Modelling Epidemics with Exponential Functions

Biology meets Maths: Exponential functions aren’t just abstract curves on a graph — they describe some of the most important processes in nature and society. One of the clearest examples is how infectious diseases spread through a population. By modelling epidemics mathematically, students can see how small changes in rate lead to dramatic differences in outcome.

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The Concept

When a disease spreads, the number of infected people can grow rapidly because each infected individual passes it on to more than one other person. This creates an exponential pattern, expressed as:

$$N(t) = N_0 e^{rt}$$

where:

• $N_0$ = initial number of infected people

• $r$ = rate of infection

• $t$ = time (days, weeks, etc.)

• $N(t)$ = total number of infected individuals at time $t$

The model predicts fast early growth that later slows as immunity builds or control measures reduce the spread.

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Classroom Activity

Students can use spreadsheet or Python tools to:

• Plot infection growth for different $r$ values

• Compare exponential and logistic models (with a population limit)

• Discuss how interventions such as vaccination or isolation alter the shape of the curve

This turns a simple mathematical equation into a real-world tool for understanding public health.

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Skills Highlight

• Applying exponential growth models to real-world contexts

• Analysing how rate constants affect curve steepness

• Using technology to visualise and interpret data

• Understanding the limitations of models and the effect of assumptions

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Why It Works in Teaching

Modelling epidemics gives exponential functions real meaning. Students see that what starts as a small number can grow rapidly under the right conditions — and that mathematics helps predict, explain, and manage such events.