturbation
of the equilibrium is symmetrical.
References
[1] Brauer, F. (2002). A Model for an SI Disease in an Age-Structured Population.
Discrete and Continuous Dynamical Systems – Series B . 2 , 257-264.
[2] Diekmann, O. & J.A.P. Heesterbeek (2000). Mathematical Epidemiology of
Infectious Diseases . John Wiley & Son. New York.
[3] Esteva, L. & C. Vargas (1998). Analysis of a Dengue Disease Transmission
Model, Math. Biosci. 150 , 131-151.
[4] Supriatna, A.K. & E. Soewono (2003). Critical Vaccination Level for Dengue
Fever Disease Transmission. SEAMS-GMU Proceedings of International
Conference 2003 on Mathematics and Its Applications , pages 208-217.
[5] Soewono, E.
& A.K. Supriatna (2001). A Two-dimensional Model for
Transmission of Dengue Fever Disease. Bull. Malay. Math. Sci. Soc. 24, 49-57.
[6] Soewono, E. & A.K. Supriatna. A Paradox of Vaccination Predicted by a Simple
Host-Vector Epidemic Model (to appear in an Indian Journal of Mathematics).
ASEP K. S UPRIATNA: Department of Mathematics, Universitas Padjadjaran, Jl. Raya
Bandung-Sumedang km 21, Sumedang 45363, Indonesia. Phone/Fax: +62 +22
7794696
E DY S OEWONO: Department of Mathematics & Center of Mathematics, Institut
Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia. Phone/Fax: +62
+22 250 8126