A Mathematical Model For Disease Transmission With Age-structure.

numbers of contact between susceptibles in group i and the infectives from groups i and j. Here
β ij denotes the constant of proportionality. In the following sections we investigate the effects of

A Mathematical Model for Disease Transmission
with Age-Structure1

mortality rate

µ i and infection rate β ij to the equilibrium values S i* and I i* , to the basic

reproductive numbers, and to the minimum vaccination levels.

Asep K. Supriatna
2.

Department of Mathematics, Padjadjaran University
Km 21 Jatinangor, fax 022-7794696, Sumedang 45363 INDONESIA

The effects of mortality and infection rates to the equilibrium values

the equilibriums

Abstract
Many diseases are known to have specific properties, such as one might be easier to transmit to
adults while the other might be easier to infect children. For this reason, incorporating agestructure in modelling the disease transmission will give a better insight into how the disease
spreads. However, many mathematical models have ignored this age-structure in analysing the
disease transmission. In this paper we discuss a simple mathematical model for disease
transmission with the inclusion of age-structure. Some classical questions such as the equilibrium
population size, the basic reproductive number and the critical vaccination level are investigated.
The results show that some properties to some extent are the generalization of the non-agestructure results while some are quite different from the known non-age-structure results.
Keywords: Disease transmission, Age structure, Minimum vaccination level

S i* and I i* . The effects will be investigated by looking at three different

assumptions, namely:
1. Both groups are incorrectly considered to be un-coupled (there is no cross-infection between
groups) and β i are calculated just before the mixing happens (pre-mixing)

The Model
Suppose that we divide a population into two groups, adults (group 1) and children (group
2). Among the simplest epidemic model in incorporating this division can be done by modifying
the simple SIR model described in [1; p.43] to obtain the following equations:

Both groups are incorrectly considered to be un-coupled and

3.

mixing happens (post-mixing)
Both groups are correctly considered to be coupled (there is cross-infection between groups).
To find the equilibriums of equations (1)-(4) we simplify the notations as follows:

I ime* = I i . The LHS of equations (1)-(4) are set to be zero. Next we substitute
β 11 S1 I 1 + β 12 S1 I 2 and β 21 S 2 I 1 + β 22 S 2 I 2 from equations (3) and (4) into equations (1) and
I1 =

B1 − µ1 S1
B − µ2 S2
and I 2 = 2
.
(α 1 + µ1 )
(α 2 + µ 2 )

No.

β i values

1

β ii

B1 −
I

e*
1

=

S 1e* =
2

( β ii + β ij ) / 2

respectively. Furthermore, the infection rate in group i is assumed to be proportional to the

1

Presented in the International Conference on Statistics and Mathematics and its Application in
the Development of Science and Technology, Bandung, October 4-6, 2004. The publication of this
paper was funded by the Indonesian Government through the scheme of Penelitian Hibah
Bersaing XII (SPK No. 011/P4T/DPPM/PHB/III/2004).

1

( S 2e* , I 2e* ) values
µ +α2
S 2e* = 2
β 22

( S1e* , I 1e* ) values
µ + α1
S1e* = 1

β 11

4

where S1 and I1 , respectively, denote the numbers of susceptibles and infectives for the adults, and
S2 and I2 , respectively, denote the numbers of susceptibles and infectives for the children. Let us
assume that the birth rate, mortality rate and recovery rate for each group be Bi, µ i and α i ,

5

The complete equilibriums are shown in Table 1.

2
3

S ime* = S i

and

1.

1

β i are calculated just after the

2.

(2) to obtain

dS1
= B1 − β 11 S1 I 1 − β 12 S1 I 2 − µ1 S1 ,
dt
dS 2
= B2 − β 21 S 2 I 1 − β 22 S 2 I 2 − µ 2 S 2 ,
dt
dI 1
= β 11 S1 I 1 + β 12 S1 I 2 − µ1 I 1 − α 1 I 1 ,
dt
dI 2
= β 21 S 2 I 1 + β 22 S 2 I 2 − µ 2 I 2 − α 2 I 2 .

dt

S i* and I i*

In this section we discuss the effects of the mortality and infection rates to the values of

