SIR Model

A SIR Model is a epidemiological model of an infection spreading through a population. SIR stands for susceptible, infected, recovered. Infected individuals are sick and may get those they interact with sick. Susceptible individuals may be made sick by infected individuals. Recovered individuals are immune to getting sick.

The model exists on a mathematical graph of nodes and edges. The nodes represent people and the edges represent the contacts between people.

An infection rate parameter is required for the model. The infection rate is a number $\alpha > 0$. When an individual gets infecected, the time it takes for them to infect a susceptible neighbor is a random variables $X$ with an exponential distribution that has mean $1/\alpha.$

Another commonly used parameter is a recovery rate which is the rate at which an individual recovers from the illness. The recovery rate is a number $\beta > 0.$ When an individual becomes sick, the time it takes them to recover is a random variable $Y$ with exponential distribution with mean $1/\beta.$

Example:

In a fully connected graph, every individual can infect every other individual. In this case, the rate at which a susceptible individual becomes infected is the $alpha$ times the number of infected individuals (since every infected individual can make them sick). The rate at which infected individuals become recovered is $\beta$ times the number of infected individuals since every infected individual is recovering at rate $\beta.$

Let $S_t$ be the number of susceptible individuals at time $t,$ $I_t$ be the number of infected individuals at time $t,$ and $R_t$ be the number of recovered individuals at time $t.$ Let $N$ be the number of individuals in the population. Then, based on the rate of change described above, we get the following set of differential equations: $$S_t' = -\alpha S_t I_t$$ $$I_t' = \alpha S_t I_t - \beta I_t$$ $$R_t' = \beta I_t$$

The rates are an approximation since the times at which an individual gets sick or recovers is random. However, if the population is large enough then by the strong law of large numbers the solutions to these differential equations will closely approximate the number of susceptible, infected, and recovered individuals with a high degree of accuracy.

Interactive Tool

There is a tool to create various graphs and test infection, recovery, and even immunization rates here: