Simple SIR Epidemiological Model

An epidemiological model for the spread of an infection in a small, closed or about-closed system.

Model description

This is a compartmental model for the spread of an infection in a closed or about-closed (relative to the population size, the fraction of individuals joining and/ or leaving the population is very small) community. S, I, and R represent the susceptible, infected, and recovered compartments, respectively.

If we let s, i, and r represent the fraction of the total population in the respective compartments above, then the following differential equations apply to the system:

$$ \begin{align*} \frac{ds}{dt} &= -\beta si \\ \frac{di}{dt} &= \beta si - \gamma i \\ \frac{di}{dt} &= \gamma i \\ \end{align*} $$

where $t$ is the time variable. The first of these equations suggests that the rate at which individuals move from compartment S to I (contract the infection) is directly proportional to the populations in these compartments, with a proportionality constant $\beta$. The second, that infected individuals recover at a rate proportional to their number and a constant $\gamma$. The second and third equations dictate that their respective compartments experience growth in populations resulting from the movement of individuals from the previous compartment.

The parameter $\beta$ is calculated as $\frac{1}{n}$, that susceptible persons interact with $1$ individual every $n$ days. If this rate is high, say, one contact every six hours, then $n$ is small, and in this example is $0.25$ or quarter a day i.e. $\frac{1}{0.25} = 4$ contacts a day.

$\gamma$ is the time between recoveries in days. It is calculated as $\frac{1}{n}$ where $n$ is the average number of days before a recovery.

Assumptions

The fraction of individuals joining and/ or leaving the population is 0 or very small.
Time is a continuous variable.
There are no deaths nor births.
Permanent immunity is developed upon recovery.
Infected individuals immediately become infectious.