Comparing Simulations with Solutions for Classic Differential Equation

Context at NCSSM is a public boarding school 650 11 th and 12 th

Context at NCSSM Graph Theory/Introduction to Proof Combinatorics I & II Number Theory or

Modeling with Differential Equations One Trimester Course (10 Weeks) Focus on Modeling with Modeling

Coupled Equations Parameters are Probabilities

Probabilistic Model in Deterministic Form What does this equation say about Probability? How can

With N in the population, we have… Solve using separation of variables and partial

How do we check the solution? Does our solution satisfy the differential equation? Does

What are we really modeling in this problem?

Physical Simulation Take class into the hall, give each students a die, and meet

But if we did the experiment again, we would have the same solution, but

How in well does our Differential Equation do in capturing what we see? If

What do you see in the data vs the solution model

Suppose we had only the data? How should we estimate the parameter values?

How does the Simulation Change if we use an SIR model instead?

Daniel J. Teague NC School of Science and Mathematics teague@ncssm. edu

Slides: 30

Download presentation

Comparing Simulations with Solutions for Classic Differential Equation Models Daniel J. Teague NC School of Science and Mathematics teague@ncssm. edu

Context at NCSSM is a public boarding school 650 11 th and 12 th grade students from every congressional district Trimester Schedule Elective Courses: Mathematical Modeling with Differential Equations Introduction to Complex Systems Structure and Dynamics of Modern Networks Advance Probability Models

Context at NCSSM Graph Theory/Introduction to Proof Combinatorics I & II Number Theory or Group Theory Research in Mathematics

Modeling with Differential Equations One Trimester Course (10 Weeks) Focus on Modeling with Modeling and Interpreting Equations and Solutions: not solving lots of different classes of DE’s Separation of Variables, Integrating Factor, Second Order Equations, Systems of Linear Equations, Numerical Solutions

Coupled Equations Parameters are Probabilities

Infectious Disease Models

Probabilistic Model in Deterministic Form What does this equation say about Probability? How can we interpret and model the probability parameter k?

With N in the population, we have… Solve using separation of variables and partial fractions

How do we check the solution? Does our solution satisfy the differential equation? Does our solution model our observations?

What are we really modeling in this problem?

Physical Simulation Take class into the hall, give each students a die, and meet and infect each other with the appropriate probabilities.

But if we did the experiment again, we would have the same solution, but different data. How different? Let’s milk this problem for a couple of days.

Netlogo Simulations

How in well does our Differential Equation do in capturing what we see? If we have the data, do we solve the equation or fit a model to the data?

What do you see in the data vs the solution model

If we slide the model forward 3 ticks…

Suppose we had only the data? How should we estimate the parameter values?

Fit a Logistic Model to the Data

Residual Plot

DE vs LS

How does the Simulation Change if we use an SIR model instead?

Netlogo

Daniel J. Teague NC School of Science and Mathematics teague@ncssm. edu