Modelling catastrophic extinction in stochastic birth-death process: Analytical insights, estimation, and efficient simulation

A comprehensive analytical and computational framework is developed for the linear birth-death process (LBDP) with catastrophic extinction (BDC process), a continuous-time Markov model that incorporates sudden extinction events into the classical LBDP. Despite its conceptual simplicity, the underlying BDC process poses substantial challenges in deriving exact transition probabilities and performing reliable parameter estimation, particularly under discrete-time observations. While previous work established foundational properties using spectral methods and probability generating functions (PGFs), explicit analytical expressions for transition probabilities and theoretical moments have remained unavailable, limiting practical applications in extinction-prone systems. This limitation is addressed by reparameterising the PGF through functional restructuring, yielding exact closed-form expressions for the transition probability function and the theoretical moments of the discretely observed BDC process, with results validated through comprehensive numerical experiments for the first time. Three parameter estimation approaches tailored to the BDC process are introduced and evaluated: maximum likelihood estimation (MLE), generalised method of moments (GMM), and an embedded Galton-Watson (GW) approach, with trade-offs between computational efficiency and estimation accuracy examined across diverse simulation scenarios. To improve scalability, a Monte Carlo simulation framework based on a hybrid tau-leaping algorithm is formulated, specifically adapted to extinction-driven dynamics, offering a computationally efficient alternative to the exact stochastic simulation algorithm (SSA). The proposed methodologies offer a tractable and scalable foundation for incorporating the BDC process into applied stochastic models, particularly in ecological, epidemiological, and biological systems where populations are susceptible to sudden collapse due to catastrophic events such as host mortality or immune response.