Computational and statistical methods for mathematical biologists and epidemiologists.
Objectives:
This course is meant to provide students in applied mathematics with the broad skill-set needed to optimize model parameters to relevant biological or epidemic data. The course will almost entirely be based on material posted on this website.
Upon completing this course:
Students will gain a basic understanding of applied statistics, and will be functional in R.
Students will learn how to read in, manipulate, and export data in R, and will be able to create publication-quality plots in R. Students will be familiar with several different parameter optimization methods, and will understand the underlying assumptions of each.
List of course modules:
- Good work habits, and requirements for homework
- Literature searches with Google Scholar
- Elements of scientific papers
- The basics of the R statistical programming language
- Difference between statistical and mathematical models
- Numerically solving systems of non-linear ODE’s in R: Euler’s method
- Numerical methods to solve non-linear ODE’s
- Numerically solving systems of non-linear ODE’s in R: 4th order Runge-Kutta using the deSolve library
- Extracting data from graphs in published literature
- Online sources of free data
- SIR disease model with age classes
- SIR modelling of influenza with a periodic transmission rate
- Fitting the parameters of an SIR model to influenza data using Least Squares and the Monte Carlo parameter sweep method
- An overview of goodness of fit statistics, and methods to fit parameters of mathematical models to data
- Fitting the parameters of an SIR model to influenza outbreak incidence count data with the Monte Carlo method: a comparison of Least Squares, Pearson chi-square weighted least squares, Poisson negative log-likelihood, and Negative Binomial negative log-likelihood
- Estimating parameter confidence intervals when using the Monte Carlo parameter sweep optimization method: the fmin+1/2 method
- A better method for estimation of confidence intervals compared to the fmin+1/2 method: the weighted mean method
- Temporal and geospatial patterns in threats to Jewish community centers: an example of contagion in social behaviours?
- Incorporating prior parameter estimates and their uncertainties into your likelihood fits
- Producing well written manuscripts in a timely fashion
- Giving a good presentation
Course expectations:
There will be regular homework projects assigned throughout the course, which will be worth 50% of the grade. Students are strongly encouraged to work together in groups to discuss issues related to the course and resolve problems. However, plagiarism of code will not be tolerated.
The culmination of the course will be a group term project (two to three students collaborating together, with the project worth 50% of the final grade) that requires the development of an R program to solve a system of ordinary differential equations that describes the dynamics of disease spread, interacting biological populations, etc. The students will then optimize the parameters of their model to data that the student has identified as being appropriate to describe with their model. The students will write-up the results of their project in a format suitable for publication, using the format required by a journal they have identified as being appropriate for the topic. A cover letter written to the editor of the journal is also required. However, submission for publication is not required, but encouraged if the analysis is novel.
Students are responsible for locating and obtaining sources of data, and developing an appropriate model for the project, so this should be something they begin to think about very early in the course.
This course has no associated textbook, due to the unique nature of the course content. Instead the course content consists of the modules that appear on this website. A textbook that students may find useful is Statistical Data Analysis, by G. Cowan
Students are expected to bring their laptops to class. Before the course begins, students are expected to have downloaded the R programming language onto their laptop from http://www.r-project.org/ (R is open-source free software).
Final project write-ups will be due Friday, April 15th. Each of the project groups will perform an in-class 20 min presentation on Monday, April 24th, 2017 and Wed, April 26th, 2017.
During the week of April 17th, project groups will meet with Dr. Towers to discuss their final project write-ups, and their upcoming presentation. By Friday, April 28th, all group members are to submit to Prof Towers a confidential email, detailing their contribution to the group project, and detailing the contributions of the other group members.