Modelling the evolution of the Belgian
population, eigenvalues and
eigenvectors
Johan Deprez
ICTMA-14, Hamburg, July 2009
www.ua.ac.be/johan.deprez
About myself
teacher educator
• (future) mathematics
teachers in upper
secondary school
10 years of experience
• now about half-time
mathematics teacher
• tertiary education
• introductory
mathematics course for
Bachelor students in
applied economics
20 years of experience
• now about half-time
researcher: only for a small part
of my professional time
A teaching sequence, incl. experiences in
different contexts
model for evolution of the Belgian population
which serves as an introduction to
the mathematical theory of matrices,
eigenvalues and eigenvectors
used in different contexts:
• in my own teaching of mathematics
• in my mathematics teacher education course
• during the ‘Science Week’ with secondary
Content of the talk
1. Teaching sequence
1. Calculations with authentic data 2. The matrix model
3. Two observations concerning the long term evolution of the population
4. Mathematical treatment of the observations 5. Eigenvalues and eigenvectors
2. Experiences with the teaching sequence in different
contexts
1. during Science Week 2. in teacher education
3. in my own mathematics teaching
Teaching sequence: 1. Calculations with
authentic data
population byage and sex
survival rates
fertility rates data from the
Belgian national statistics
institute
I don’t use data concerning migration!
Teaching sequence: 1. Calculations with
authentic data
Questions like:
• How many men of age 35 on 1st Jan. 2003?
• How many women of age 35 on 1st Jan. 2010?
• How many births in 2003? How many boys/girls?
• How many boys of age 3 on 1st Jan. 2010?
Students are aware of assumptions/simplifications:
• constant survival and fertility rates
• (no migration)
Teaching sequence: 2. The matrix model
Age 1 Jan. 2003 Fertility rate Survival rate
0-19 I 2 407 368 0.43 0.98 20-39 II 2 842 947 0.34 0.96 40-59 III 2 853 329 0.01 0.83 60-79 IV 1 840 102 0 0.30 80-99 V 410 944 0 0 TOTAL 10 354 690 Simplified data:
• age groups of 20 years • male+female
• rounded to 2 decimals
based on the
NIS-data (not trivial!) a little bit manipulated...
(to obtain a nice eigenvalue)
important consequence: you can calculate the evolution of the
population in steps of 20 years only!
Calculations are no longer messy if you work recursively: • first calculate the population in 2023
• then calculate the population in 2043 • ...
rates over periods of 20 years!
Teaching sequence: 2. The matrix model
944 410 102 840 1 329 853 2 947 842 2 368 407 2 0 30 . 0 0 0 0 0 0 83 . 0 0 0 0 0 0 96 . 0 0 0 0 0 0 98 . 0 0 0 01 . 0 34 . 0 43 . 0 102 840 1 30 . 0 329 853 2 83 . 0 947 842 2 96 . 0 368 407 2 98 . 0 329 853 2 01 . 0 947 842 2 34 . 0 368 407 2 43 . 02003
2023
number in I in 2023: number in II in 2023: number in III in 2023: number in IV in 2023: number in V in 2023: 329 853 2 01 . 0 947 842 2 34 . 0 368 407 2 43 . 0 368 407 2 98 . 0 947 842 2 96 . 0 329 853 2 83 . 0 102 840 1 30 . 0 This calculation corresponds to a matrix calculation!
Age 1 Jan. 2003 Fertility rate Survival rate 0-19 I 2 407 368 0.43 0.98 20-39 II 2 842 947 0.34 0.96 40-59 III 2 853 329 0.01 0.83 60-79 IV 1 840 102 0 0.30 80-99 V 410 944 0 0 TOTAL 10 354 690
Teaching sequence: 2. The matrix model
V
IV
III
II
I
0
30
.
0
0
0
0
0
0
83
.
0
0
0
0
0
0
96
.
0
0
0
0
0
0
98
.
0
0
0
01
.
0
34
.
0
43
.
