TOPIC:

MODELING ENROLLMENT

TRENDS

OUTLINE

1-INTRODUCTION2-THE MODEL3-DIAGRAM4-DERIVATION OF THE MODEL5-ANALYSIS OF THE MODEL6-CONCLUSION7-REFERENCES

1-INTRODUCTION

Predicting the enrollment of students at a University is a challenge that we sometimes Predicting the enrollment of students at a University is a challenge that we sometimes face. The enrollment to a program is also challenging and our decisions could be face. The enrollment to a program is also challenging and our decisions could be influenced by some factors. Along with the diminishing number of students, it has influenced by some factors. Along with the diminishing number of students, it has been observed that some program at the University still have lot of students, been observed that some program at the University still have lot of students, whereas other suffer from a great loss of students.whereas other suffer from a great loss of students.

The model that we constructed has its source in the Article “A model of student The model that we constructed has its source in the Article “A model of student migration” from the Authors Scheurle, J. and Seydel, R. (University of Germany), migration” from the Authors Scheurle, J. and Seydel, R. (University of Germany), June 1, 1999. The enrollment to a program is determined by the choice of a student June 1, 1999. The enrollment to a program is determined by the choice of a student to follow either option 1 (“easier” program defined as program where subjects are to follow either option 1 (“easier” program defined as program where subjects are “easy” with easy examination and favorable grade) or option 2 (“harder” program “easy” with easy examination and favorable grade) or option 2 (“harder” program where subjects are “hard” and professors are exhaustive, ambitious, helpful.), and where subjects are “hard” and professors are exhaustive, ambitious, helpful.), and also by the advertisement (communication).also by the advertisement (communication).

The reasons of such trends must be expected to be related to size of the market, and to The reasons of such trends must be expected to be related to size of the market, and to psychological effects. Our goals are to see how the advertisement influences the psychological effects. Our goals are to see how the advertisement influences the choice of students and to try to know which option is favorable to graduation.choice of students and to try to know which option is favorable to graduation.

This model is similar to the epidemic model that we covered in class. This model is similar to the epidemic model that we covered in class.

2-THE MODEL

To set up the model, we divide students into two groups: one group following option 1 and the other one following option 2. The freshman students initially enter option 1.

-Students in option 1 can migrate into option 2 while those in option 2 can only move to option 1 if they are disappointed

-The graduation from any option is conditioned by a parameter.

Variables -Independent variables: “t” is the time scaled in semesters. -Dependent variables: X(t) is the number of who enroll in option 1 Y(t) is the number of students who are enrolled in option 2Parameters -”a” is the factor measuring the contact rate and the

effectiveness of the communication between X(t) and Y(t). -ß is the drop-out rate from option 2 back to option 1 -λ is the number of freshman students at the beginning of

the semester. - is the rate of graduation of option 1 - is the rate of graduation of option 22d

1d

3-DIAGRAM

1d

X(t)X(t)λ

2d Y(t)Y(t)

X(t)Y(t)a βY(t)

Option 1

Option 2

4-DERIVATION OF THE MODEL

Discrete form of the differential equations

Done by the authors.

Continuous form of the differential equations

Done by the presenter.

1

2

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

dX taX t Y t Y t d X t

dtdY t

aX t Y t Y t d Y tdt

0 0( )X t x 0 0( )Y t y

( ) 0X t ( ) 0Y t

1 1 2 2 1 1

2 1 2 2 2 2

X ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

j j j j j

j j j j j

a t X t X t X t d X t

X a t X t X t X t d X t

1 2( ) 0; ( ) 0

with

X t X t

System differential of the enrollment trends.

1

2

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

dX taX t Y t Y t d X t

dtdY t

aX t Y t Y t d Y tdt

0 0( )X t x 0 0( )Y t y

( ) 0X t ( ) 0Y t

removed

removed

NBThe differential system of equations above is similar to the epidemic model with

I=X, S=Y S-------> I--------> S

5-ANALYSIS OF THE MODEL

Quality analysis:

Equilibrium point:

V= (1)

W= (2)

The stability matrix is based on the Jacobian matrix J.

Det ( J ) =

In the neighborhood of V the mathematical model is

stable for with Y=0

In the neighborhood of W the mathematical model is

stable for Y> 0 or

1

;0d

2 12

2

1;

d dd

a d a

2 1 1 2( ) ( )a Yd Xd d d

1 2( )a d d

1 2( )a d d

20 40 60 80 100 120

0.001

0.002

0.003

0.004

a

NBWe can notice that the more freshmen are in the program, less the advertisement will be efficient.

What is the influence of the communication toward X and Y? We will give some simulation to explain it.

Simulations or experiments

case2 1d d

10 20 30 40t

50

100

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xy

(0) 200X (0) 20Y

10 20 30 40t

50

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xy

a 0.001

20 40 60 80t

100

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400

500

xy

Case: 2 1d d

50;

d1 0.3;

d2 0.1;

0.1;

a 0.002

20 40 60 80 100t

50

100

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xy

10 20 30 40t

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xy

0.001a

0.002a

0.003a

Case : 2 1d d

50;

d1 0.2;

d2 0.2;

0.1;

a 0.002

20 40 60 80 100t

50

100

150

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xy

10 20 30 40t

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xy

a 0.003

20 40 60 80 100t

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xy

0.002a

0.001a

Phase Diagrams

Case: 2 1d d

50;

d1 0.1;

d2 0.3;

0.1;

220 240 260 280x

40

60

80

100

y

0.002a

340 360 380 400 420 440x10

1520253035

y

0.001a

Case: 2 1d d

50;

d1 0.3;

d2 0.1;

0.1

120 140 160 180 200x

50

100

150

200

y

0.002a

Case: 2 1d d

50;

d1 0.2;

d2 0.2;

0.1;

a 0.002

160 170 180 190 200x

40

60

80

100

y

6-CONCLUSION

1 2( )a d d

The study of this model help to understand the influence of advertisement during the enrollment of students to a program The choice of some parameter can influence positively or negatively the results expected.This model seems to be more realistic than precise, when the advertisement is relatively low ( a=0.001) the value of Y is greater than X, but When the advertisement is relatively high ( a =0.003) the value of Y is less than X. In addition there in no student in option 2 when As suggestions:We can improve the model by taking care that the previous inequality does not become true. This leads to the following strategies:-advertisement towards more frehsmen students entering option 1. -making students in option 2 feel happier (decrease ).-use the same rate of graduation to help students finish quicker.-model 3 or more options.-prevent drop-out from entering option 2.-build a periodical rate of freshman students enrolling.

7-REFRENCES

*Seydel, R and Scheurle, J.(1999) “A model of student migration”, International journal of Bifurcation and Chaos, Vol. 10, No 2 (2000) 477-480.

*Feictinger, G. (1992) “Limit cycles in dynamical economic systems” Ann.Operations Res. 37, 313-344.

*Kengne, E. (1998) “Ordinary differential equations”, University of Dschang (Cameroon)

*Harlan, S. “Math 5270 class notes”, University of Minnesota Duluth, 2006.

For more simulations go to http://www.d.umn.edu/~mbunt001