Extinction Probability in A Birth-Death Process with Killing

Abstract

We study birth-death processes on the nonnegative integers, where {1, 2, . . . } is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

ORTHOGONAL POLYNOMIALS ON ℝ+ AND BIRTH-DEATH PROCESSES WITH KILLING

“…Some of the results mentioned in this paper are quoted from the recent papers [5], [6] and [7], which deal with birth-death processes with killing. It is the purpose of this note to collect these results, and to elaborate on them from the perspective of orthogonal polynomials.…”

A Continuous-Time Ehrenfest Model with Catastrophes and Its Jump-Diffusion Approximation

“…Similarly as (7), from (39) we obtain the following relations for the moments of X(t), for k = 1, 2, . .…”

Stochastic dynamics of Francisella tularensis infection and replication

“…Rupture of the host cell is a "catastrophe" event that affects every bacterium in the cell at the same time. Catastrophes have been considered mathematically as an extension of birth-and-death processes [43,44], including scenarios where a subset of the population is removed [45][46][47][48][49][50][51], or where a catastrophic event kills the entire population [52][53][54][55][56][57]. Our interest here is in the process depicted PLOS COMPUTATIONAL BIOLOGY in Fig 2, where two distinct absorbing states exist, both of which represent the loss of all intracellular bacteria.…”