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.