A THRESHOLD NUMBER FOR DENGUE DISEASE
ENDEMICITY IN AN AGE STRUCTURED MODEL
1Asep K. Supriatnaa & Edy Soewonob
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.
1
Introduction
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 non-increasing 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.
2
Host-Vector Model with Monotonic Non-Increasing
Survival Rate
The model discussed here is analogous to the following age-unstructured host-vector SI model:
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).
A.K. SUPRIATNA & E. SOEWONO
,
,
,
,
V V H V V V V
V H V V V V
H H V H H H H
H V H H H H
I
I
S
I
dt
d
S
I
S
B
S
dt
d
I
I
S
I
dt
d
S
I
S
B
S
dt
d
µ
β
µ
β
µ
β
µ
β
−
=
−
−
=
−
=
−
−
=
where
=
HS
the number of susceptibles in the host population=
VS
the number of susceptibles in the vector population=
HI
the number of infectives in the host population=
VI
the number of infectives in the vector population=
HB
host recruitment rate;B
V=
vector recruitment rate=
H
µ
host death rate;µ
V=
vector death rate=
H
β
the transmission probability from vector to host=
Vβ
the transmission probability from host to vectorThe model above based on the assumption that the host population
N
H and thevector population
N
V each are divided into two compartments,S
H andI
H forthe host, and
S
V andI
Vfor the vector.An analogous age-structured one for the above model is made by generalizing the model in [1]. Suppose that there exists
Q
H(a
)
, a function of age describing the fraction of human population who survives to the age of a or more, such that,1
)
0
(
=
H
Q
andQ
H(a
)
is a non-negative and monotonically non-increasing for∞
≤
≤
a
0
. If it is assumed that human life expectancy is finite, then∫
∞∞
<
=
0
Q
H(
a
)
da
L
and∫
∞∞
<
0
aQ
H(
a
)
da
1Let
N
H=
S
H+
I
H. Further, let also assume thatN
H(0)(
t
)
,S
H(0)(
t
)
, and)
(
) 0 (t
I
H denotes, respectively, the numbers ofN
H(0) who survive at timet
, thenumbers of
S
H(0) who survive at timet
, and the numbers ofI
H(0) who survive at timet
. Then we have∫
+
=
tH H H
H
t
N
t
B
Q
a
da
N
0 ) 0
(
(
)
(
)
)
(
. 2Since the per capita rate of infection in human population at time t is
β
HI
V(t
)
, the number of susceptibles at time t is given by∫
∫
+
=
t −− I sdsH H H
H
t
S
t
B
Q
a
e
da
S
t
a t HV
0
) (
) 0
(
(
)
(
)
)
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 −− I sdsH H
H
t
I
t
B
Q
a
e
da
I
t
a t HV
0
) (
) 0
(
(
)
(
)
1
)
(
β . 4It is clear that
0
)
(
lim
) 0
(
=
∞
→
N
t
t
H ,(
)
0
lim
) 0(
=
∞
→
S
t
t
H , and(
)
0
lim
) 0(
=
∞
→
I
t
t
H . 5Analogously, we can derive similar equations for the mosquitoes, which are
∫
+
=
tV V V
V
t
N
t
B
Q
a
da
N
0 ) 0
(
(
)
(
)
)
(
, 6∫
∫
+
=
t −− I sdsV V V
V
t
S
t
B
Q
a
e
da
S
t
a t VH
0
) (
) 0
(
(
)
(
)
)
(
β , 7∫
∫
−
+
=
t −− I sdsV V V
V
t
I
t
B
Q
a
e
da
I
t
a t VH
0
) (
) 0
(
(
)
(
)
1
)
(
β . 8It is also clear that
0
)
(
lim
) 0
(
=
∞
→
N
t
t
V ,(
)
0
lim
) 0(
=
∞
→
S
t
t
V , and(
)
0
lim
) 0(
=
∞
→
I
t
t
V . 9Hence, equations (3), (4), (7), and (8) constitute an age-structured of a host-vector SI model.
