# Mathematical Modelling: Analysis and Applications

By Prof. Ameeya Kumar Nayak   |   IIT Roorkee
Learners enrolled: 2036
This course provides introduction of mathematical modeling and analysis in biological sciences. It is designed for students in both applied mathematics and bio-medical / biological sciences. The major content of this course is chosen from population dynamics. This course covers the fundamentals of deterministic models in both discrete and continuous time domain. This course includes both linear and non-linear models with sufficient amount of theoretical background. The relevant concepts and solution methods of various difference and differential equations are provided. We have also focused on graphical solution for clear analysis of nature of models.

INTENDED AUDIENCE: UG/PG students of technical universities/colleges

PREREQUISITES:         Basic Calculus

INDUSTRY SUPPORT :
Nil

Summary
 Course Status : Completed Course Type : Core Duration : 4 weeks Start Date : 14 Sep 2020 End Date : 21 Sep 2020 Exam Date : 18 Dec 2020 IST Enrollment Ends : 25 Sep 2020 Category : Mathematics Credit Points : 1 Level : Undergraduate/Postgraduate

### Course layout

Week 1  : Overview of mathematical modeling, types of mathematical models and methods to solve the same; Discrete time linear models – Fibonacci rabbit model, cell-growth model, prey-predator model; Analytical solution methods and stability analysis of system of linear difference equations; Graphical solution – cobweb diagrams; Discrete time age structured model – Leslie Model; Jury’s stability test; Numerical methods to find eigen values – power method and LR method.

Week 2  :  Discrete time non-linear models- different cell division models, prey-predator model; Stability of non-linear discrete time models; Logistic difference equation; Bifurcation diagrams.

Week 3  :  Introduction to continuous time models – limitations & advantage of discrete time model, need of continuous time models; Ordinary differential equation (ODE) – order, degree, solution and geometrical significance; Solution of first order first degree ODE – method of separation of variables, homogeneous equation, Bernoulli equation; Continuous time models – model for growth of micro-organisms, chemostat; Stability and linearization methods for system of ODE’s.

Week 4  :  Continuous time single species model – Allee effect; Qualitative solution of differential equations using phase diagrams; Continuous time models – Lotka Volterra competition model, prey-predator models.

### Books and references

1.J.N. Kapur, Mathematical Models in Biology and Medicine, East-West Press Private limited.
2.Leah, Edelstein, Keshet, Mathematical Models in Biology, SIAM publications.
3.J.D. Murray, Mathematical Biology Vol. I, II, 3rd edition, Springer publications.

### Prof. Ameeya Kumar Nayak

IIT Roorkee
Dr. Ameeya Kumar Nayak is Associate Professor in Department of Mathematics at IIT Roorkee and actively involved in teaching and research in the direction of numerical modeling of fluid flow problems for last ten years. His research interests are in the fundamental understanding of species transport in macro and micro-scale confinements with applications in biomedical devices and micro electro mechanical systems. He has authored and co-authored more than 32 peer-reviewed journal papers, which includes publications in Springer,ASME, American Chemical Society and Elsevier journals. He is also active in writing book chapter with reputed international publication house.

### Course certificate

• The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
• The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
• Date and Time of Exams:  18 December 2020, Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
• The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
• Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE:
• Average assignment score = 25% of average of best 3 assignments out of the total 4 assignments given in the course.
• Exam score = 75% of the proctored certification exam score out of 100
• Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75.
• If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.
• Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and  IIT Roorkee. It will be e-verifiable at nptel.ac.in/noc
• Only the e-certificate will be made available. Hard copies will not be dispatched.