0
V
IV
III
II
I
to
L
from
944 410 102 840 1 329 853 2 947 842 2 368 407 2 ) 0 ( X population 1st Jan. 2003 Leslie matrix survival rates fertility rates)
0
(
)
1
(
L
X
X
)
1
(
)
2
(
L
X
X
X
(
n
)
L
X
(
n
1
)
population in 2023: population in 2043: in general:assumptions/simplifications: we use the same fertility and survival rates in every step, no migration, ...
Teaching sequence: 3. Two observations
concerning the long term evolution
long term: graphs of all age groups show a common regularity babyboom babyboom babyboom ‘short’ term:
chaos Does it make sense to study
the long term evolution? It does, in my opinion...
formula describing the regularity?
Teaching sequence: 3. Two observations
concerning the long term evolution
After … periods I II III IV V 0 1 -15.7% -17.0% -4.3% +28.7% +34.3% 2 -16.1% -15.7% -17.0% - 4.3% +28.7% 3 -15.9% -16.1% -15.7% -17.0% -4.3% 4 -16.0% -15.9% -16.1% -15.7% -17.0% 5 -16.0% -16.0% -15.9% -16.1% -15.7% 6 -16.0% -16.0% -16.0% -15.9% -16.1% After … periods I ... 0 2003 2 407 368 ... 1 2023 2 030 304 ... 2 2043 1 702 458 ... ... ... ... .... -15.7% -16.1%
Teaching sequence: 3. Two observations
concerning the long term evolution
After … periods I II III IV V 0 1 -15.7% -17.0% -4.3% +28.7% +34.3% 2 -16.1% -15.7% -17.0% - 4.3% +28.7% 3 -15.9% -16.1% -15.7% -17.0% -4.3% 4 -16.0% -15.9% -16.1% -15.7% -17.0% 5 -16.0% -16.0% -15.9% -16.1% -15.7% 6 -16.0% -16.0% -16.0% -15.9% -16.1%
1st observation: In the long run, the number of individuals in each age group decreases by 16% per period of 20 years, i.e. the number in each age group decays exponentially with
Teaching sequence: 3. Two observations
concerning the long term evolution
after ... periods 0-19 (I) 20-39 (II) 40-59 (III) 60-79 (IV) 80-99 (V)
0 23.25% 27.46% 27.56% 17.77% 3.97% 1 20.22% 23.50% 27.19% 23.59% 5.50% 2 19.06% 22.27% 25.35% 25.36% 7.95% 3 18.91% 22.04% 25.24% 24.84% 8.98% 4 18.91% 22.07% 25.20% 24.95% 8.87% 5 18.91% 22.06% 25.22% 24.90% 8.91% 6 18.91% 22.06% 25.21% 24.92% 8.89% 7 18.91% 22.06% 25.22% 24.91% 8.90% 8 18.91% 22.06% 25.21% 24.92% 8.90% 9 18.91% 22.06% 25.21% 24.91% 8.90% 10 18.91% 22.06% 25.21% 24.91% 8.90%
2nd observation: in the long run the age distribution stabilizes
long term age distribution
Teaching sequence: 4. Mathematical
treatment of the observations
Central question:
How can you calculate the long term growth
rate and the long term age distribution in a
‘mathematical’ way?
Teaching sequence: 4. Mathematical
treatment of the observations
in the long run: X(n) 0.84·X(n-1), or: X(n+1) 0.84·X(n)
hence: L·X(n) 0.84·X(n) for large n
LT age distribution:
1st observation (long term growth factor):
2nd observation (long term age distribution):
and the approximation improves indefinitely if n increases indefinitely
)
(
)
(
lim
n
t
n
X
X
n
Teaching sequence: 4. Mathematical
treatment of the observations
(1) LT age distribution X is a solution of the
system L·X = 0.84·X
(2) ... and satisfies the condition that the sum of its
elements is equal to 1 (100%)
Combination of the two observations:
X
X
L
n
t
n
X
n
t
n
X
L
n
t
n
X
n
t
n
X
L
n
X
n
X
L
n n
84
.
0
)
(
)
(
lim
84
.
0
)
(
)
(
lim
)
(
)
(
84
.
0
)
(
)
(
)
(
84
.
0
)
(
This system has an infinite number of solutions.
