Convergences for a Virus-like Evolving Population driven by Mutually-exciting Hawkes Processes

TL;DR

This paper introduces a stochastic model for virus-like evolving populations using mutually-exciting Hawkes processes for both birth and death events, analyzing conditions for Markov property and population convergence.

Contribution

It develops a novel birth-death model with Hawkes processes, providing conditions for Markov property and insights into population phase transitions.

Findings

Identifies conditions for Markov property in the model

Establishes convergence results for the population dynamics

Detects a phase transition at a critical fitness level

Abstract

This paper presents a stochastic model motivated by the study of a virus-like evolving population with different mutation rates. This is a continuous time birth-death model: the birth processes are mutually-exciting Hawkes processes and the death process is also a Hawkes process. This structure for the births and the deaths does not allow, in general, to get the Markov property of the processes involved. But considering the couple given by the Hawkes processes and their intensities we are able to deduce the necessary and sufficient conditions for the Markov property of the couple. This property is the main tool to get the convergence results describing the behaviour of the population, and the existence of a phase transition at a critical fitness level.