A note on fractional linear pure birth and pure death processes in epidemic models

TL;DR

This paper explores the application of fractional linear birth and death processes to epidemic modeling, revealing how these processes can describe different epidemic regimes and potentially extend to tumor growth modeling.

Contribution

It demonstrates the use of fractional birth and death processes in epidemic models without relying on empirical distributions, linking subcritical and supercritical regimes to fractional death and birth processes.

Findings

Subcritical regimes correspond to fractional death processes.

Supercritical regimes correspond to fractional birth processes.

Potential application to tumor growth modeling.

Abstract

In this note we highlight the role of fractional linear birth and linear death processes recently studied in \citet{sakhno} and \citet{pol}, in relation to epidemic models with empirical power law distribution of the events. Taking inspiration from a formal analogy between the equation of self consistency of the epidemic type aftershock sequences (ETAS) model, and the fractional differential equation describing the mean value of fractional linear growth processes, we show some interesting applications of fractional modelling to study \textit{ab initio} epidemic processes without the assumption of any empirical distribution. We also show that, in the frame of fractional modelling, subcritical regimes can be linked to linear fractional death processes and supercritical regimes to linear fractional birth processes. Moreover we discuss a simple toy model to underline the possible application of these stochastic growth models to more general epidemic phenomena such as tumoral growth.