A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate.
A Condition for Stability in an SIR Age Structured Disease Model with
Decreasing Survival Rate
A.K. Supriatna1, Edy Soewono2
1Department of Mathematics, Universitas Padjadjaran, km 21 Bandung-Sumedang 45363, Indonesia
fax: 062-22-7794696, email: asupriat@yahoo.com.au; aksupriatna@bdg.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:
N H = Total number of individuals in the
population
S H = The number of susceptible individuals in
the population
I H = The number of infective individuals in the
population
RH = The number of recover or immune
individual in the population
BH = The recruitment rate or the birth rate
β H = The transmission probability of the disease
We assume that the population N H is
divided into three compartments, S H , I H , and
RH , such that N H = S H + I H + RH . To include age
structure, 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 decreasing for 0 ≤ a ≤ ∞ . If it
is assumed that life expectancy is finite, then
∫
∞
0
QH (a)da = L < ∞ and
∫
∞
0
aQH (a)da < ∞ .
1
Further, let also assume that N H (0) (t ) ,
S H (0) (t ) ,
I H (0) (t ) ,
and
RH (0) (t ) denotes,
respectively, the numbers of N H (0), S H (0),
I H (0), and RH (0) who survive at time t . Then
we have
N H (t ) = N H (0) (t ) + ∫ BH QH (a)da .
t
0
2
Since the per capita rate of infection in the
population at time t is β H I H (t ) , then the number
of susceptible at time t is given by
t
β H I H ( s ) ds
−
S H (t ) = S H (0) (t ) + ∫ BH QH (a )e ∫t −a
da .
t
3
If the rate of recovery is γ then the number of
infective at time t is given by
0
t
t
t
−
β H I H ( s ) ds ⎤ − ∫ γ H ds
⎡
t −a
I H (t ) = I H (0) (t ) + ∫ BH QH (a ) ⎢1 − e ∫t −a
da
⎥e
0
⎣
⎦
t
t
β H I H ( s ) ds ⎤ − aγ
−
⎡
H
= I H (0) (t ) + ∫ BH QH ( a ) ⎢1 − e ∫t −a
da.
⎥e
0
⎣
⎦
4
The Equilibrium of the system is given by
( S H* , I H* , RH* ) with I H* satisfying
I H* = ∫ BH QH (a ) ⎡1 − e − β H I H a ⎤ e − aγ H da .
11
0
⎣
⎦
It is easy to see that ( S H* , I H* , RH* ) = ( N H* , 0, 0) is the
disease-free equilibrium. To find a non-trivial
equilibrium (an endemic equilibrium), we could
observe the following.
∞
β H BH
RH (t ) = N H (0) (t ) + ∫ BH QH (a )da
t
t
−
β H I H ( s ) ds
− S H (0) (t ) − ∫ BH QH (a )e ∫t −a
da
f
t
0
t
( I H*
lim
5
*
IH
It is clear that
lim
lim
N H (0) (t ) = 0 ,
S H (0) (t ) = 0 ,
t→∞
t→∞
lim
lim
I H (0) (t ) = 0 , and
RH (0) (t ) = 0 .
t→∞
t→∞
=
6
Hence, equations (3), (4), (5), and (6) constitute an
SIR age structured disease model.
*
IH
∞
β H I H ( s ) ds
−
S H (t ) = ∫ BH QH ( a )e ∫t −a
da ,
, say
⎡1 − e− β H I H* a ⎤ e− aγ H
⎣⎢
⎦⎥
*
IH
→ 0+
∫
∞
0
→ ∞−
lim
0
QH ( a )
da
13
*
→∞
IH
∫
∞
0
⎡1 − e − β H I H* a ⎤ e − aγ H
⎢
⎥⎦
QH (a ) ⎣
da
*
βH I H
14
aQH (a )e − aγ H da
*
)
f (I H
β B
− H H
∫
∞
0
1 − e− βH I H a
*
QH (a )
*
βH IH
e − aγ H da
15
*
7
8
0
t
∞
β H I H ( s ) ds ⎤
−
⎡
− aγ H
⎤⎦ da
RH (t ) = ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎡⎣1 − e
0
⎣
⎦
=
β H BH
lim
=0
t
∞
β H I H ( s ) ds ⎤ − aγ
−
⎡
H
I H (t ) = ∫ BH QH (a) ⎢1 − e ∫t−a
da
⎥e
0
⎣
⎦
∫
12
*
f (I H
)
lim
In this section we will show that there is a
threshold number for the model discussed above.
Let us consider the following limit system of
equations which has the same behavior with the
system (3) to (6) whenever t → ∞ :
0
→ 0+
= β H BH
III. The Existence of a Threshold
Number
∞
) = β H BH
∞
I H*
β H I H*
This function is monotonically decreases with
0
N H (t ) = ∫ BH QH (a)da ,
0
⎡1 − e− β H I H* a ⎤ e− aγ H
⎢
⎥⎦
QH (a ) ⎣
da = 1.
