Asymptotic Absorption-Time Distributions in Extinction-Prone Markov Processes.

Area of Science:

• Stochastic processes
• Mathematical biology
• Probability theory

Background:

• Birth-death Markov chains are fundamental models in various scientific fields.
• Understanding absorption-time distributions is crucial for analyzing system dynamics and predicting outcomes.
• Extinction-prone chains, common in biological and chemical systems, exhibit specific behaviors near absorbing boundaries.

Purpose of the Study:

• To characterize the absorption-time distributions for birth-death Markov chains with an absorbing boundary.
• To identify the types of asymptotic distributions for extinction-prone chains.
• To explore the implications of these distributions in diverse scientific models.

Main Methods:

• Mathematical analysis of birth-death Markov chains.
• Characterization of asymptotic distributions near an absorbing boundary.
• Application of theoretical findings to specific models of evolution, epidemics, and chemical reactions.

Main Results:

• Absorption-time distributions for extinction-prone chains can be Gaussian, Gumbel, or skewed.
• Skewed distributions arise when dynamics significantly decelerate near the boundary.
• New results are established for absorption-time distributions in several applied models.

Conclusions:

• The study provides a comprehensive characterization of absorption-time distributions for extinction-prone Markov chains.
• The findings offer new insights into the dynamics of systems modeled by these chains.
• Applications include understanding African sleeping sickness and other evolutionary or epidemic processes.