Birth-death processes

Integral functionals of Markov processes are widely used in stochastic modeling for applications in ecology, evolution, infectious disease epidemiology, and operations research. The integral of a stochastic process is often called the “cost” or “reward” accrued by the process. Many important stochastic counting models can be written as general birth-death processes (BDPs), which are continuous-time Markov chains on the non-negative integers in which only jumps to adjacent states are allowed, and there are no jumps down from zero. While there has been considerable progress in understanding general BDPs, work on integral functionals of BDPs has been limited to simple models and moments. In this paper, we show how to compute the distribution function and probability density of integral functionals for any general BDP. The method allows routine evaluation of full probability distributions for prospective modeling in several popular biological models and provides a framework for computing likelihoods that may be useful for statistical inference in a variety of applied scenarios. We provide examples of previously intractable integrals of BDPs from biology, queuing theory, population genetics, finance, and infectious disease epidemiology. In the final example, we use concepts from operations research to show how to select a control parameter to obtain exact probabilistic bounds on the total cost of an epidemic.