I

=

B2 −
I

µ1 (µ 1 + α 1 )
( β 11 + β 12 ) / 2
µ1 + α1

=

e*

2

µ1 + α 1
( β 11 + β 12 ) / 2

B1 −
e*
1

µ1 ( µ1 + α 1 )
β 11
µ1 + α 1

S 2e* =

µ2 +α2
( β 21 + β 22 ) / 2

B2 −
I

e*
2

=

µ 2 (µ 2 + α 2 )
β 22
µ2 + α2

µ 2 (µ 2 + α 2 )
( β 21 + β 22 ) / 2
µ2 +α 2

 e* B 2 − µ 2 S 2em* 
 e* B1 − µ 1 S1em* 
 S 2m ,

 S 1m ,





+
µ 2 + α 2 
µ
α
1
1



Table 1: The equilibrium values for three different assumptions. 1. Separated groups (premixing); 2. Separated groups (post-mixing); 3. Coupled groups. In the coupled group model, the
equilibriums are in implicit form.
3

β ii dan β ij

2

To compare the values of the equilibriums we substitute back the resulting values of
and

I 2 into equations (1) and (2), which were equating to zero beforehand, to obtain
B − µ 1 S1
B − µ2S2
B1 − β 11 S1 1
− µ 1 S1 = 0 ,
− β 12 S1 2
(α 1 + µ1 )
(α 2 + µ 2 )
B − µ1 S 1
B − µ2 S2
B2 − β 21 S 2 1
− β 22 S 2 2
− µ2S2 = 0 .
(α 1 + µ1 )
(α 2 + µ 2 )

The last two equations were solved (in this case we use Maple V), by considering
and

6

each other. We will show that S1 ≠ S 2 . We will also show that conclusion if we incorrectly
assume that there is no cross-infection between groups will significantly different from the case if
we correctly assume that indeed there is a cross-infection between the groups.
Note that from ∆S = S1 − S 2 we can derive

7

provided

I1

result if we incorrectly assume that there is no cross-infection between the groups, that is

α1 = α 2 = α

B1 = B2 = B , to find an implicit form
X 2 + Y2
S2 =
,
β 12 µ 2 (α + µ1 ) S1

S1e* =

8

In which

X 2 = −(Bα 2 + Bαµ 2 + Bαµ1 + Bµ1 µ 2 − β 11 BαS1 − β 11 Bµ 2 S1 − β 12 BαS1 − β 12 Bµ1 S1 )

and

Y2

(

The expression of

X2

2

2

2

X 2 can be simplified into

X 2 and Y2 yields
= − B((α + µ1 )(α + µ 2 ) − S1 ( β 11 (α + µ 2 ) + β 12 (α + µ1 ) ) ,

The next simplifications of

Y2

= − S1 (− (α + µ1 ) µ1 (α + µ 2 ) + β 11 N 1 µ1 (α + µ 2 ) ) .

9
10

Furthermore, to compare the values of equilibrium of the different groups we let

∆S = S1 − S 2 , so ∆S = S1 −

in [2].

3.

The minimum vaccination level
Let us assume that a portion pi of susceptible population is vaccinated so that the
dynamics is given by equations (1) to (4) except
11

Equilibrium values of the system, now are given by

= − B(α (α + µ 2 ) + µ1 (α + µ 2 ) − S1 ( β 11α + β 11 µ 2 + β 12α + β 12 µ1 ) ) .
X2

µ1 + α µ 2 + α
<
= S 2e* . We give a numerical example to illustrate this phenomenon
β
β

dS i
= (1 − pi ) Bi − β ii S i I i − β ij S i I j − µ i S i .
dt

)

= − − µ1α 2 S1 − αµ1 µ 2 S1 − αµ1 S1 − µ1 µ 2 S1 + β 11αµ1 S1 + β 11 µ1 µ 2 S1 .
2

S1 − S 2 = ( B − µ1 S1 )[(α + µ 2 )((α + µ1 ) − S1 β 11 )] + S1 β 12 (α + µ1 )( µ 2 S1 − B) > 0
( B − µ1 S1 ) = 0 and ( µ 2 S1 − B) > 0 . Alternatively B / S1 = µ1 and B / S1 < µ 2 ,
means that µ1 < µ 2 . This concludes that µ1 < µ 2 ⇒ S1 > S 2 . This result contradicts the

X 2 + Y2
. The RHS can be written as
β 12 µ 2 (α + µ1 ) S1

(1 − p i ) Bi
−1
*
(
1
)
(1 − pi ) R0 i − 1
p
B
S
µ i S vi*
µ
−
−
i
i
i vi
12
I vi* =
,
= µ i S vi*
= µ i S vi*
µi + α i
µi + α i
µi + α i
S 0i
B /µ
with R0i = i * i = *
. It is clear that eventually there will be no infective if
S vi
S vi
1
(1 − p i ) R0i ≤ 1 means that pi ≥ 1 −
. We conclude that the critical vaccination level for
R0i
each group is given by p ci = 1− 1 / R0i . This value has the same expression as the critical
vaccination level for many known structured or un-structured population models [3].