*
βH I H
The LHS of (12) is a function of
Furthermore, considering that
RH (t ) = N H (t ) − S H (t ) − I H (t ) then we have
t
⎡
−
β H I H ( s ) ds ⎤ − aγ
H
− I H (0) (t ) − ∫ BH QH (a) ⎢1 − e ∫t − a
⎥ e da
0
⎣
⎦
t
t
⎡
−
β H I H ( s ) ds ⎤
− aγ H
⎤
= RH (0) (t ) + ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎣⎡1 − e
⎦ da.
0
⎣
⎦
∫
∞
*
9
t
10
Equations (7) show that the value of N H (t ) is
constant, hence the equations for the agestructured SIR model reduce to three equations,
(8) to (10).
Therefore, a unique non-trivial value of I H occurs
if and only if
R0 = β H BH
∫
∞
0
aQH (a )e− aγ H da > 1 .
16
Since the LHS of (16) determines the occurrence
of the non-trivial value of I H* , then it will be
refereed as a threshold number R0 of the model.
Hence,
an
endemic
equilibrium
( S H* , I H* , RH* ) ≠ ( N H* , 0, 0) occurs if and only
if R0 > 1 .
IV. The Stability of the Equilibria
To investigate the stability of the
equilibria we use the method in [1] and use the
lemma therein.
β H I H ( s ) ds
−
t
1 − e ∫t −a
≤ ∫t − a β H I H ( s )ds
t
LEMMA 4.1. (BRAUER, 2001). Let f (t ) be a
bounded non-negative function which satisfies an
estimate of the form
f (t ) ≤ f 0 (t ) + ∫ f (t − a) R(a)da ,
t
0
where f 0 (t ) is a non-negative function with
limt →∞ f 0 (t ) = 0 and R(a) is a non-negative
function
∫
with
limt →∞ f (t ) = 0 .
∞
0
R(a)da < 1.
≤ a β H supt − a ≤ s ≤ t I H ( s)
Hence we have,
I H (t ) = I H (0) (t )
t
−
β H I H ( s ) ds
+ ∫ BH QH (a)(1 − e ∫t −a
)e − aγ H da
t
≤ I H (0) (t )
0
t
+ ∫ BH QH (a)(a β H supt − a ≤ s ≤t I H ( s))e − aγ H da 19
Then
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 ) + ∫ supt − a ≤ s ≤t f ( s ) R(a)da .
And
RH (t ) = RH (0) (t )
t
−
β H I H ( s ) ds ⎤
⎡
− aγ H
⎤⎦ da
+ ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎡⎣1 − e
0
⎣
⎦
≤ RH (0) (t )
t
t
0
17
+ ∫ BH QH (a )(a β H supt − a ≤ s ≤ t I H ( s )) ⎡⎣1 − e − aγ H ⎤⎦ da
0
The following lemma is the extension of Brauer’s
lemma.
LEMMA 4.2. Let f j (t ), j = 1, 2 be bounded non-
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤t f1 ( s) R1 (a)da ,
t
f 2 (t ) ≤ f 20 (t ) + ∫ supt − a ≤ s ≤t f1 ( s) R2 (a)da ,
0
t
0
f j 0 (t ) is
non-negative
with
limt →∞ f j 0 (t ) = 0 and R j (a) is non-negative with
∫
∞
0
t
Moreover,
∫
negative functions satisfying
where
18
R j (a)da < 1. Then limt →∞ f j (t ) = 0, j = 1, 2 .
∞
f1 (t ) ≤ sup{ f10 (t ), f 20 (t )}
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a )}da
t
0
f 2 (t ) ≤ sup{ f10 (t ), f 20 (t )}
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a )}da
t
0
sup{ f1 (t ), f 2 (t )} ≤ sup{ f10 (t ), f 20 (t )}
and hence,
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a)}da
lim
I H (0) (t ) = 0 and
t→∞
BH QH (a)a β H e− aγ H da = R0 < 1
then
using
Lemma 4.1 we conclude that limt →∞ I H (t ) = 0 .
Next,
let
us
see
the
0
expression ∫ BH QH (a)a β H ⎡⎣1 − e − aγ H ⎤⎦ da
0
∞
which,
if R0 S = ∫ BH QH (a)a β H da , can be written in the
∞
R0S − R0 . Hence, if
0
∫
form
PROOF.
since
20
R0 S < 1 + R0
then
BH QH (a)a β H ⎡⎣1 − e− aγ H ⎤⎦ da < 1 (Appendix 1).
lim
Furthermore, since
RH (0) (t ) = 0 then using
t→∞
Lemma 4.2 we conclude that limt →∞ RH (t ) = 0 .
Consequently,
limt →∞ S H (t ) = limt →∞ ( N H (t ) − RH (t ) − I H (t )) = N H*
This shows that the disease-free equilibrium
( S H* I H* , IV* ) = ( N H* 0, 0) is globally stable.
∞
0
t
0
From Lemma 4.1 we conclude that
limt →∞ sup{ f1 (t ), f 2 (t )} = 0 , and this is suffice to
show that limt →∞ 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 R0 < 1 .
Consider the following inequalities.
4.2.
The stability of the endemic
The
endemic
equilibrium
equilibrium.
