Survival Rate

Point. 1 Introduction Reducing the number of dengue fever disease prevalence is regarded as an important public health concern in Indonesia, and in ma...

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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. Supriatnaa & Edy Soewonob

d I H = β H S H IV − µ H I H , dt d IV = βV SV I H − µV IV , dt

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.

SH SV IH IV BH

µH βH βV

= the number of susceptibles in the host population = the number of susceptibles in the vector population = the number of infectives in the host population = the number of infectives in the vector population = host recruitment rate; BV = vector recruitment rate = 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 the

and

IH

for

for the vector.

An analogous age-structured one for the above model is made by generalizing the model in [1]. Suppose that there exists QH (a ) , a function of age describing the fraction of human population who survives to the age of a or more, such that,

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

numbers of at time



S H (0) who survive t . Then we have

at time

t , and

1

N H ( 0 ) (t ) , S H ( 0) (t ) ,

N H (0) who survive the numbers of

for

at time

and

t , the

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

Threshold Number of a Dengue Model

See also [2]. The number of human infectives is

A.K. SUPRIATNA & E. SOEWONO

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

0

t − β V I H ( s ) ds   IV (t ) = IV ( 0 ) (t ) + ∫ BV QV (a ) 1 − e ∫t −a  da . 0  

V

dF1 ( F2 ( I H )) >0 dI H

10

).

d 2 F1 ( F2 ( I H )) < 0. dI H2

and

14 15

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

The existence of the corresponding non-trivial value of

18

IV* follows immediately. The

LHS of (18) will be refereed as a threshold number R0 of the model. We conclude that an endemic equilibrium

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

* H

free equilibrium. To find a non-trivial equilibrium (an endemic equilibrium), we could observe the following

9

3 The existence of a threshold number

2

V

Therefore, a unique non-trivial value of

Hence, equations (3), (4), (7), and (8) constitute an age-structured of a host-vector SI model.



0

* a − βV I H

* − 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-

8

lim lim lim NV ( 0) (t ) = 0 , SV ( 0) (t ) = 0 , and IV ( 0) (t ) = 0 . t→∞ t →∞ t→∞

N H (t ) = ∫ BH QH (a )da ,



*

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

( 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

∞ − β H I V ( s ) ds   I H (t ) = ∫ BH QH (a ) 1 − e ∫t −a  da . 0   t

11

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

Threshold Number of a Dengue Model

A.K. SUPRIATNA & E. SOEWONO

Further, we generalize Lemma 1 using a similar argument as in [1] as follows. Lemma 2. Let

I H (t ) = I H* + v(t ) and IV (t ) = IV* + u (t ) , and substitute these quantities into equation (4) to obtain the following calculations:

f j (t ), j = 1, 2 be bounded non-negative functions satisfying

t

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 , t

0

t

f 2 (t ) ≤ f 20 (t ) + ∫ sup t − a ≤ s ≤ t f1 ( s ) R2 (a )da with



f j 0 (t ) is ∞ 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

where

*

0

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 .

is globally stable.



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

Next define

*

0



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 . *

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*

that

( I , I ) appears only if R0 > 1 . Let us see the

and

* H

* V

IV* , respectively, by

v(t )

and

u (t ) .

Define

limt → ∞ v(t ) = 0 .

This

shows

that

lim t →∞ I H (t ) = I H* . The fact that

lim t→∞ I V (t ) = IV* can be shown analogously. Hence, we conclude that the endemic equilibrium

( 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. SUPRIATNA: Department of Mathematics, Universitas Padjadjaran, Jl. Raya Bandung-Sumedang km 21, Sumedang 45363, Indonesia. Phone/Fax: +62 +22 7794696 EDY SOEWONO: Department of Mathematics & Center of Mathematics, Institut Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia. Phone/Fax: +62 +22 250 8126