Disease Spread (SIR Model Preview)

Calculus 2 · Axiom Academy

Disease Spread (SIR Model Preview) Three differential equations split a town into the Susceptible, the Infected, and the Recovered — and decide whether an outbreak takes off or fizzles. A new bug lands in a town of 400 people: 10 are sick, 390 can still catch it. Each day the infected pass it on at rate and recover at rate — and those two numbers settle everything that follows. Pick how contagious the bug is, hit Go, and play the whole epidemic. Three coupled rates run at once: susceptibles fall , the infected rise, peak, and fall , the recovered pile up . That's dS/dt = − SI , dI/dt = SI − I , dR/dt = I , live. The tipping point: will it take off? Forget the whole curve for a second — just look at day one. The infected count climbs only if each sick person infects more than one other before recovering. Drag the contagiousness and watch the very first move of dI/dt flip from shrinking to growing. The divide sits exactly at R₀ = 1 . It's logistic growth in disguise Here's the punchline the curve hides. New cases per day are SI — proportional to the infected I (who's spreading it) and the susceptible S (who's left to catch it). That double-barrelled term is exactly the logistic law S·I(1−I/···) : slow when either side is small, fastest in the middle. Slide the day and watch new cases rise to a peak, then fall — the same S-curve story as population growth.

This is the written version of the interactive lesson above. See the full Calculus 2 course.