TOPIC:

MODELING ENROLLMENT

TRENDS

OUTLINE

•1-ABSTRACT•2-INTRODUCTION•3-THE MODEL•4-VARIABLES & PARAMETERS•5-DIAGRAM•6-DERIVATION OF THE MODEL•7-ANALYSIS OF THE MODEL•8-CONCLUSION•9-CLASSIFICATION•10-SUGGESTIONS•11-REFERENCES

1-ABSTRACT

The enrollment as a student at the University is a challenge that we sometimes face, but enroll to a program seems again more fastidious and challenging and our decision could be 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, whereas other suffer from a great loss of students. In particular, the “hard” program appears to suffer more from loss than the “easier” one. These appellations are relative to each individual, but the mathematical model that we constructed will help to understand trends such as reported above. The reasons for such trends must be expected to be related to the size of the market, and to psychological effects.

2-INTRODUCTION

• Our model has its source in the Article “A model of student migration” From the Authors J.Scheurle and R. Seydel, University of Koln ( Germauny), June 1, 1999. The enrollment Trends is determined by the choice of a student to follow either Option 1 (“easier” program defined as program where subject are easy with easy examination and favorable grade) or Option 2 (“harder” program defined as program where subjects are hard and professor are exhaustive, ambitious but very helpful), and also by the advertisement (communication)

• Our goals are to see how the advertisement influences the choice of the students, also to know in which option student must graduate faster.

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

3-THE MODEL

• To set up the model, we divide students into freshman students who can enroll only in Option1; into students enrolling in option1 who can migrate into option 2 and students who are in option2 who can only move to option1 when they are disappointed.

• The graduation to any program is conditioned by a parameter.

4-VARIABLES & PARAMETERS

• Independent variables:• -” t” is the time scaled in semesters.• Dependent variables: • -X(t) is the number of students who enroll in option1 at the time

t and Y(t) is the number of students who are enrolled in option2.• Parameters: • -“a” is the factor measuring the contact rate and the

effectiveness of the communication.• -λ is the number of freshman students at the beginning of the

semester.• - is the rate of success for option1• - is the rate of success for option2.• -ß is the rate of disappointment.1d

2d

5-DIAGRAM

1d

X(t)X(t)λ

2d Y(t)Y(t)

X(t)Y(t)a βY(t)

Option 1

Option 2

6-DERIVATION OF THE MODEL

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 x0 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

7-ANALYSIS OF THE MODEL

• Quality analysis:• Equilibrium point:

• V= (1)

• W= (2)

• The stability matrix is based on the Jacobian matrix J. Det ( J ) =

• Using relation (2), Det( J ) = In the neighborhood of W the mathematical model is stable for Y>0 or

• Using relation (1), Det( J ) = In the neighborhood of V the mathematical model is stable for with Y=0

• Chart of •

1

;0d

2 12

2

1;

d dd

a d a

( )a f

2 1 1 2( ) ( )w wa Y d X d d d = 1 2( )a d d

2 1 1 2( ) ( )v va Y d X d d d = 1 2( )a d d

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

1 2( )a d d

1 2( )a d d

1 2( )d da

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

150

200

250

300

xy

10 20 30 40t

50

100

150

200

250

300

xy

a 0.001

20 40 60 80t

100

200

300

400

500

xy

P1 PlotEvaluatext. nsoln,t, 0, 80, PlotRange 0, 500,AxesLabelt, x, TextStyleFontSize 15, FontWeight Bold;

Comments

Despite the fact that the option 2 has a “harder” program it seems that the advertisement has more impact on students inoption2.

The graph of X change while the value of “a” increases. There will be more students in option 2 from the 10th to 16th semester when the communication is high

P2 PlotEvaluateyt. nsoln,t, 0, 80, PlotRange 0, 100,AxesLabelt, y, TextStyleFontSize 15, FontWeight Bold;

Case: 2 1d d

50;

d1 0.3;

d2 0.1;

0.1;

a 0.002

20 40 60 80 100t

50

100

150

200

250

300

xy

a 0.003

10 20 30 40t

50

100

150

200

250

300

xy

a 0.001

P1 PlotEvaluatext. nsoln,t, 0, 90, PlotRange 0, 200,AxesLabelt, x, TextStyleFontSize 15, FontWeight Bold;

P2 PlotEvaluateyt. nsoln,t, 0, 90, PlotRange 0, 40,AxesLabelt, y, TextStyleFontSize 15, FontWeight Bold;

CommentsIn this case, despite the fact that the rate of success is high in option1,

it seems that students in option2 graduate or move more when the advertisement is increasing.

Case : 2 1d d

50;

d1 0.2;

d2 0.2;

0.1;

a 0.002

20 40 60 80 100t

50

100

150

200

250

300

xy

a 0.003

10 20 30 40t

50

100

150

200

250

300

xy

a 0.001

20 40 60 80 100t

50

100

150

200

250

300

xy

CommentsThe number of students increases in option2 when the advertisement is high, while the number of students in option1 decreases.

Phase Diagrams

Case: 2 1d d

50;

d1 0.1;

d2 0.3;

0.1;a 0.002

220 240 260 280x

40

60

80

100

y

( ; )dX dY

vdt dt

a 0.001

340 360 380 400 420 440x10

1520253035

y

Case: 2 1d d

50;

d1 0.3;

d2 0.1;

0.1a 0.002

120 140 160 180 200x

50

100

150

200

y

a 0.001170 175 180 185 190 195 200

x

5

10

15

20

y

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

a 0.001

210 220 230 240x

5

10

15

20

y

8-CONCLUSION

The study of this model help to understand the influene of advertisement during the enrollment of students to a program The choice of some parameter can influence positively or negatively the results expected. However, the advertisement is influenced by the number of freshmen students coming into the program. By increasing the number of freshmen we decrease the advertisement therefore the number Y will be affected.

9-CLASSIFICATION

• This model seems to be more realistic than precise, when and a=0.001 the value of Y is greater than X, but when and a =0.003 the value of Y is less than X. In addition

there in no student in option2 when (3).1 2( )d da

10-SUGGESTIONS

• We can improve the model by taking care that the inequality (3) does not become true. This leads to the strategies

• advertising towards more freshmen entering option1 (decreases )

• make students in option2 feel happier (decrease ) • use the same rate ( ) of success to help

students finish quicker 2 1d d

11-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. (2006) class notes.