A Fractional-Order Infectivity and Recovery SIR Model

Christopher N Angstmann,Bruce I Henry,Anna V Mcgann

doi:10.3390/fractalfract1010011

Christopher N Angstmann, Bruce I Henry + Show 1 more

Open Access

https://doi.org/10.3390/fractalfract1010011

Copy DOI

Journal: Fractal and Fractional	Publication Date: Nov 17, 2017
Citations: 23	License type: CC BY 4.0

Affiliation: UNSW Sydney

Abstract

The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.

Highlights

The classic Susceptible Infectious Recovered (SIR) epidemiological model was originally introduced by Kermack and McKendrick in 1927 [1]
The SIR model was generalised in the following decade by Kermack and McKendrick to allow for age dependencies in disease transmission [2,3]
The model can be constructed as a directed continuous-time random walk (CTRW) through the SIR compartments [10,11,13]

Summary

Introduction

The classic SIR epidemiological model was originally introduced by Kermack and McKendrick in 1927 [1]. Fractional derivatives have been incorporated into compartment models by replacing the integer-order derivatives with Caputo derivatives [6]. Whilst such models may be able to be fitted to data, the underlying assumptions, such as the positivity of certain parameters and dimensional agreement, can be violated [13,16,17]. Fractional derivatives have recently been included in compartment models in a physically consistent way by modelling the dynamics of transitions between compartments as a stochastic process, a CTRW [10,11,13].

Derivation

Fractional Infectivity and Recovery SIR

Reduction to Classic and Fractional Recovery SIR Models

Equilibrium State Analysis

Summary and Discussion