( S H* , I H* , RH* ) appears only if R0 > 1 . Let us see the
perturbations of I H* and RH* , respectively, by u (t )
and v (t ) . Define I H (t ) = I H* + u (t ) and substitute
this quantity into equation (4) to obtain the
following calculations.
I H* + u (t ) = I H (0) (t )
−
β H [ I H* + u ( s )] ds ⎞ − aγ
⎛
H
+ ∫ BH QH (a) ⎜1 − e ∫t −a
da
⎟e
0
⎝
⎠
u (t ) = − I H* + I H (0) (t )
RH* + v(t ) = RH (0) (t )
t
β H [ I H + u ( s )] ds
−
) ⎡⎣1 − e− aγ H ⎤⎦ da
+ ∫ BH QH (a )(1 − e ∫t −a
0
t
t
t
t
β H I H* ds − ∫ β H u ( s ) ds ⎞ − aγ
−
⎛
H
e t −a
da
+ ∫ BH QH (a) ⎜1 − e ∫t −a
⎟e
0
⎝
⎠
t
v(t ) = − RH* + RH (0) (t )
t
−
β H I H* ds − ∫ β H u ( s ) ds ⎞
⎛
− aγ H
e t −a
+ ∫ BH QH (a ) ⎜1 − e ∫t −a
⎟ ⎣⎡1 − e
⎦⎤ da
0
⎝
⎠
t
t
= − ∫ BH QH (a)(1 − e − β H I H a )e− aγ H da + I H (0) (t )
∞
*
β H u ( s ) ds ⎞ − aγ
−
⎛
H
da
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟e
0
⎝
⎠
∞
∞
*
− ∫ BH QH (a )(1 − e − β H I H a )da
t
t
*
t
−
β H u ( s ) ds ⎞
*
⎛
− aγ H
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟ ⎣⎡1 − e
⎦⎤ da
0
⎝
⎠
t
v(t ) = − ∫ BH QH (a)(1 − e− β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
∞
− ∫ BH QH ( a)(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da
t
t
β H u ( s ) ds ⎞ − aγ
−
*
⎛
H
da
+ ∫ BH QH (a) ⎜ 1 − e − β H I H a e ∫t −a
⎟e
0
⎝
⎠
u (t ) = − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
*
t
β H u ( s ) ds ⎞ − aγ
*
−
⎛
H
da
+ ∫ BH QH (a )e − β H I H a ⎜ 1 − e ∫t −a
⎟e
0
⎝
⎠
t
t
−
β H u ( s ) ds ⎞
*
⎛
− aγ H
⎤⎦ da
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟ ⎡⎣1 − e
0
⎝
⎠
t
v(t ) = − ∫ BH QH (a)(1 − e− β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
∞
t
β H u ( s ) ds ⎞
−
*
⎛
− aγ H
⎤⎦ da
+ ∫ BH QH (a )e − β H I H a ⎜1 − e ∫t −a
⎟ ⎡⎣1 − e
0
⎝
⎠
t
≤ − ∫ BH QH (a )(1 − e − β H I H a ) ⎣⎡1 − e − aγ H ⎦⎤ da + RH (0) (t )
≤ − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
∞
+ ∫ BH QH (a)e
*
a
− βH IH
0
β H ae
− aγ H
t
t
supt − a ≤ s ≤ t u ( s )da
Hence, we have
u (t ) ≤ − ∫ BH QH ( a)(1 − e − β H I H a )e − aγ H da + I H (0) (t )
*
v(t )
+ ∫ sup t − a ≤ s ≤t u ( s ) BH QH ( a )e − β H I H a β H ae − aγ H da
t
t
By
f1 (t ) = u (t ) ,
defining
R1 (a) = BH QH (a)e
*
− βH IH
a
β H ae − aγ ,
f10 (t ) = − ∫ BH QH (a)(1 − e
∞
t
and
H
*
a
− βH I H
)e − aγ H da + I H (0) (t )
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤t f1 ( s ) R1 (a)da .
, then we have
By
that R1 (a ) is non-negative with
∫
0
R1 (a)da < 1
(see Appendix 2). Then by Lemma 4.1 we have
limt →∞ f1 (t ) = 0 , means that limt →∞ I H (t ) = I H* .
Next, define RH (t ) = RH* + v(t ) and
substitute these quantities into equation (5) to
obtain the following calculations:
∞
*
+ ∫ supt − a ≤ s ≤ t u ( s ) BH QH ( a )e − β H I H a β H a ⎡⎣1 − e− aγ H ⎤⎦ da
t
*
defining f1 (t ) = u (t ) ,
R2 ( a ) = BH QH ( a )e
*
a
− βH IH
f 2 (t ) = v (t )
β H a ⎡⎣1 − e − aγ ⎤⎦ ,
f 20 (t ) = − ∫ BH QH (a )(1 − e
∞
t
H
− β H I H* a
and
) ⎡⎣1 − e − aγ H ⎤⎦ da + I H (0) (t )
f 2 (t ) ≤ f 20 (t ) + ∫ supt − a ≤ s ≤t f1 ( s ) R2 (a)da .
