modelling the evolution of the belgian population, eigenvalues and eigenvectors johan deprez...

Modelling the evolution of the Belgian population, eigenvalues and

eigenvectors

Johan DeprezICTMA-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

school students

Content of the talk

1. Teaching sequence1. Calculations with authentic data2. The matrix model3. Two observations concerning the long term evolution of

the population4. Mathematical treatment of the observations5. Eigenvalues and eigenvectors

2. Experiences with the teaching sequence in different contexts1. during Science Week2. in teacher education3. in my own mathematics teaching

3. The model of the Belgian statistical institute

Teaching sequence: 1. Calculations with authentic data population by

age 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)

Calculations become messy!

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!


944410

1028401

3298532

9478422

3684072

030.0000

0083.000

00096.00

000098.0

0001.034.043.0

102840130.0

329853283.0

947842296.0

368407298.0

329853201.0947842234.0368407243.0

20032023

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:

329853201.0947842234.0368407243.0

368407298.0 947842296.0

329853283.0

102840130.0

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


V

IV

III

II

I

030.0000

0083.000

00096.00

000098.0

0001.034.043.0VIVIIIIII

toL

from

944410

1028401

3298532

9478422

3684072

)0(X

population 1st Jan. 2003

Leslie matrix

survival rates

fertility rates

)0()1( XLX )1()2( XLX )1()( nXLnX

population in 2023:

population in 2043:

in general:

assumptions/simplifications: we use the same fertility and survival rates in every step, no migration, ...

Evolution per age class

0

500000

1000000

1500000

2000000

2500000

3000000

0 1 2 3 4 5 6 7 8 9 10 11 12

after ... periods

I

II

III

IV

V

Teaching sequence: 3. Two observations concerning the long term evolution

Evolution per age class

0

500000

1000000

1500000

2000000

2500000

3000000

0 1 2 3 4 5 6 7 8 9 10 11 12

after ... periods

I

II

III

IV

V

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?


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%


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 growth rate 0.84 (= the long term growth rate).


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(LT = long term)

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?


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

nt

nXX

n

(where t(n) is the total population after n periods)


(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:

XXLnt

nX

nt

nXL

nt

nX

nt

nXL

nXnXL

nn

84.0)(

)(lim84.0

)(

)(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:


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-En)=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

the long term age distribution

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)


• 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.


• 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


• exam♦ students master the mathematical concepts (and

applications) as good (or bad) as with traditional teaching

♦ some typical errors are related to the use of the example• only positive eigenvalues?• one eigenvector versus all eigenvectors

The model of the Belgian statistical institute

• ... is much more refined♦ age groups of 1 year and distinction between

male and female♦ projections for smaller geographic entities

(“arrondissement”, about 50 for the whole country)

♦ birth and fertility rates are not constant♦ migration is taken into account

• relatively short term: projections up to year 2050

Thank you for your attention!

modelling the evolution of the belgian population, eigenvalues and eigenvectors johan deprez...

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