Reed–Frost model

Description

This is an example of a "chain binomial" model, a simplified, iterative model of how an epidemic will behave over time.

The Reed-Frost model is one of the simplest stochastic epidemic models. It was formulated by Lowell Reed and Wade Frost in 1928 (in unpublished work) and describes the evolution of an infection in generations. Each infected individual in generation t (t = 1,2,...) independently infects each susceptible individual in the population with some probability p. The individuals that become infected by the individuals in generation t then constitute generation t + 1 and the individuals in generation t are removed from the epidemic process.

The Reed–Frost model is based on the following assumptions:

1. The infection is spread directly from infected individuals to others by a certain type of contact (termed "adequate contact") and in no other way.
2. Any non-immune individual in the group, after such contact with an infectious individual in a given period, will develop the infection and will be infectious to others only within the following time period; in subsequent time periods, he is wholly and permanently immune.
3. Each individual has a fixed probability of coming into adequate contact with any other specified individual in the group within one time interval, and this probability is the same for every member of the group.
4. The individuals are wholly segregated from others outside the group. (It is a closed population.)
5. These conditions remain constant during the epidemic.

The following parameters are set initially:

- Size of the population- Number of individuals already immune- Number of cases (usually set at 1)- Probability of adequate contact

With this information, a simple formula allows the calculation of how many individuals will be infected, and how many immune, in the next time interval. This is repeated until the entire population is immune, or no infective individuals remain. The model can then be run repeatedly, adjusting the initial conditions, to see how these affect the progression of the epidemic.

The probability of adequate contact corresponds roughly with R₀, the basic reproduction number—in a large population when the initial number of infecteds is small, an infected individual is expected to cause R 0 = ln ⁡ ( 1 / ( 1 − p ) ) new cases.

Mathematics

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

I t + 1 = S t B ( 1 − ( 1 − p ) I t ) , S t + 1 = S t − I t + 1

with initial number of susceptible and infected individuals ( S 0 , I 0 ) given. Here, 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),

I t + 1 = S t ( 1 − ( 1 − p ) I t ) , S t + 1 = S t ( 1 − p ) I t