, then we have
t
0
∞
− ∫ BH QH ( a)(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + I H (0) (t )
0
t
We see that limt →∞ f10 (t ) = 0 and it can be shown
≤
t
*
0
*
0
Hence, we have
∞
*
+ ∫ BH QH ( a)e − β H I H a β H a ⎡⎣1 − e− aγ H ⎤⎦ supt − a ≤ s ≤ t u ( s )da
*
t
*
t
t
t
*
0
t
∞
*
t
0
∞
*
0
t
u (t ) = − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
t
= − ∫ BH QH (a )(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
*
0
t
*
0
We can show that in Appendix 3 that
limt →∞ f 20 (t ) = 0 and R2 (a) is non-negative with
∫
∞
R2 (a)da < 1 . Then by Lemma 4.2 we have
limt →∞ f 2 (t ) = 0 , means that lim t →∞ RH (t ) = RH* .
Finally, since limt →∞ N H (t ) is a constant,
lim t →∞ RH (t ) = RH* , and limt →∞ I H (t ) = I H* then
0
( S H* , I H* , RH* ) is globally stable.
V. Concluding Remarks
In this paper we have discussed an agestructured SIR disease model with a decreasing
survival rate. We found a threshold number for the
existence and uniqueness of an endemic
equilibrium,
that
is,
R0 = β H BH
∫
∞
0
aQH (a )e − aγ H da . As is the case of
the SI disease model discuss in [1], an endemic
equilibrium appears if R0 > 1 and disappears if
R0 < 1 . In the SI disease model, the threshold for
the existence of the equilibrium is also the
threshold for the stability of the equilibrium.
However, in our case in which there is a recover
compartment, there is an additional condition for
the equilibrium to be stable. Here we found that
there is a stable endemic equilibrium if R0 > 1
and R0 S = β H BH ∫ QH (a)ada < 1 + R0 , and there
∞
is a stable disease-free equilibrium if R0 < 1 and
0
R0 S = β H BH ∫ QH (a)ada < 1 + R0 . We notice that
∞
0
this condition is consistent with that in [1] if the
recovery rate γ = 0 , since in this case R0 is
equivalent to R0S . Hence, we conclude that the SI
model in [1] is naturally nested in the SIR model
discussed in this paper.
VI. References
[1] F. Brauer. A model for an SI disease in an
age-structured population. Discrete and
Continuous Dynamical Systems – Series B. 2
(2002), 257-264.
[2] S. Busenberg, M. Ianelli, and H.R. Thieme.
Global behavior of an age-structured epidemic
model, SIAM J. Math. Anal. 22 (1991),
1065-1080.
[3] H.W. Hethcote. An age-structured model for
pertussis transmission. Math. Biosc. 145
(1997), 89-136.
Appendix 1
∫
∞
0
∞
R0 S < 1 + R0
then
Proof:
It is straightforward from the definition that
R0 = ∫ BH QH (a)a β H e− aγ H da
∞
R0 S = ∫ BH QH (a)a β H da .
0
∞
0
We
claim
that
R j (a) is non-negative with
∫
limt →∞ f j 0 (t ) = 0 and
∞
0
R j (a)da < 1 .
Proof:
It is clear that, if X , Y ∈ {H , V } with X ≠ Y then
limt →∞ f j 0 (t ) =
limt →∞ − ∫ BX QX (a )(1 − e− β X IY a )da + I X (0) (t ) = 0.
∞
*
∫
t
To witness that
R1 (a ) as follows.
Define
∫
∞
0
R j (a)da < 1 , let us proceed for
1 − e − β H ax a
da
0
βH x
which is a decreasing function of x . We see that
g ( x) = BH β H
g (− IV* )
∞
= BH β H
e − aγ H QH (a )e − β H IV a
*
∫
∞
0
e
− aγ H
1 − e− β H IV a a
*
QH (a)
β H IV*
and
da .
Since IV* is an equilibrium value, then we have
g ( − IV* ) = 1 .
Furthermore,
g (0) = BH β H
=
∫
∫
∞
0
∫
∞
0
e − aγ H aQH (a )e − β H IV a da
*
R1 (a )da.
g is decreasing function then
Considering that
∞
0
R1 (a)da = g (0) < g (− IV* ) = 1 .
Appendix 3
If
∫
∞
0
R0 S < 1 + R0
R2 (a)da = ∫ BH QH (a)e
∞
0
*
− βH IH
a
is less than one.
∫
∞
BH QH (a )e − β H I H a β H a ⎡⎣1 − e − aγ H ⎤⎦ da
*
≤ ∫ BH QH (a ) β H a ⎡⎣1 − e − aγ H ⎤⎦ da < 1.
0
0
∞
then
β H a ⎡⎣1 − e− aγ ⎤⎦ da
Proof:
Using the result in Appendix 1 we have
R1 (a)da = ∫ BH QH (a) β H a ⎡⎣1 − e − aγ H ⎤⎦ da < 1 .
