The SIR model of spreading of diseases is the following system of differential equations: \[ \begin{aligned} \frac{\dd s}{\dd t} &= -\beta\cdot s\cdot i \\[0.25cm] \frac{\dd i}{\dd t} &= \beta\cdot s\cdot i - \gamma\cdot i \\[0.25cm] \frac{\dd r}{\dd t} &= \gamma\cdot i \end{aligned} \] with parameters \(\beta\) and \(\gamma\). We have: \[\beta=\gamma\cdot R_0\quad\text{with}\quad R_0=\text{the basic reproduction number}\] and \[\gamma=\frac{1}{D}\quad\text{with}\quad D=\text{the average duration that an infected person can infect others.}\] Because \[s+i+r=1\] we actually do not need the last equation in the above system and we deal with a twodimensional system, and in particular with a so-called Lotka-Volterra system.

Below are simulations to play with.