A fractional-order infectivity SIR model

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. •A physical basis for fractional derivatives in modelling disease processes.•Correcting ad hoc fractional compartment models.•Physical derivation of an SIR model with fractional order derivatives.•Derivation of master equations for SIR models from continuous time random walks.•Equivalence between fractional and structured evolution equations.