The SIR epidemic model from a PDE point of view

We present a derivation of the classical Susceptible-Infected-Removed (SIR) and Susceptible-Infected-Removed-Susceptible (SIRS) models through a mean-field approximation from a discrete version of SIR(S). We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR(S) model. Moreover, for the SIRS model, we show that the long time limit of the SIRS model will be a Dirac measure supported on the corresponding isolated equilibria. For the SIR model, we show that the long time limit is a Radon measure supported in a segment of nonisolated equilibria.