### ACM 10070 Mathematical Modelling in the Sciences

This is a discontinued module. However, material related to it is posted here, as an archive.

**Description:** This module is an introduction to the mathematical modeling of phenomena in many branches of the physical and biological science. The models are formulated in terms of difference and differential equations and the students are introduced to the basic properties of these equations and certain analytical (i.e. "pen and paper") techniques for solving them.
[Mathematical background] Linear and non-linear ODEs, the order of a differential equation, separability of ODEs, homogeneity and inhomogeneity of ODEs, [Solution methods] substitution, separation of variables, integrating-factor technique, the exponential substitution for second-order linear homogeneous problems, the criterion for the existence and uniqueness of solutions, [Qualitative methods] Vector fields, simple vector fields for autonomous ODEs, fixed points and bifurcations, [Modelling techniques] Dimensional analysis, the scientific method, "theory" versus "model", [Applications] These include (but are not necessarily limited to) population models, fisheries models with harvesting, drug delivery, interest rates, epidemics, the fluid analogy of electrical circuits, RC and LRC circuits, [Discrete systems] discrete population dynamics, the Fibonacci sequence, properties of discrete maps (fixed points, orbits, stability), chaos in discrete maps, cellular automata.

**Learning Outcomes:** On completion of this module students should be able to:

- Solve linear first-order ODEs using the integrating-factor technique
- Solve separable ODEs by integration
- Solve arbitrary ODEs given a trial solution
- Plot slope fields for autonomous ODEs
- Identify the fixed points and the stability properties of autonomous ODEs, carry out a simple bifurcation analysis of such ODEs
- Construct and solve simple continuous mathematical models. The models may involve cooling, radioactive decay, population dynamics, harvesting, predation, disease outbreaks, drug delivery, interest rates, laser operation, or electrical circuits
- Construct and solve simple discrete mathematical models
- Compute the fixed points and periodic orbits of discrete autonomous maps and find their stability.

### Module materials

**Recommended movies:**

- Introduction to Differential Calculus
- The Chain Rule
- Best straight-line approximation - called the "linear approximation" in the movie
- Introduction to Integral Calculus
- Integration by u-substitution
- Integration by parts
- The integrating-factor method - This video works through the theory of the integrating-factor method and lots of examples. Just be careful! It uses f(x) instead of P(x) and g(x) instead of Q(x). Otherwise it's great!
- The formula for an annuity mortgage with discrete compounding of interest. Another good one! Only be careful of the notation - i for the interest rate (compounded on a discrete basis), M for the repayment amount, and A for the amount owing.
- The one-step binomial model - This video works through the same example from Hull's book that appears in my notes. The movie uses Delta instead of N for the weighting given to shares in the portfolio but apart from notation the method is the same.

**Computational Resources:**

- Wolfram Alpha Plotting
- Wolfram Alpha - plot two functions at once
- Excel spreadsheet - The population dynamics of Genovia (12/10/2016)
- Simple numerical SIR model implemented in Excel - This is a numerical implementation of the Kermack and McKendrick disease model based on Susceptible, Infected and Recovered (dead) sub-populations. All parameters are adjustable so the various effects mentioned in class can be investigated, in particular the threshold point at which the outbreak becomes an epidemic.

### Check your homework marks

Your marks in the homeworks and the midterm test can be checked here - just enter the module code and your student number. All marks out of 20.

### Application of discrete maps to wealth accumulation in the long run

For a topical application of discrete maps to wealth accumulation as a function of savings preferences, see the last chapter of the lecture notes, available here. The particular model in this chapter is based on the work of
Piketty and Zucman, in the