A fractional order recovery SIR model from a stochastic process

TL;DR

This paper derives a fractional order SIR epidemic model from a stochastic process, linking fractional derivatives to power law recovery times, and demonstrates its numerical stability and convergence through simulations.

Contribution

It introduces a novel fractional order SIR model derived from stochastic processes, connecting fractional derivatives to disease recovery time distributions.

Findings

Fractional derivatives naturally arise with power law recovery times.

The model aligns with classical SIR models and extends to discrete time.

Simulations show convergence to equilibrium and increased infecteds as fractional order decreases.

Abstract

Over the past several decades there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an an-hoc manner. These models may be mathematically interesting but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.