Mathematical calculation of LT age distribution if
the long term growth factor is known:
Teaching sequence: 4. Mathematical
treatment of the observations
The system LX=0.84X has an infinite number of solutions. This characterizes the number 0.84!
The long term growth factor is the (strictly positive) number for which the system LX= X has an infinite number of solutions, i.e. for which det(L-
En)=0.Teaching sequence: 5. Eigenvalues and
eigenvectors
A a square matrix (n n)
A number is an
eigenvalue
of A iff det (A-E
n)=0.
this means that the system AX = X has an infinite number of solutions
A column matrix X (≠ 0) is an
eigenvector
of A
corresponding to the eigenvalue iff AX = X.
unlike the example: eigenvalues may be negative!
Teaching sequence: 5. Eigenvalues and
eigenvectors
Theorem (for Leslie matrices having two consecutive non-zero fertility rates)
(1) L has exactly one strictly positive, real eigenvalue 1.
(2) One of the eigenvectors of L corresponding to the eigenvalue 1 is a column matrix X consisting of strictly positive numbers
adding up to 1.
(3) For every (realistic) initial age distribution, X(n)/t(n) (where t(n) is the total population after n steps) converges to X.
the long term growth rate
Experiences with the teaching sequence:
1. During the ‘Science Week’
• secondary school students visit universities
and attend workshops concerning a scientific
subject
• my workshop:
♦ part 1, 2 and 3 of the teaching sequence. incl.
dependency ratio (approximated by
#(I+IV+V)/#(II+III))
♦ spreadsheet: how to reach a stable population or
a socially acceptable dependency ratio (by
manipulating birth rates and survival rates,
changing age of retirement, taking migration into
account, ...)?
Experiences with the teaching sequence:
2. In mathematics teacher education
• most students have Master degree in mathematics,
some (more and more) have Master degree in a
‘related’ subject
• students work through the whole teaching
sequence:
♦ workshop (part 1, 2 and 3)
♦ homework (part 4, text with explanation and exercises)
• positive reactions of students:
♦ “Now I see why eigenvalues and eigenvectors are useful.” ♦ students report that the teaching sequence stimulates
critical thinking: subtle relation between mathematics and reality
♦ Remarks of students show that the teaching sequence
makes them think about the evolution of our population, i.e. “Are right-wing parties (who promote having more children) right?”
Experiences with the teaching sequence:
3. In my own mathematics teaching
• end of introductory mathematics course for Bachelor students in applied economics
• context is not ideal
♦ students are not so good in mathematics ♦ groups of 40-60 students
♦ last topic of the year: no possibility to give students a home work, lack of time, ...
• after theory about matrices, linear systems and determinants, but before the theory of eigenvalues and eigenvectors
• part 2-5 of the teaching sequence
• in combination with a simpler application (consumers
switching between different brands of a product, simpler LT behaviour)
• examination is not about modelling, but about (applications of) mathematics (slightly new contexts)
Experiences with the teaching sequence:
3. In my own mathematics teaching
• questionnaire filled in by 20 randomly
chosen students
• Do the students find the teaching sequence
♦ instructive?
♦ interesting?
♦ difficult?
• General appreciation of the teaching
sequence by the students?
Experiences with the teaching sequence:
3. In my own mathematics teaching
• Do the students find the teaching sequence
instructive?
♦ yes!
♦ i.e. The students find that the teaching sequence shows • ... that mathematics can be used to describe reality.
• ... that mathematical models are always a simplification of reality.
• ... that matrices are useful. ♦ less convincing:
• The teaching sequence shows that eigenvalues and eigenvectors are useful. (two questions with different answers)
• I learnt more about the evolution of the Belgian population.
Experiences with the teaching sequence:
3. In my own mathematics teaching
• Do students find the teaching sequence interesting?
♦ mixed opinions!
• Do students find the teaching sequence (too)
difficult?
♦ no!
♦ more difficult than the other examples in the course
♦ mixed opinions concerning whether the presentation of the example could be understood during class
♦ the example can certainly be understood after personal study at home
• General appreciation
♦ students find that more examples of this type should be treated in the course
♦ appreciation correlates negatively to experienced rate of difficulty
