Everyone is in exactly one of three buckets, and people only ever move forward:
not yet affected → sick right now → past it. That is the classic SIR model,
three lines of calculus written in 1927:
dS/dt = −b · S · I / N
dI/dt = +b · S · I / N − I / D
dR/dt = +I / D
b is the spread rate (how many new cases one sick person starts per day when
everyone around them is still healthy) and D is how many days a person stays contagious.
Their product is the famous R0 = b · D — the number of people one case passes it to at
the very beginning. Above 1 the curve grows, below 1 it fades.
The slowdown simply multiplies b from the chosen day onward. Nothing else
changes — same crowd, same illness — which is exactly why the effect is so easy to read.
Solved with fourth-order Runge–Kutta at a tenth-of-a-day step over 360 days,
recomputed from scratch on every slider move. No libraries, no network, about a hundred lines
of arithmetic.
What to notice. The S·I/N term is why the curve eventually falls even with
no slowdown at all: the disease runs out of people to reach. And it is why a late slowdown
does almost nothing — by then the crowd has already done the flattening for you.