Epidemics and the SIR Model Simulation

About this experiment

The SIR model splits a population into three groups: people who can still catch a disease (susceptible), people who have it and can pass it on (infectious) and people who have recovered and are immune. Three simple rates are enough to reproduce the rise and fall of an epidemic. Change the basic reproduction number R₀, vaccinate part of the town before the outbreak or add a period of social distancing, and compare the curve with what would have happened without any measures.

Learning goals

Describe the three compartments of the SIR model and the flows between them
Explain the meaning of the basic reproduction number R₀ and the effective reproduction number
Explain why an epidemic peaks and declines before everyone has been infected
Calculate the herd immunity threshold and test it with vaccination
Explain what "flattening the curve" means and why the timing of measures matters

How to use it

1Press Play with the default settings. On which day does the number of infectious people peak? What is the effective R on that day?
2Reset and set R₀ to 1.2, then 4. How do the height of the peak and the total number ever infected change?
3Set R₀ to 2.5 and read the herd immunity threshold. Vaccinate a little less and then a little more than that. What happens to the outbreak?
4Remove vaccination and switch on social distancing from day 20 for 60 days. Compare the solid curve with the dashed one.
5Set Fewer contacts to 70 %, start on day 0 and keep it for 60 days. What happens after the distancing ends, and why?
6Find a set of measures that keeps the peak below the health care capacity line.

The model

\frac{d S}{d t} = - β S I, \frac{d I}{d t} = β S I - γ I, \frac{d R}{d t} = γ I

S, I and R are fractions of the population; β is the contact rate and 1/γ the infectious period

R_{0} = \frac{β}{γ}, R_{eff} = R_{0} S

Each case infects R₀ others in a fully susceptible population, fewer as immunity builds up

p_{c} = 1 - \frac{1}{R _{0}}

Herd immunity threshold: the immune fraction at which each case infects fewer than one other

ln \frac{S _{\infty}}{S _{0}} = - R_{0} (S_{0} + I_{0} - S_{\infty})

Final size relation: not everyone is infected, even with no measures at all

Assumptions

Everyone mixes with everyone else equally. Real contact patterns depend on age, households, schools and work.
Recovered and vaccinated people are immune for the rest of the outbreak, and vaccination is fully effective.
There are no births, deaths from other causes, travel or new variants, and the outbreak starts with 0.1 % of people infectious.
The model is deterministic: the same settings always give the same curve. Real outbreaks in small towns have a lot of chance in them.
The town picture shows the current fractions. Who is infected first is drawn spreading outwards for illustration; the SIR model itself has no geography.

Try this in class

Q1Why does the epidemic start to decline while there are still many susceptible people?
Q2Measles has an R₀ of about 15. What fraction of children must be vaccinated to stop it spreading?
Q3Distancing that starts too late hardly changes the peak. Use the simulation to explain why.
Q4Flattening the curve may not reduce the total number of people infected. Why can it still save lives?
Q5Which assumptions of the model would you change first to make it more realistic for a school?

Frequently asked questions

What is the difference between R₀ and the effective R?

R₀ is the average number of people one case infects when everyone is susceptible. The effective R also takes into account immunity and measures at a given moment. The epidemic grows while the effective R is above 1 and shrinks when it falls below 1.

Why is the peak reached when the effective R equals 1?

The number of infectious people stops rising exactly when each case is replaced by one new case. In the SIR model that happens when the susceptible fraction drops to 1/R₀.

Is this model used for real diseases?

Yes, as a starting point. Public health models add more compartments (for example people who are infected but not yet infectious, called SEIR), age groups, contact networks and random effects, but the ideas of R₀, herd immunity and flattening the curve all come from this simple model.