Thought Toys · Exhibit 16

The epidemic threshold

A single number decides everything: whether a new infection sputters out after a handful of cases, or sweeps through a whole population. Slide it across one and watch the curve flip. Then vaccinate just enough people to break the chain — and find the exact fraction where the outbreak can no longer grow.

Share of the population over time — susceptible, infected, recovered
susceptible infected now recovered / immune over capacity

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0.5 · dies out4 · highly contagious

What you're seeing

Everyone starts susceptible (blue). A few people are infected and, while they're infectious, each passes it on at a rate set by how contagious the bug is and how long they stay sick. Infected people (amber) eventually recover (green) and can't catch it again. That's the whole model: three buckets, people flowing S → I → R.

The one number that rules it is R₀ — the average number of people a single case infects in a fully susceptible crowd. Below 1, each case replaces itself with less than one new case and the chain dies: the amber curve only sinks. Above 1, it grows faster than it fades, and you get the familiar wave that rises, peaks, and burns out only when it runs low on susceptible people. Nudge R₀ through 1.0 and watch a non-event become an epidemic.

Now the hopeful part. You don't have to make everyone immune to stop it — only enough that each case can't find more than one new victim. Drag "vaccinated" up, or hit snap to herd immunity: the curve collapses the instant the immune share passes 1 − 1/R₀. The dashed line is an illustrative care capacity — keeping the amber peak under it is the whole point of "flattening the curve."

The rule, exactly. Fractions of the population obey dS/dt = −βSI,  dI/dt = βSI − γI,  dR/dt = γI with recovery rate γ = 1/(infectious period), transmission β = R₀·γ, and R₀ = β/γ. Infections grow only while the effective reproduction number R₀·S > 1, so immunizing a fraction 1 − 1/R₀ drives it below 1 — herd immunity. Integrated with RK4 (Δt = 0.2 day). Counter-example, verified in node: with R₀ < 1 there is no outbreak at all — the infected curve never rises above its seed, it only declines (a negative control: the wave needs R₀·S to cross 1, not merely a few infected people).

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