Asymptotic persistence time formulae for multitype birth–death processes

• Published in: Journal of applied probability Vol. 60; no. 3; pp. 895 - 920
• Main Authors: Ball, Frank G, Clancy, Damian
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2023
• Subjects: Asymptotic properties; Extinction; Infections; Markov processes; Ordinary differential equations; Original Article; Probability distribution functions; Random variables; 92D30; 60J27; Large deviations; population processes; 92D40; stochastic epidemic models
• ISSN: 0021-9002; 1475-6072
• DOI: 10.1017/jpr.2022.102

We consider a class of processes describing a population consisting of k types of individuals. The process is almost surely absorbed at the origin within finite time, and we study the expected time taken for such extinction to occur. We derive simple and precise asymptotic estimates for this expected persistence time, starting either from a single individual or from a quasi-equilibrium state, in the limit as a system size parameter N tends to infinity. Our process need not be a Markov process on $ {\mathbb Z}_+^k$ ; we allow the possibility that individuals’ lifetimes may follow more general distributions than the exponential distribution.