0
If
Appendix 2
H
Decreasing Survival Rate
A.K. Supriatna1, Edy Soewono2
1Department of Mathematics, Universitas Padjadjaran, km 21 Bandung-Sumedang 45363, Indonesia
fax: 062-22-7794696, email: asupriat@yahoo.com.au; aksupriatna@bdg.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:
N H = Total number of individuals in the
population
S H = The number of susceptible individuals in
the population
I H = The number of infective individuals in the
population
RH = The number of recover or immune
individual in the population
BH = The recruitment rate or the birth rate
β H = The transmission probability of the disease
We assume that the population N H is
divided into three compartments, S H , I H , and
RH , such that N H = S H + I H + RH . To include age
structure, 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 decreasing for 0 ≤ a ≤ ∞ . If it
is assumed that life expectancy is finite, then
∫
∞
0
QH (a)da = L < ∞ and
∫
∞
0
aQH (a)da < ∞ .
1
Further, let also assume that N H (0) (t ) ,
S H (0) (t ) ,
I H (0) (t ) ,
and
RH (0) (t ) denotes,
respectively, the numbers of N H (0), S H (0),
I H (0), and RH (0) who survive at time t . Then
we have
N H (t ) = N H (0) (t ) + ∫ BH QH (a)da .
t
0
2
Since the per capita rate of infection in the
population at time t is β H I H (t ) , then the number
of susceptible at time t is given by
t
β H I H ( s ) ds
−
S H (t ) = S H (0) (t ) + ∫ BH QH (a )e ∫t −a
da .
t
3
If the rate of recovery is γ then the number of
infective at time t is given by
0
t
t
t
−
β H I H ( s ) ds ⎤ − ∫ γ H ds
⎡
t −a
I H (t ) = I H (0) (t ) + ∫ BH QH (a ) ⎢1 − e ∫t −a
da
⎥e
0
⎣
⎦
t
t
β H I H ( s ) ds ⎤ − aγ
−
⎡
H
= I H (0) (t ) + ∫ BH QH ( a ) ⎢1 − e ∫t −a
da.
⎥e
0
⎣
⎦
4
The Equilibrium of the system is given by
( S H* , I H* , RH* ) with I H* satisfying
I H* = ∫ BH QH (a ) ⎡1 − e − β H I H a ⎤ e − aγ H da .
11
0
⎣
⎦
It is easy to see that ( S H* , I H* , RH* ) = ( N H* , 0, 0) is the
disease-free equilibrium. To find a non-trivial
equilibrium (an endemic equilibrium), we could
observe the following.
∞
β H BH
RH (t ) = N H (0) (t ) + ∫ BH QH (a )da
t
t
−
β H I H ( s ) ds
− S H (0) (t ) − ∫ BH QH (a )e ∫t −a
da
f
t
0
t
( I H*
lim
5
*
IH
It is clear that
lim
lim
N H (0) (t ) = 0 ,
S H (0) (t ) = 0 ,
t→∞
t→∞
lim
lim
I H (0) (t ) = 0 , and
RH (0) (t ) = 0 .
t→∞
t→∞
=
6
Hence, equations (3), (4), (5), and (6) constitute an
SIR age structured disease model.
*
IH
∞
β H I H ( s ) ds
−
S H (t ) = ∫ BH QH ( a )e ∫t −a
da ,
, say
⎡1 − e− β H I H* a ⎤ e− aγ H
⎣⎢
⎦⎥
*
IH
→ 0+
∫
∞
0
→ ∞−
lim
0
QH ( a )
da
13
*
→∞
IH
∫
∞
0
⎡1 − e − β H I H* a ⎤ e − aγ H
⎢
⎥⎦
QH (a ) ⎣
da
*
βH I H
14
aQH (a )e − aγ H da
*
)
f (I H
β B
− H H
∫
∞
0
1 − e− βH I H a
*
QH (a )
*
βH IH
e − aγ H da
15
*
7
8
0
t
∞
β H I H ( s ) ds ⎤
−
⎡
− aγ H
⎤⎦ da
RH (t ) = ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎡⎣1 − e
0
⎣
⎦
=
β H BH
lim
=0
t
∞
β H I H ( s ) ds ⎤ − aγ
−
⎡
H
I H (t ) = ∫ BH QH (a) ⎢1 − e ∫t−a
da
⎥e
0
⎣
⎦
∫
12
*
f (I H
)
lim
In this section we will show that there is a
threshold number for the model discussed above.
Let us consider the following limit system of
equations which has the same behavior with the
system (3) to (6) whenever t → ∞ :
0
→ 0+
= β H BH
III. The Existence of a Threshold
Number
∞
) = β H BH
∞
I H*
β H I H*
This function is monotonically decreases with
0
N H (t ) = ∫ BH QH (a)da ,
0
⎡1 − e− β H I H* a ⎤ e− aγ H
⎢
⎥⎦
QH (a ) ⎣
da = 1.
*
βH I H
The LHS of (12) is a function of
Furthermore, considering that
RH (t ) = N H (t ) − S H (t ) − I H (t ) then we have
t
⎡
−
β H I H ( s ) ds ⎤ − aγ
H
− I H (0) (t ) − ∫ BH QH (a) ⎢1 − e ∫t − a
⎥ e da
0
⎣
⎦
t
t
⎡
−
β H I H ( s ) ds ⎤
− aγ H
⎤
= RH (0) (t ) + ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎣⎡1 − e
⎦ da.
