SIR Epidemic Model Simulator

What the SIR model is

The SIR model splits a population into people who are susceptible (S), infectious (I) and recovered (R), and moves them between those groups at two rates: new infections at βSI/N per day and recoveries at γI per day. This simulator solves those equations, or the SEIR version with a latent stage, for the R₀, infectious period and vaccination you set, and reports the peak of the outbreak, how many people are infected in the end and the herd immunity threshold, 1 − 1/R₀.

In plain text the equations are dS/dt = −βSI/N, dI/dt = βSI/N − γI and dR/dt = γI. N is the population, β the transmission rate per day and γ = 1/D the recovery rate for a mean infectious period of D days. The basic reproduction number is R₀ = β/γ: how many people one case infects when everyone else can catch it. R is often read as removed rather than recovered, since someone who dies of the infection leaves the chain of transmission in the same way. Kermack and McKendrick published it in 1927, as the case with constant rates of a more general theory of epidemics.

How to use it

Set R₀ and the infectious period, play the outbreak and watch the squares. Each is a quarter of a percent of the population: blue while susceptible, amber while infected and green once recovered, with the vaccinated drawn as slashed green outlines. The scrubber moves through the days, and the readouts beside the scene give the results for the whole outbreak:

• Peak infectious and Peak on day: the most people infectious at once, and when.
• Total infected: everyone infected at some point, the final size of the epidemic.
• Herd immunity threshold: the immune share, 1 − 1/R₀, above which an outbreak cannot grow.
• R on day 0: R₀ times the share still susceptible after vaccination. The outbreak grows only if it is above 1.
• Early doubling time: how quickly the number infected doubles while nearly everyone is still susceptible, the same exponential growth as the cell doubling time calculator works with.

If a problem gives you β and γ rather than R₀, set R₀ to β/γ and the infectious period to 1/γ days. The rates the equations are using are printed under the parameters, so you can check the match.

Worked example: R₀ = 2.5 in a town of 100,000

Take the defaults: 100,000 people, 10 of them infectious on day 0, nobody vaccinated, R₀ = 2.5 and an infectious period of 5 days. Then γ = 1/5 = 0.2 per day and β = R₀γ = 0.5 per day.

• Herd immunity threshold. 1 − 1/2.5 = 0.6, so 60 percent.
• Early growth. On day 0, R = 2.5 × 99,990/100,000 = 2.49975, so r = γ(R − 1) = 0.29995 per day and the number infected doubles every ln 2/r = 2.31 days at first.
• The peak. The number infectious is largest when S has fallen to N/R₀ = 40,000. The first integral of the equations gives I_max = I₀ + S₀ − (N/R₀)(1 + ln(R₀S₀/N)), here 10 + 99,990 − 40,000 × (1 + ln 2.49975), which is 23,352 people, 23.4 percent of the town, on day 32.1.
• The total. The final size relation ln(s₀/s∞) = R₀(s₀ + i₀ − s∞), with s₀ = 0.9999 and i₀ = 0.0001, gives s∞ = 0.10734: 10,734 people are never infected and 89,266 are, 89.3 percent.

Now switch the model to SEIR with a 5 day latent period, keeping R₀ at 2.5. The doubling time lengthens to 5.96 days, the peak falls to 11,489 and moves to day 80.1, and the total infected is 89,266 again. Go back to SIR and vaccinate 40 percent instead: R on day 0 is 1.5, the peak is 3,788 on day 75.1, and 34,980 people are infected in all, 58.3 percent of the 60,000 who were not vaccinated.

Why the outbreak turns while most people are still susceptible

Rewrite the second equation as dI/dt = γI(R₀S/N − 1). The number infectious grows while R₀S/N is above 1 and falls once it drops below 1. That product is the effective reproduction number R, the number of people each case infects now rather than at the start, and it falls because every infection uses up a susceptible person.

So the peak comes exactly when the susceptible share reaches 1/R₀: 40 percent for R₀ = 2.5. The dashed line on the first plot marks that level, and in the SIR model the infectious curve peaks where the susceptible curve crosses it. Read the other way, at that moment 60 percent of the town can no longer catch it, the 23 percent infectious and the 37 percent recovered, which is the herd immunity threshold: the peak is the moment the population crosses it.

