Appendix 2 — Mathematical Modelling

What is Mathematical Modelling?

Mathematical modelling translates a real-world problem into mathematical language, solves the mathematical problem, and interprets the solution back in the real-world context. It is a cycle, not a one-step task.

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Steps in Mathematical Modelling

1. Identify the real-world problem and define the goal.
2. Make simplifying assumptions about the situation.
3. Formulate the mathematical problem using variables, equations, or inequalities.
4. Solve the mathematical problem using appropriate techniques (algebra, calculus, etc.).
5. Interpret the solution in the original real-world context.
6. Validate the model by comparing predictions with actual data. If the match is poor, revise the assumptions and repeat from step 2.

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Key Example from the Textbook: Population Growth

Problem: The population of a city in 2000 was 5,00,000. It grew to 6,00,000 by 2005. Predict the population in 2010.

Step 1 – Identify: Find $P(2010)$.

Step 2 – Assumptions: the population grows continuously, at a rate proportional to the current population (Malthusian model), with no external factors (migration, epidemics).

Step 3 – Formulate: Let $t = 0$ correspond to the year 2000 and $P(t)$ be the population at time $t$ (in years), with $P(0) = 5{,}00{,}000$ and $P(5) = 6{,}00{,}000$. The differential equation is

where $k$ is the constant growth rate.

Step 4 – Solve: The general solution is

Using $P(5) = 6{,}00{,}000$:

For $t = 10$:

Step 5 – Interpret: The model predicts a population of 7,20,000 in 2010.

Step 6 – Validate: Compare with actual 2010 census data; a mismatch means the assumptions (e.g. constant growth rate) need revision.

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Important Notes for Exams …