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∫
∞=
0
(
)
)
(
t
B
Q
a
da
N
H H H , 10∫
∞ −
∫
−
=
−0
) (
1
)
(
)
(
t
B
Q
a
e
da
I
t
a t HIV sds
H H H
β
. 11
Similarly, equations (6) and (8) can be written as
∫
∞=
0
(
)
)
(
t
B
Q
a
da
N
V V V . 12∫
∞ −
∫
−
=
−0
) (
1
)
(
)
(
t
B
Q
a
e
da
I
t
a t VIHsds
V V V
β
. 13
Equations (10) and (12) show that the value of
N
H(t
)
andN
V(t
)
are constants, hence the equations for the age-structured host-vector SI model reduce to two equations, (11) and (13).A.K. SUPRIATNA & E. SOEWONO
The Equilibrium of the system is given by
(
I
*H,
I
V*)
satisfying[
1
]
(
)
)
(
*1 0
* *
V a
I H
H
H
B
Q
a
e
da
F
I
I
=
−
HV=
∫
∞ −β, 14
[
1
]
(
)
)
(
*2 0
* *
H a
I V
V
V
B
Q
a
e
da
F
I
I
=
−
VH=
∫
∞ −β .15 The last equations can be reduced as
∫
∞ − ∫ −
−
=
=
∞ −0
) 1 )( ( *
2 1
* 0
*
1
)
(
))
(
(
F
I
B
Q
a
e
da
F
I
a da e a Q B
H H H
H
a H I V V V
H β
β
, 16
Note that
F
1!
F
2 is bounded. It is easy to see that(
,
)
(
0
,
0
)
* *=
V HI
I
is thedisease-free equilibrium. To find a non-trivial equilibrium (an endemic equilibrium), we could observe the following
0
))
(
(
21
>
H H
dI
I
F
dF
and 1
(
22(
))
0
2<
HH
dI
I
F
F
d
. 17
Therefore, a unique non-trivial value of
I
H*occurs if and only if1
)
(
)
(
)
0
(
0 0
2
1
>
=
B
B
∫
∞aQ
a
∫
∞aQ
a
da
da
dI
F
dF
V H
V H V H H
β
β
!
. 18The existence of the corresponding non-trivial value of
I
V*follows immediately. The LHS of (18) will be refereed as a threshold number R0 of the model. We concludethat an endemic equilibrium
(
*,
*)
≠
(
0
,
0
)
V HI
I
occurs if and only ifR
0>
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.
Lemma 1 (Brauer, 2001). Let
f
(t
)
be a bounded non-negative function whichsatisfies an estimate of the form
∫
−
+
≤
tda
a
R
a
t
f
t
f
t
f
0
0
(
)
(
)
(
)
)
(
,where
f
0(
t
)
is a non-negative function withlim
t→∞f
0(
t
)
=
0
andR
(a
)
is anon-negative function with
(
)
1
.
0
<
∫
∞R
a
da
Thenlim
t→∞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
∫
−≤≤+
≤
tt s a
t
f
s
R
a
da
t
f
t
f
0
0
(
)
sup
(
)
(
)
)
Threshold Number of a Dengue Model
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 s at
f
s
R
a
da
t
f
t
f
0 2 1
10
1
(
)
(
)
sup
(
)
(
)
,∫
−≤≤+
≤
t t s at
f
s
R
a
da
t
f
t
f
0 1 2
20
2
(
)
(
)
sup
(
)
(
)
where
f
j0(
t
)
is non-negative withlim
t→∞f
j0(
t
)
=
0
andR
j(a
)
is non-negativewith
(
)
1
.
0
<
∫
∞da
a
R
j Thenlim
t→∞f
j(
t
)
=
0
,
j
=
1
,
2
.4.1
The stability of the disease-free equilibrium
We investigate the stability of the disease-free equilibrium for the case of
R
0<
1
. Consider the following inequalities.)
