A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate
A.K. Supriatna
1
, Edy Soewono
2
1Department of Mathematics, Universitas Padjadjaran, km 21 Bandung-Sumedang 45363, Indonesia fax: 062-22-7794696, email:
asupriatyahoo.com.au ; aksupriatnabdg.centrin.net.id
2 Financial and Industrial Mathematics Group ITB, Bandung 40132, Indonesia
Abstract
In this paper we present an SIR model for disease transmission with an assumption that individuals in the under-laying demographic population experience a monotonically decreasing survival
rate. We show that the results in an analogous SI model for disease transmission are the special case of the SIR model in this paper. We found that there is a threshold for the disease transmission determining the
existence and the absence of the endemic equilibrium. We investigate the stability of this equilibrium via a Gronwall-like inequality theorem. Unlike in the SI model, the threshold for the existence is not equivalent
to the threshold for the stability of the equilibrium. We provide an additional condition which consistently generalizes the results in the SI model..
Keywords
: Disease Modeling, SIR Model, Threshold Number, Stability of an Equilibrium Point
I. Introduction
Age structure is among the important factors affecting the dynamics of a population in
relation to the spread of contagious diseases. To study the effect of age structure in the dynamics of
contagious diseases, at least there are two approaches, first by developing a population
model with continuous age [1,2] and second by developing a population with age groups [3]. A
model of SI disease transmission is studied in [1] and a model SIS disease transmission is studied in
[2] by assuming continuous age.
An SI model only fits to diseases that cause an infective individual remains infective for
life. To increase realism, in this paper we present a model for an SIR disease transmission by
assuming continuous age. Here we assume individuals in the under-laying population
experience a monotonically decreasing survival rate as their age goes by. We also assume that
there is a density-dependent but age-independent birth rate. We show that there is an endemic
threshold, below which the disease will stop, and above which the disease will stay endemic. The
results in the SI disease model in [1] generalize into the SIR model.
II. The Mathematical Model
The model discussed here is the generalization of the model in [1] to include an R
compartment as an attempt to increase the realism of the model. Throughout the paper we use the
following notations:
H
N =
Total number of individuals in the population
H
S =
The number of susceptible individuals in the population
H
I =
The number of infective individuals in the population
H
R =
The number of recover or immune individual in the population
H
B =
The recruitment rate or the birth rate
H
β
=
The transmission probability of the disease We assume that the population
H
N
is divided into three compartments,
H
S
,
H
I
, and
H
R
, such that
H H
H H
N S
I R
= + +
. To include age structure, suppose that there exists
, a function of age describing the fraction of human
population who survives to the age of a or more, such that,
H
Q a
1
H
Q =
and is a non-negative
and monotonically decreasing for . If it
is assumed that life expectancy is finite, then
H
Q a
a ≤ ≤ ∞
H
Q a da
L
∞
= ∞
∫
and .
H
aQ a da
∞
∞
∫
1 Further, let also assume that
, ,
, and denotes,
respectively, the numbers of
H
N t
H
S t
H
I t
H
R t
H
N
0,
H
S
0,
H
I
0, and
H
R
0 who survive at time
t
. Then we have
t H
H H
H
N t
N t
B Q a da
= +
∫
. 2
Since the per capita rate of infection in the population at time t is
H H
I t
β , then the number
of susceptible at time t is given by
t H H
t a
t I
s ds H
H H
H
S t
S t
B Q a e
da
β
−
−
∫ =
+
∫
. 3
If the rate of recovery is γ then the number of
infective at time is given by
t
1 1
.
t t
H H H
t a t a
t H H
t a H
t I
s ds ds
H H
H H
t I
s ds a
H H
H
I t
I t
B Q a
e e
da I
t B Q
a e
e da
β γ
β γ
− −
−
− −
− −
⎡ ⎤
∫ ∫
= +
− ⎢
⎥ ⎣
⎦ ⎡
⎤ ∫
= +
− ⎢
⎥ ⎣
⎦
∫ ∫
4 Furthermore, considering that
then we have
H H
H H
R t
N t
S t
I t
= −
−
1 1
1 .
t H H
t a t
H H t a
H t
H H t a
H
t H
H H
H t
I s ds
H H
H t
I s ds
a H
H H
t I
s ds a
H H
H
R t
N t
B Q a da
S t
B Q a e
da I
t B Q
a e
e da
R t
B Q a
e e
da
β β
γ β
γ
− −
−
− −
− −
−
= +
∫ −
− ⎡
⎤ ∫
− −
− ⎢
⎥ ⎣
⎦ ⎡
⎤ ∫
⎡ ⎤
= +
− −
⎢ ⎥ ⎣
⎦ ⎣
⎦
∫ ∫
∫ ∫
5 It is clear that
lim
H
N t
t =
→ ∞
,
lim
H
S t
t =
→ ∞
,
lim
H
I t
t =
→ ∞
, and
lim
H
R t
t =
→ ∞
. 6
Hence, equations 3, 4, 5, and 6 constitute an SIR age structured disease model.
III. The Existence of a Threshold Number
