Chapter 16 — Appendix A.2: Mathematical Modelling

According to the appendix, mathematical modelling is an attempt to study some part (or form) of a real-life problem in mathematical terms. It is the process of converting a physical situation into mathematics using suitable conditions and assumptions, providing a representation and solution for problems that are difficult to tackle directly.

The four stages are: (1) understanding the problem — identifying parameters involved; (2) formulation — finding relevant factors and writing a mathematical description such as an equation or inequality; (3) finding the solution — solving the equation or applying a theorem; and (4) interpretation/validation — checking computed results against real-world data and describing the solution in context.

Formulation consists of (1) identifying the relevant factors or parameters — determining which variables actually influence the outcome (for example, showing that the mass of a pendulum's bob does not appreciably affect its period, whereas length does); and (2) mathematical description — expressing the relationship among the identified parameters as an equation, inequality, or geometric figure.

The simple pendulum example walks through all four stages. Understanding: the period T of oscillation must be found. Formulation: experiments show mass is not essential but length l is; plotting T against l gives a parabola-like curve leading to T = 2π√(l/g). Solution: the formula is applied to compute T for specific lengths (e.g., l = 225 cm gives T = 3.04 s). Validation: computed values are compared with experimentally measured periods; small errors (e.g., 0.011 s) arise because string mass and air resistance were neglected.

The Konigsberg bridge problem asks whether one can walk through the town crossing each of its seven bridges exactly once. Euler (1736) modelled the town as a network of vertices (land areas) and arcs (bridges). He proved the walk is impossible because all four vertices have an odd number of arcs joining them. Since a valid Euler path can have at most two odd-degree vertices (a start and an end), four odd vertices make the walk impossible. This work founded graph theory.

A farm requires at least 800 kg of special food daily — a mix of corn (protein 9%, fibre 2%, Rs 10/kg) and soyabean (protein 60%, fibre 6%, Rs 20/kg) — with at least 30% protein and at most 5% fibre in the mix. The problem is formulated as minimising z = 10x + 20y subject to three linear constraints and solved graphically. The minimum daily cost is Rs 11,294, achieved with approximately 470.6 kg corn and 329.4 kg soyabean.

The model assumes constant birth rate b and death rate d with no migration. Starting from P(t+1) = P(t) + bP(t) − dP(t), it derives the exponential formula P(t) = P(0)·r^t where r = 1 + b − d is the growth rate (the Malthusian parameter). For an example with P(0) = 250,000,000, b = 0.02, d = 0.01, the model predicts a population of approximately 276,155,531 after 10 years.

In the population growth model in this appendix, the constant r = 1 + b − d is called the growth rate or the Malthusian parameter, in honour of Robert Malthus who first brought this model to popular attention. It combines the birth rate b and death rate d into a single multiplier applied at each time step.

The appendix lists four applications: (1) analysing blood flow to find how constrictions in blood vessels affect circulation; (2) simulating the trajectory of a cricket ball to decide LBW decisions (the third-umpire review system); (3) weather prediction by measuring temperature, air pressure, humidity, and wind speed at multiple stations and feeding the data into computers; and (4) estimating national rice yield by sampling representative fields and applying statistical techniques.

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