Epidemic model: how SIR and SEIR curves rise, peak and fall

Guide · Updated 2026-11-10

How compartmental outbreak models work, what R0, the latent and infectious periods and contact cuts do to the curve, and a worked SIR example with vaccination.

Open the Epidemic model Free, in your browser, nothing uploaded.

The epidemic model tool runs a small outbreak inside your browser. You describe a disease (how catching it is, how long people take to become infectious and how long they stay that way), choose a population and a vaccinated share, and it draws how many people are susceptible, exposed, infectious and immune on every day. It is meant for students of biology, public health and maths. It is a teaching model, not a forecast, and not medical advice. To work out the vaccine coverage that stops spread, open the herd immunity; to draw a real outbreak from case dates, use the epi curve; and to make your own charts from the daily numbers, try the publication plot maker.

What it is

A compartmental model sorts every person into one box at a time and moves people between the boxes at set rates. The boxes are:

• S, susceptible: can still catch it.
• E, exposed: infected, but not yet passing it on (only in SEIR and SEIRS).
• I, infectious: can pass it on.
• R, recovered or immune, which also holds anyone vaccinated at the start.

New infections happen at the rate β S I / N, where N is the population. The transmission rate β is R0 / D, where R0 is how many people one case infects when everyone is susceptible and D is the infectious period in days. Exposed people become infectious at the rate 1 / (latent period), and infectious people recover at the rate 1 / D. In SEIRS, immune people slowly drift back to S at the rate 1 / (immunity duration), so the disease can return.

The key number is the R at the start: R0 times the share susceptible. If it is above 1, each case more than replaces itself and cases grow. As S shrinks, the effective R falls, and when it drops below 1 the curve peaks and turns down. The herd immunity threshold, 1 − 1/R0, is the immune share at which that happens.

How to use it

1. Pick SIR, SEIR or SEIRS in the Model box. The latent period box appears for SEIR and SEIRS, and “Immunity lasts” only for SEIRS.
2. Enter the population, R0 and the infectious period in days, plus the latent period if the model has one.
3. Set the initial infectious cases, the vaccinated at the start (%) and the days to run (up to 3,650).
4. To test distancing or closures, type lines in Interventions as from day, to day, % contact cut. Cuts that overlap multiply together.
5. Read the result card: the peak day and peak size, the share infected during the run, the final-size answer, R at the start, the threshold and the share still susceptible. Two charts follow: the four boxes over time and new infections per day.
6. Use Show as to switch the charts between people and per cent, then download the daily numbers as CSV or the full report as an image, PDF or Word file.

A worked example

Take a made-up illness in a town of 20,000 people. Choose SIR, set R0 = 1.8, an infectious period of 4 days, 5 initial cases, 0 % vaccinated and 250 days.

• Transmission rate: β = 1.8 ÷ 4 = 0.45 per day; recovery rate γ = 1 ÷ 4 = 0.25 per day.
• Herd immunity threshold: 1 − 1/1.8 = 44.4 %.
• The tool puts the peak on day 39.2 with 2,361 people infectious at once (11.8 % of the town).
• By day 250, 73.3 % have been infected, the same as the final-size equation, and 26.7 % never catch it.

Now vaccinate 25 % at the start. R at the start drops to 1.8 × 14,995 ÷ 20,000 = 1.35. The peak moves to day 71.1 and shrinks to 558 infectious people, and only 35.2 % of the town is infected over the run.

At 50 % vaccinated, R at the start is 0.90, below 1. The tool reports no outbreak: cases fall from day 0.

Back at 0 %, add the line 25, 55, 30 (contacts cut by 30 % from day 25 to day 55). During the cut, R0 is effectively 1.8 × 0.7 = 1.26. The peak comes slightly earlier, on day 38.0, but at only 825 people, and the total infected falls to 62.0 %. By day 55 enough people are immune that it does not take off again.

Finally switch to SEIR with a 2-day latent period and no cut. The peak slides to day 66.7 and drops to 1,566, but the final share infected stays at 73.3 %. A latent stage slows an outbreak without changing how many it reaches.

Where it is used

• Public health teaching. Courses use SIR and SEIR to explain R, peaks and herd immunity before moving to models with age groups and contact surveys.
• Planning exercises. Comparing peak size with and without a contact cut shows why spreading cases out matters for hospital beds, even when the total barely moves.
• Ecology and animal health. The same equations describe diseases in livestock, wildlife and plants.

Common mistakes

• Treating the curve as a prediction. Real outbreaks depend on age, places, testing and behaviour. Use the model to compare scenarios, not to forecast a date.
• Stopping the run too early. If the result says “still rising at the end”, raise the days to run before reading the peak.
• Expecting a contact cut to end an outbreak. A short cut started early often only delays the peak. Try different start days and lengths to see the difference.

Questions people ask

Why does a latent period change the peak but not the total? It delays each generation of cases, so the curve stretches out. Each case still infects the same number of people on average, so the final size stays the same.

What happens if immunity wears off? Pick SEIRS and run for a few years. People return to the susceptible box, and the outbreak comes back in smaller waves until it settles at a steady level.