0
⎣
⎦
∫
∞
*
9
t
10
Equations (7) show that the value of N H (t ) is
constant, hence the equations for the agestructured SIR model reduce to three equations,
(8) to (10).
Therefore, a unique non-trivial value of I H occurs
if and only if
R0 = β H BH
∫
∞
0
aQH (a )e− aγ H da > 1 .
16
Since the LHS of (16) determines the occurrence
of the non-trivial value of I H* , then it will be
refereed as a threshold number R0 of the model.
Hence,
an
endemic
equilibrium
( S H* , I H* , RH* ) ≠ ( N H* , 0, 0) occurs if and only
if R0 > 1 .
IV. The Stability of the Equilibria
To investigate the stability of the
equilibria we use the method in [1] and use the
lemma therein.
β H I H ( s ) ds
−
t
1 − e ∫t −a
≤ ∫t − a β H I H ( s )ds
t
LEMMA 4.1. (BRAUER, 2001). Let f (t ) be a
bounded non-negative function which satisfies an
estimate of the form
f (t ) ≤ f 0 (t ) + ∫ f (t − a) R(a)da ,
t
0
where f 0 (t ) is a non-negative function with
limt →∞ f 0 (t ) = 0 and R(a) is a non-negative
function
∫
with
limt →∞ f (t ) = 0 .
∞
0
R(a)da < 1.
≤ a β H supt − a ≤ s ≤ t I H ( s)
Hence we have,
I H (t ) = I H (0) (t )
t
−
β H I H ( s ) ds
+ ∫ BH QH (a)(1 − e ∫t −a
)e − aγ H da
t
≤ I H (0) (t )
0
t
+ ∫ BH QH (a)(a β H supt − a ≤ s ≤t I H ( s))e − aγ H da 19
Then
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 ) + ∫ supt − a ≤ s ≤t f ( s ) R(a)da .
And
RH (t ) = RH (0) (t )
t
−
β H I H ( s ) ds ⎤
⎡
− aγ H
⎤⎦ da
+ ∫ BH QH (a ) ⎢1 − e ∫t −a
⎥ ⎡⎣1 − e
0
⎣
⎦
≤ RH (0) (t )
t
t
0
17
+ ∫ BH QH (a )(a β H supt − a ≤ s ≤ t I H ( s )) ⎡⎣1 − e − aγ H ⎤⎦ da
0
The following lemma is the extension of Brauer’s
lemma.
LEMMA 4.2. Let f j (t ), j = 1, 2 be bounded non-
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤t f1 ( s) R1 (a)da ,
t
f 2 (t ) ≤ f 20 (t ) + ∫ supt − a ≤ s ≤t f1 ( s) R2 (a)da ,
0
t
0
f j 0 (t ) is
non-negative
with
limt →∞ f j 0 (t ) = 0 and R j (a) is non-negative with
∫
∞
0
t
Moreover,
∫
negative functions satisfying
where
18
R j (a)da < 1. Then limt →∞ f j (t ) = 0, j = 1, 2 .
∞
f1 (t ) ≤ sup{ f10 (t ), f 20 (t )}
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a )}da
t
0
f 2 (t ) ≤ sup{ f10 (t ), f 20 (t )}
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a )}da
t
0
sup{ f1 (t ), f 2 (t )} ≤ sup{ f10 (t ), f 20 (t )}
and hence,
+ ∫ supt − a ≤ s ≤ t sup{ f1 ( s ), f 2 ( s )}sup{R1 (a ), R2 (a)}da
lim
I H (0) (t ) = 0 and
t→∞
BH QH (a)a β H e− aγ H da = R0 < 1
then
using
Lemma 4.1 we conclude that limt →∞ I H (t ) = 0 .
Next,
let
us
see
the
0
expression ∫ BH QH (a)a β H ⎡⎣1 − e − aγ H ⎤⎦ da
0
∞
which,
if R0 S = ∫ BH QH (a)a β H da , can be written in the
∞
R0S − R0 . Hence, if
0
∫
form
PROOF.
since
20
R0 S < 1 + R0
then
BH QH (a)a β H ⎡⎣1 − e− aγ H ⎤⎦ da < 1 (Appendix 1).
lim
Furthermore, since
RH (0) (t ) = 0 then using
t→∞
Lemma 4.2 we conclude that limt →∞ RH (t ) = 0 .
Consequently,
limt →∞ S H (t ) = limt →∞ ( N H (t ) − RH (t ) − I H (t )) = N H*
This shows that the disease-free equilibrium
( S H* I H* , IV* ) = ( N H* 0, 0) is globally stable.
∞
0
t
0
From Lemma 4.1 we conclude that
limt →∞ sup{ f1 (t ), f 2 (t )} = 0 , and this is suffice to
show that limt →∞ 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 R0 < 1 .
Consider the following inequalities.
4.2.
The stability of the endemic
The
endemic
equilibrium
equilibrium.
( S H* , I H* , RH* ) appears only if R0 > 1 . Let us see the
perturbations of I H* and RH* , respectively, by u (t )
and v (t ) . Define I H (t ) = I H* + u (t ) and substitute
this quantity into equation (4) to obtain the
following calculations.
