The SIR epidemic model from a PDE point of view

We present a derivation of the classical SIR model through a mean-field approximation from a discrete version of SIR. We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR model. Moreover, we show that the long time limit of the evolution will be a Dirac measure. The exact position will depend on the well-know $R_0$ parameter, and it will be supported on the corresponding stable SIR equilibrium.