Reed–Frost model

Let ${\displaystyle I_{t}}$ represent the number of cases of infection at time ${\displaystyle t}$. Assume all cases recover or are removed in exactly one time-step. Let ${\displaystyle S_{t}}$ represent the number of susceptible individuals at time ${\displaystyle t}$. Let ${\displaystyle {\mathcal {B}}(x)}$ be a Bernoulli random variable that returns ${\displaystyle 1}$ with probability ${\displaystyle x}$ and ${\displaystyle 0}$ with probability ${\displaystyle 1-x}$. Making use of the random-variable multiplication convention, we can write the Reed–Frost model as

${\displaystyle {\begin{aligned}I_{t+1}&=\sum _{k=0}^{S_{t}}{\mathcal {B}}(1-(1-p)^{I_{t}}),\\S_{t+1}&=S_{t}-I_{t+1}\end{aligned}}}$

with initial number of susceptible and infected individuals ${\displaystyle (S_{0},I_{0})}$ given. Here, ${\displaystyle p}$ is the probability that a person comes in contact with another person in one time-step and that that contact results in disease transmission.

The deterministic limit is (found by replacing the random variables with their expectations),

${\displaystyle {\begin{aligned}I_{t+1}&=S_{t}\,(1-(1-p)^{I_{t}}),\\S_{t+1}&=S_{t}\,(1-p)^{I_{t}}\end{aligned}}}$