I H* + u (t ) = I H (0) (t )
−
β H [ I H* + u ( s )] ds ⎞ − aγ
⎛
H
+ ∫ BH QH (a) ⎜1 − e ∫t −a
da
⎟e
0
⎝
⎠
u (t ) = − I H* + I H (0) (t )
RH* + v(t ) = RH (0) (t )
t
β H [ I H + u ( s )] ds
−
) ⎡⎣1 − e− aγ H ⎤⎦ da
+ ∫ BH QH (a )(1 − e ∫t −a
0
t
t
t
t
β H I H* ds − ∫ β H u ( s ) ds ⎞ − aγ
−
⎛
H
e t −a
da
+ ∫ BH QH (a) ⎜1 − e ∫t −a
⎟e
0
⎝
⎠
t
v(t ) = − RH* + RH (0) (t )
t
−
β H I H* ds − ∫ β H u ( s ) ds ⎞
⎛
− aγ H
e t −a
+ ∫ BH QH (a ) ⎜1 − e ∫t −a
⎟ ⎣⎡1 − e
⎦⎤ da
0
⎝
⎠
t
t
= − ∫ BH QH (a)(1 − e − β H I H a )e− aγ H da + I H (0) (t )
∞
*
β H u ( s ) ds ⎞ − aγ
−
⎛
H
da
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟e
0
⎝
⎠
∞
∞
*
− ∫ BH QH (a )(1 − e − β H I H a )da
t
t
*
t
−
β H u ( s ) ds ⎞
*
⎛
− aγ H
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟ ⎣⎡1 − e
⎦⎤ da
0
⎝
⎠
t
v(t ) = − ∫ BH QH (a)(1 − e− β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
∞
− ∫ BH QH ( a)(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da
t
t
β H u ( s ) ds ⎞ − aγ
−
*
⎛
H
da
+ ∫ BH QH (a) ⎜ 1 − e − β H I H a e ∫t −a
⎟e
0
⎝
⎠
u (t ) = − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
*
t
β H u ( s ) ds ⎞ − aγ
*
−
⎛
H
da
+ ∫ BH QH (a )e − β H I H a ⎜ 1 − e ∫t −a
⎟e
0
⎝
⎠
t
t
−
β H u ( s ) ds ⎞
*
⎛
− aγ H
⎤⎦ da
+ ∫ BH QH (a ) ⎜1 − e − β H I H a e ∫t −a
⎟ ⎡⎣1 − e
0
⎝
⎠
t
v(t ) = − ∫ BH QH (a)(1 − e− β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
∞
t
β H u ( s ) ds ⎞
−
*
⎛
− aγ H
⎤⎦ da
+ ∫ BH QH (a )e − β H I H a ⎜1 − e ∫t −a
⎟ ⎡⎣1 − e
0
⎝
⎠
t
≤ − ∫ BH QH (a )(1 − e − β H I H a ) ⎣⎡1 − e − aγ H ⎦⎤ da + RH (0) (t )
≤ − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
∞
+ ∫ BH QH (a)e
*
a
− βH IH
0
β H ae
− aγ H
t
t
supt − a ≤ s ≤ t u ( s )da
Hence, we have
u (t ) ≤ − ∫ BH QH ( a)(1 − e − β H I H a )e − aγ H da + I H (0) (t )
*
v(t )
+ ∫ sup t − a ≤ s ≤t u ( s ) BH QH ( a )e − β H I H a β H ae − aγ H da
t
t
By
f1 (t ) = u (t ) ,
defining
R1 (a) = BH QH (a)e
*
− βH IH
a
β H ae − aγ ,
f10 (t ) = − ∫ BH QH (a)(1 − e
∞
t
and
H
*
a
− βH I H
)e − aγ H da + I H (0) (t )
f1 (t ) ≤ f10 (t ) + ∫ supt − a ≤ s ≤t f1 ( s ) R1 (a)da .
, then we have
By
that R1 (a ) is non-negative with
∫
0
R1 (a)da < 1
(see Appendix 2). Then by Lemma 4.1 we have
limt →∞ f1 (t ) = 0 , means that limt →∞ I H (t ) = I H* .
Next, define RH (t ) = RH* + v(t ) and
substitute these quantities into equation (5) to
obtain the following calculations:
∞
*
+ ∫ supt − a ≤ s ≤ t u ( s ) BH QH ( a )e − β H I H a β H a ⎡⎣1 − e− aγ H ⎤⎦ da
t
*
defining f1 (t ) = u (t ) ,
R2 ( a ) = BH QH ( a )e
*
a
− βH IH
f 2 (t ) = v (t )
β H a ⎡⎣1 − e − aγ ⎤⎦ ,
f 20 (t ) = − ∫ BH QH (a )(1 − e
∞
t
H
− β H I H* a
and
) ⎡⎣1 − e − aγ H ⎤⎦ da + I H (0) (t )
f 2 (t ) ≤ f 20 (t ) + ∫ supt − a ≤ s ≤t f1 ( s ) R2 (a)da .