(
sup
)
(
1
e
()I
s
ds
a
H t astI
Vs
t a t H V ds
s I
t
a t HV
≤ ≤ − − −
≤
∫
≤
∫
−
−ββ
β
. 20)
(
sup
)
(
1
e
()I
s
ds
a
V t astI
Hs
t a t V H ds
s I
t
a t VH
≤ ≤ − − −
≤
∫
≤
∫
−
−ββ
β
. 21Hence we have,
∫
∫
≤ ≤ − −+
≤
∫
−
+
=
− t V t s a t H H H Ht I sds
H H H H
da
s
I
a
a
Q
B
t
I
da
e
a
Q
B
t
I
t
I
t a t HV0 ) 0 ( 0 ) ( ) 0 (
))
(
sup
)(
(
)
(
)
1
)(
(
)
(
)
(
β
β 22∫
∫
≤ ≤ − −+
≤
∫
−
+
=
− t H t s a t V V V Vt I sds
V V V V
da
s
I
a
a
Q
B
t
I
da
e
a
Q
B
t
I
t
I
t a t V H0 ) 0 ( 0 ) ( ) 0 (
))
(
sup
)(
(
)
(
)
1
)(
(
)
(
)
(
β
β 23If further we assume that
(
)
1
0
<
∫
∞aB
Hβ
HQ
Ha
da
and(
)
1
0
<
∫
∞aB
Vβ
VQ
Va
da
, then using Lemma 2 we conclude thatlim
t→∞I
H(
t
)
=
0
andlim
t→∞I
V(
t
)
=
0
. This shows that the disease-free equilibrium(
*,
*)
(
0
,
0
)
=
V HI
I
is globally stable.4.2
The stability of the endemic equilibrium
The endemic equilibrium
(
I
*H,
I
V*)
appears only ifR
0>
1
. Let us see the perturbations ofI
H* and* V
I
, respectively, byv
(t
)
andu
(t
)
. DefineA.K. SUPRIATNA & E. SOEWONO
)
(
)
(
*t
v
I
t
I
H=
H+
and(
)
*(
)
t
u
I
t
I
V=
V+
, and substitute these quantities into equation (4) to obtain the following calculations:∫
−
∫
+
=
+
t −− I +usdsH H H
H
v
t
I
t
B
Q
a
e
da
I
t
a t H V
0 )] ( [ ) 0 ( *
)
1
)(
(
)
(
)
(
* β∫
∫
∫
−
∫
+
+
−
−
=
∫
∫
−
+
+
−
=
− − − − − ∞ − − −t Ia usds
H H H a I H H
t Ids usds
H H H H
da
e
e
a
Q
B
t
I
da
e
a
Q
B
da
e
e
a
Q
B
t
I
I
t
v
t a t H V H V H t a t H t a t HV0 ) ( ) 0 ( 0 0 ) ( ) 0 ( * * * *
1
)
(
)
(
)
1
)(
(
1
)
(
)
(
)
(
β β β β β∫
∫
∫
−
∫
+
−
−
+
−
−
=
− − − − ∞ −t Ia usds
H H
t I a
H H H t a I H H
da
e
e
a
Q
B
da
e
a
Q
B
t
I
da
e
a
Q
B
t
v
t a t H V H V H V H 0 ) ( 0 ) 0 ( * * *1
)
(
)
1
)(
(
)
(
)
1
)(
(
)
(
β β β β∫
∫
∫
∫
≤ ≤ − − ∞ − − − ∞ −+
+
−
−
≤
∫
−
+
+
−
−
=
− t t s a t H a I H H H t a I H Ht Ia usds
H H H t a I H H
da
s
u
a
e
a
Q
B
t
I
da
e
a
Q
B
da
e
e
a
Q
B
t
I
da
e
a
Q
B
t
v
V H V H t a t H V H V H 0 ) 0 ( 0 ) ( ) 0 ()
(
sup
)
(
)
(
)
1
)(
(
1
)
(
)
(
)
1
)(
(
)
(
* * * *β
β β β β βHence, we have
∫
∫
− ≤ ≤ − ∞ −+
+
−
−
≤
t H a I H H t s a t H t a I HH
Q
a
e
da
I
t
u
s
B
Q
a
e
ada
B
t
v
HV HV0 ) 0 ( * *
)
(
)
(
sup
)
(
)
1
)(
(
)
(
β ββ
Next define
f
(
t
)
=
v
(
t
)
, 0(
)
(
)(
1
)
(0)(
)
*
t
I
da
e
a
Q
B
t
f
H t a I H H V H+
−
−
=
∫
∞ −β , anda
e
a
Q
B
a
R
H a I H H V Hβ
β *)
(
)
(
=
− . It can be shown that(
)
1
0
<
∫
∞R
a
da
. If)
(
)
(
t
u
t
v
=
, that is, the perturbation is symmetrical, then by Lemma 1 we conclude thatlim
t→∞v
(
t
)
=
0
. This shows that*
)
(
lim
t→∞I
Ht
=
I
H. The fact that *)
(
lim
t→∞I
Vt
=
I
Vcan be shown analogously. Hence, we conclude that the endemicequilibrium
(
*,
*)
≠
(
0
,
0
)
V HI
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. SUPRIATNA: Department of Mathematics, Universitas Padjadjaran, Jl. Raya Bandung-Sumedang km 21, Sumedang 45363, Indonesia. Phone/Fax: +62 +22 7794696