Herd immunity, and why an epidemic overshoots it

Reaching the threshold stops the growth, not the epidemic. At the peak more people are infectious than at any other moment, and each of them still infects R of the people left, fewer than one but not zero, so infections carry on down the far side of the curve. The final size is therefore well above the threshold. For an outbreak started by a single case in a large population where everyone can catch it, the final size relation and the first integral give:

Vaccination works on the same threshold from the other side. Immunise a random share above 1 − 1/R₀ before the first case and R on day 0 is below 1, so the outbreak never grows. Try it at the defaults. At 59 percent R on day 0 is 1.02, and a slow outbreak peaks at 22 people infectious on day 284 and infects 2,314 in all. At 60 percent R is 0.99975: the number infectious never rises, though the first ten still pass it on, and 888 people are infected in all, counting them. At 61 percent the total is 340.

SIR or SEIR: a latent period moves the peak, not the total

The SEIR model adds an exposed class E of people who have been infected but cannot yet infect anyone: dE/dt = βSI/N − σE, and infectious people now arrive at σE per day, where 1/σ is the mean latent period. R₀ is still β/γ, because everyone who is exposed goes on to become infectious.

The latent stage slows the start. Wallinga and Lipsitch (2007) showed that when both stages last exponentially distributed times, as they do here, the early growth rate r and R₀ are linked by R₀ = (1 + r/σ)(1 + r/γ), against R₀ = 1 + r/γ without one. At the defaults that takes the growth rate from 0.3 to 0.116 per day. What it does not change is the total: the final size relation holds unchanged with a latent stage, as Ma and Earn showed in 2006, so SIR and SEIR with the same R₀ infect the same number of people.

Run the relation the other way and it matters for estimates. The same early growth rate means a larger R₀ with a latent stage than without one, so fitting SIR to an outbreak that has a latent period underestimates R₀. Wearing, Rohani and Keeling (2005) found that ignoring the latent period, or assuming exponentially distributed periods once it is included, always underestimates R₀ from outbreak data.

Reading the two plots

The first plot is the share of the population in each compartment, with the vaccinated as a flat dashed line when there are any and 1/R₀ dashed in grey. The second is new infections per day, which is what an epidemic curve of cases by date of infection shows. It peaks before the number infectious does, because each new infection needs a susceptible person as well as an infectious one: at the defaults new infections peak on day 28.4 and the number infectious on day 32.1.

What this model leaves out

Chance. The equations move fractions of a population, so an outbreak with R above 1 always takes off. Real outbreaks start with a few people, and a few people can all recover before passing it on. The radioactive decay simulator shows the same gap between a smooth average and a handful of random events.

Structure. Everyone meets everyone with the same β. Households, schools, ages and travel all make contacts uneven, and with uneven contacts the effect of immunity depends on who has it, so one threshold for a random share no longer tells the whole story.

Time beyond one wave. There are no births, no deaths from other causes and no waning immunity, so the model gives one wave and then stops. Recovery also happens at a constant rate, which makes the infectious period exponentially distributed, the same memoryless shape as radioactive decay in the half-life calculator. Wearing, Rohani and Keeling (2005) call that unrealistic for most infections, whose periods cluster more tightly around their mean.

People reacting. Nobody changes their behaviour, no policy changes during the outbreak, and nothing is seasonal. Those are among the reasons real epidemic curves are rarely the single smooth hump drawn here.

Common mistakes

• Treating R₀ as a fixed property of a pathogen. It is set by how infectious a contact is, how many contacts people have and how long they stay infectious, so the same infection has a different R₀ in a different population.
• Confusing R₀ with R. R₀ assumes everyone else is susceptible. R is R₀ times the share who still are, and it is R that has to be below 1 for cases to fall.
• Expecting the epidemic to stop at the herd immunity threshold. It peaks there and then overshoots it, which is why 89 percent are infected when R₀ = 2.5 and the threshold is 60 percent.
• Using the infectious period as the recovery rate. γ is one over the period: 5 days is 0.2 per day, not 5.
• Assuming the latent period changes the total. It changes when and how high the peak is, not how many are infected in the end.
• Treating the peak in new infections and the peak in people infectious as one day. New infections per day peak first, while the number currently infectious is still rising.
• Taking the curve as a forecast. This is the smooth behaviour of an idealised population. It shows why epidemics rise and fall, not when a real one will.