1. TOPIC: MODELING ENROLLMENT TRENDS

2. OUTLINE 1-INTRODUCTION 2-THE MODEL 3-DIAGRAM 4-DERIVATION OF THE MODEL 5-ANALYSIS OF THE MODEL 6-CONCLUSION 7-REFERENCES

3. 1-INTRODUCTION 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 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. 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), 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 “easy” with easy examination and favorable grade) or option 2 (“harder” program where subjects are “hard” and professors are exhaustive, ambitious, helpful.), and also by the advertisement (communication). 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 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.

4. 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 2 Parameters -”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 2

5. 3-DIAGRAM X(t) λ Y(t) X(t)Y(t) a βY(t) Option 1 Option 2

6. 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.

7. System differential of the enrollment trends. removed NB The differential system of equations above is similar to the epidemic model with I=X, S=Y S-------> I--------> S

8. 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

9. NB We 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.

10. Simulations or experiments case

12. Case:

14. Phase Diagrams Case:

18. 6-CONCLUSION 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.

19. 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.

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