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

Rahul Roy, Dharmaraja Selvamuthu, Paola Tardelli

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.

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

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.
Paper Structure (12 sections, 18 theorems, 93 equations, 1 figure)

This paper contains 12 sections, 18 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The population model
  3. Expected number of births and deaths and expected intensities
  4. Representation for the intensities
  5. Discussion on the Markov property
  6. Dynamical behaviour of the intensities
  7. Stability
  8. A virus-like evolving population
  9. Convergence results for a virus-like evolving population
  10. Proofs
  11. Proof of Theorem \ref{['EBElambda']}
  12. Proof of Theorem \ref{['Exlambda3']}

Key Result

Proposition 3.2

For $i=1,2,3$, by (generallambdai) and (lambda3), $\left(\lambda^i(t), N^i(t) \right)$ are such that the expected intensities $\mathbb{E} [\lambda^i(t)]$ satisfy the differential equations:

Figures (1)

  • Figure 1: A partition of the population at time $t$ is $X_t= \{(1,x_4), (4,x_2),(2,x_3),(3,x_1)\}$

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Remark 3.4
  • Theorem 3.5
  • ...and 33 more