A Fractional-Order Infectivity SIR Model

TL;DR

This paper derives a fractional-order SIR epidemic model from a stochastic process that accounts for infection-age dependence, connecting it to classical models and highlighting its fractional dynamics.

Contribution

It introduces a novel derivation of a fractional infectivity SIR model from a stochastic process with infection-age dependence, bridging stochastic processes and fractional differential equations.

Findings

The model incorporates power-law infectivity functions.

It reduces to the standard SIR model under certain limits.

Provides a new framework linking stochastic processes to fractional epidemic models.

Abstract

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.