, then we have
t
0
∞
− ∫ BH QH ( a)(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + I H (0) (t )
0
t
We see that limt →∞ f10 (t ) = 0 and it can be shown
≤
t
*
0
*
0
Hence, we have
∞
*
+ ∫ BH QH ( a)e − β H I H a β H a ⎡⎣1 − e− aγ H ⎤⎦ supt − a ≤ s ≤ t u ( s )da
*
t
*
t
t
t
*
0
t
∞
*
t
0
∞
*
0
t
u (t ) = − ∫ BH QH (a )(1 − e − β H I H a )e − aγ H da + I H (0) (t )
t
= − ∫ BH QH (a )(1 − e − β H I H a ) ⎡⎣1 − e − aγ H ⎤⎦ da + RH (0) (t )
*
0
t
*
0
We can show that in Appendix 3 that
limt →∞ f 20 (t ) = 0 and R2 (a) is non-negative with
∫
∞
R2 (a)da < 1 . Then by Lemma 4.2 we have
limt →∞ f 2 (t ) = 0 , means that lim t →∞ RH (t ) = RH* .
Finally, since limt →∞ N H (t ) is a constant,
lim t →∞ RH (t ) = RH* , and limt →∞ I H (t ) = I H* then
0
( S H* , I H* , RH* ) is globally stable.
V. Concluding Remarks
In this paper we have discussed an agestructured SIR disease model with a decreasing
survival rate. We found a threshold number for the
existence and uniqueness of an endemic
equilibrium,
that
is,
R0 = β H BH
∫
∞
0
aQH (a )e − aγ H da . As is the case of
the SI disease model discuss in [1], an endemic
equilibrium appears if R0 > 1 and disappears if
R0 < 1 . In the SI disease model, the threshold for
the existence of the equilibrium is also the
threshold for the stability of the equilibrium.
However, in our case in which there is a recover
compartment, there is an additional condition for
the equilibrium to be stable. Here we found that
there is a stable endemic equilibrium if R0 > 1
and R0 S = β H BH ∫ QH (a)ada < 1 + R0 , and there
∞
is a stable disease-free equilibrium if R0 < 1 and
0
R0 S = β H BH ∫ QH (a)ada < 1 + R0 . We notice that
∞
0
this condition is consistent with that in [1] if the
recovery rate γ = 0 , since in this case R0 is
equivalent to R0S . Hence, we conclude that the SI
model in [1] is naturally nested in the SIR model
discussed in this paper.
VI. References
[1] F. Brauer. A model for an SI disease in an
age-structured population. Discrete and
Continuous Dynamical Systems – Series B. 2
(2002), 257-264.
[2] S. Busenberg, M. Ianelli, and H.R. Thieme.
Global behavior of an age-structured epidemic
model, SIAM J. Math. Anal. 22 (1991),
1065-1080.
[3] H.W. Hethcote. An age-structured model for
pertussis transmission. Math. Biosc. 145
(1997), 89-136.
Appendix 1
∫
∞
0
∞
R0 S < 1 + R0
then
Proof:
It is straightforward from the definition that
R0 = ∫ BH QH (a)a β H e− aγ H da
∞
R0 S = ∫ BH QH (a)a β H da .
0
∞
0
We
claim
that
R j (a) is non-negative with
∫
limt →∞ f j 0 (t ) = 0 and
∞
0
R j (a)da < 1 .
Proof:
It is clear that, if X , Y ∈ {H , V } with X ≠ Y then
limt →∞ f j 0 (t ) =
limt →∞ − ∫ BX QX (a )(1 − e− β X IY a )da + I X (0) (t ) = 0.
∞
*
∫
t
To witness that
R1 (a ) as follows.
Define
∫
∞
0
R j (a)da < 1 , let us proceed for
1 − e − β H ax a
da
0
βH x
which is a decreasing function of x . We see that
g ( x) = BH β H
g (− IV* )
∞
= BH β H
e − aγ H QH (a )e − β H IV a
*
∫
∞
0
e
− aγ H
1 − e− β H IV a a
*
QH (a)
β H IV*
and
da .
Since IV* is an equilibrium value, then we have
g ( − IV* ) = 1 .
Furthermore,
g (0) = BH β H
=
∫
∫
∞
0
∫
∞
0
e − aγ H aQH (a )e − β H IV a da
*
R1 (a )da.
g is decreasing function then
Considering that
∞
0
R1 (a)da = g (0) < g (− IV* ) = 1 .
Appendix 3
If
∫
∞
0
R0 S < 1 + R0
R2 (a)da = ∫ BH QH (a)e
∞
0
*
− βH IH
a
is less than one.
∫
∞
BH QH (a )e − β H I H a β H a ⎡⎣1 − e − aγ H ⎤⎦ da
*
≤ ∫ BH QH (a ) β H a ⎡⎣1 − e − aγ H ⎤⎦ da < 1.
0
0
∞
then
β H a ⎡⎣1 − e− aγ ⎤⎦ da
Proof:
Using the result in Appendix 1 we have
R1 (a)da = ∫ BH QH (a) β H a ⎡⎣1 − e − aγ H ⎤⎦ da < 1 .
0
If
Appendix 2
H