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Simplest form
The SIR model is defined for a population \(N\), \(S\) the number of susceptible persons, \(I\) the number of infected people and \(R\) the number of poeple who have recovered. The following system of differential equations governs the evolution of those three variables:
\[ \frac{dS}{dt} = - \frac{\beta I S}{N} \] \[ \frac{dI}{dt} = \frac{\beta I S }{N}- \gamma I \] \[ \frac{dR}{dt} = \gamma I \]
The basic reproduction number \(R_0\) is defined by \[ R_0 = \frac{\beta}{\gamma} \]