On the extinction of Continuous State Branching Processes with catastrophes

TL;DR

This paper studies continuous state branching processes with catastrophic jumps, analyzing their extinction probabilities, asymptotic behaviors, and applications to infection spread, using stochastic differential equations and Lévy process techniques.

Contribution

It introduces a novel framework for CSBPs with multiplicative jumps, characterizes their Laplace exponents, and identifies four extinction regimes in critical and subcritical cases.

Findings

Four regimes for extinction speed in critical/subcritical cases.

Explicit characterization of the Laplace exponent via a differential equation.

Application to model infection propagation in biological systems.

Abstract

We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a L\'evy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of L\'evy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.