= S1 β 12 µ 2 (α + µ1 ) S1 + B((α + µ1 )(α + µ 2 ) − S1 ( β 11 (α + µ 2 ) + β 12 (α + µ1 ) )
+ S1 (− (α + µ1 ) µ1 (α + µ 2 ) + β 11 S1 µ1 (α + µ 2 ) )

4.

= S1 β 12 (α + µ1 )( µ 2 S1 − B) + ( B − µ1 S1 )(α + µ1 )(α + µ 2 ) + S1 β 11 (α + µ 2 )( µ1 S1 − B)
= ( B − µ1 S1 )[(α + µ1 )(α + µ 2 ) − S1 β 11 (α + µ 2 )] + S1 β 12 (α + µ1 )( µ 2 S1 − B)

= ( B − µ1 S1 )[(α + µ 2 )((α + µ1 ) − S1 β 11 )] + S1 β 12 (α + µ1 )( µ 2 S1 − B) .
It is easy to show that if µ1 = µ 2 = µ , β 12 = β 21 = 0 and β 11 = β 22 = β then
equilibrium values for both groups are equal. The discussion in [2] shows that if µ1 ≠ µ 2 and
there are no cross infection between the two group, i.e. β 12 = β 21 = 0 , then the equilibriums for

both groups can be obtained from a single (unstructured) model.
Next we see the following case of µ1 ≠ µ 2 and β 11 = β 12 = β 21 = β 22 = β ≠ 0 .
In this case we assume that the mortality of the two groups are different and each group can infect

3

The computation of the basic reproduction ratio
In this section we derive the basic reproductive ratio, R0, for the system in equations (1)
to (4). In this system there are two types of infections: type 1 infected by group 1 and type 2
infected by group 2. To find the R0, we construct the next generation matrix
where

 k11
K = 
 k 21

k12 

k 22 

k ij = expected number of new cases of type i infected by one case of type j. Hence,



B
1
 with S i denotes the disease free susceptible equilibrium, i.e S i = i .
k ij = β ij S i 
 µ j +α j 
µ
i


The full matrix is given by

4

 β 11 S1

 α 1 + µ1
K =

 β 21 S 2
α + µ
1
 1
value

β 12 S1 

α2 + µ2 
.

β 22 S 2 
α 2 + µ 2 

13

In [1] it is found that R0 is the dominant eigen value of the matrix K. Since the eigen
for 2x2 matrix is given by the root of the characteristic equation

λ2 − Trace( K )λ + Det ( K ) = 0 , it can be proved that if β ij = ai bi then
R0 =

β S
1  β 11 S1

+ 22 2
2  α 1 + µ1 α 2 + µ 2


.



14

In this case R0 for the coupled groups is the average of the R0 for the un-coupled groups. The
knowledge of R0 both for coupled or uncoupled groups, and its changes to the change of other
parameters, is very important to compare various decisions in controlling the disease. Relevant
question regarding an age-structure, such as which group should have a high priority to be
vaccinated can be address further. We examine the effects of the mortality and infection rates to
the values of R0, and hence to the critical vaccination level, by investigating various cases of β ij ,
elsewhere.

Acknowledgement
The author thanks Prof. E. Soewono of ITB and Prof. J.A.P. Heesterbeek of Utrecht
University for the fruitful discussion during the conference excursion in Bali, July 2004. Their
views and suggestions have improved the material in the previous draft of this paper. The author
also acknowledges the financial support from the Indonesian Government through the scheme of
Penelitian Hibah Bersaing XII (SPK No. 011/P4T/DPPM/PHB/III/2004).

References
[1].

Diekmann, O. & J.A.P. Heesterbeek (2000). Mathematical Epidemiology of Infectious
Diseases. John Wiley & Son. New York.

[2].

Supriatna, A.K. et al. (2004). Model Metapopulasi Epidemik (In Indonesian). 12th
National Conference of Mathematics, Bali 23-27 July 2004.

[3].

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. pp. 208-217.

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