## COMO101 Modelling and Computation

Second Semester |

COMO 101 is on blackboard: go to https://blackboard.otago.ac.nz/ for up-to-date course information, lectures, tutorial sheets, etc.

This paper provides an introduction to mathematical and computational modelling with applications in science, engineering, biomedicine and industry. Topics include the translation of real- world problems into mathematical models, and the use of simulation and numerical methods to evaluate and apply the models.

### Timetable

#### (1) Introduction to estimation and mathematical modelling (2 weeks).

Scientists often carry out “back of the envelope” calculations to get a rough answer to a problem before proceeding to more exact and involved approaches. Working with a few assumptions and a number of rough measurements, we can often make some startlingly accurate predictions and estimations. We use these examples of techniques to introduce the whole notion of mathematical modelling, its weaknesses and strengths. The examples we use range from an analysis of a potential hydro scheme in Otago harbour to how we can measure the thickness of gladwrap with a school ruler.

#### (2) Difference equations and dynamical models (4 weeks)

Difference equations are a standard technique for describing, mathematically, how quantities change over time. We introduce the basic ideas of how to set up and describe difference equations and systems of different equations, drawing on examples from population growth, epidemiology, ecology and genetics. The emphasis is on simulation of the systems and qualitative analysis, rather on analytical solutions.

#### (3) Randomness and stochastic models (3 weeks)

The growing importance of simulation is one of the most important developments in model-based inference and indeed in statistical computing in general. We review relevant ideas from probability theory, with an emphasis on how to simulate random variables and processes. We demonstrate how simulation can be used to estimate quantities and integrals.

#### (4) Data fitting and numerical methods (2 weeks)

It can come as a surprise to students that while most mathematics papers focus exact and analytical solutions, in application most equations and integrals are evaluated numerically. We introduce and review some standard techniques for solving equations numerically, optimizing and integrating. We see how these can be used to determine model parameters from data, illustrating the ideas with applications from epidemiology and biochemistry.

#### (5) Uncertainty quantification (2 weeks)

In this last section we come full circle and show how simulation can be used to assess the reliability of the models and estimates. We look at how determining whether our inferences are sensitive to small changes in parameters, and introduce the thorny area of model comparison.

### Prerequisites

None, though students will be expected to do some algebraic manipulation.

### Lectures

There are three lectures per week.

### Lecturer

Prof David Bryant, Math and Stats, Room 514, Science III (email). David's main area of research expertise is the application and development of mathematical, computational and statistical techniques in evolutionary biology and genetics.

### Tutorials

You will have a single, one hour tutorial/lab per week, held in a computer lab. You should have been assigned tutorial times. Yes, you can change them provided that the numbers in different tutorials remains fairly balanced. **Tutorials start week two.**

Your final grade will combine internal assessment and the final exam. The breakdown of assessment is:

Assignments (four) 5% each

Practical test (terms req.) 5%

Midterm test 15%

Tutorials 10%

Final Exam 50%

The practical test will be held during tutorial times in August. Students will be able to repeat the test until they obtain a pass and satisfy the terms requirement, though the original mark will be used for the grade. The Midterm (theory) test will be held immediately before or after semester break. The room for the test will be announced shortly. Assignments will be due at 8pm on Wednesdays and must be submitted electronically through Blackboard. Late assignments will not be accepted.

Assignments will include some exam-style questions and a written report developing a case study. The reports are to written according to a strict template discussed in class.

### Terms Requirement

You have to fulfill the terms requirement in order to be allowed to sit the final exam. In this paper, to pass “terms” you must pass the practical test.

### Exam format

The Como101 final examination will be three hours long. You will be permitted to take calculators into the exam, but no notes or communicating devices.

### Required text

There will be no required text. Lecture slides will be available on blackboard.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.5E + 0.2A + 0.05P + 0.1T + 0.15M**

where:

- E is the Exam mark
- A is the Assignments mark
- T is the tutorials mark
- M is the midterm mark
- P is the practical mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.