The Reed-Frost Model is a random, discrete step model of how an infection moves through a population. Proposed in 1928, it is based on the idea that at each step, infected individuals pass an infection on with fixed probability by contact with susceptible members of the population, and thereafter recover and gain immunity.

This is a kind of Compartmental or SIR Model, since the population is separated into distinct groups:

  • Susceptible — individuals who are yet to be infected,
  • Infectious — individuals who are currently infected,
  • Recovered — individuals who have been infected and are now immune.

We can simulate an epidemic using this model by starting with a single infected individual and a probability of infection p, and stepping through time week by week, updating the number of susceptible, infected, and recovered individuals.

The most important parameter is the Basic Reproduction Number, or R0, which gives the expected infections per carrier at the start of the epidemic. Limiting the spread of the disease is then a matter of reducing this value (e.g. by social distancing, vaccination).

Mathematically, for a population of size n,

R0 = np,

and reducing transmission by a factor c leads to the Effective Reproduction Number

RE = (1 − c)R0.

If we can achieve RE ≤ 1, the epidemic dies out.







Contagion!
Modelling an Epidemic
In our simulation, we will allow the expected number of infections per carrier R0 to vary from 1 to 4.

Combined with the reduction factor c, the simulation will run with the Effective Reproduction Number RE = (1 − c)R0.

Since the model is probabilistic, results can vary with the same input. In particular, the infection may die out almost immediately because of limited initial transmission.

If you select Run 10 Simulations, then each run will show the pattern of infection across 10 simulations.

Selecting the Use average infection rate checkbox removes the random component of the simulation.

Basic reproduction number R0   
Reduction factor c      RE

   

About the activity

The pattern of infection across multiple simulations, that you can study using the Run 10 Simulations option is quite interesting. There will generally be two clusters — one depending on R0, and one at 0.

In this model, the infection transmission is a Binomial process, since it is effectively an infected person meeting a susceptible person, then tossing a coin to see if the infection is passed on. The Binomial Bomber activity shows another example of binomial randomness.