  Fractional-order SIR models have become increasingly popular in the
literature in recent years, however unlike the standard SIR model, they often
lack a derivation from an underlying stochastic process. Here we derive a
fractional-order infectivity SIR model from a stochastic process that
incorporates a time-since-infection dependence on the infectivity of
individuals. The fractional derivative appears in the generalised master
equations of a continuous time random walk through SIR compartments, with a
power-law function in the infectivity. We show that this model can also be
formulated as an infection-age structured Kermack-McKendrick
integro-differential SIR model. Under the appropriate limit the fractional
infectivity model reduces to the standard ordinary differential equation SIR
model.
