Pii: S0025-5564(00)00046-8

A general integral for birth±death Markov processes is considered. A closed form expression is provided for its expected value in terms of the birth and death rates. A simple route for numerical evaluation of its variance is also suggested. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Birth±death process; Hitting time; Integral of Markov process Hern andez-Su arez and Castillo-Chavez [1] discusses a methodology for evaluating the expectation of the integral under the stochastic path of a birth±death process up to extinction time and the expected time to extinction. Their main result is a closed form expression for the expectation of the aforementioned integral. In this note we demonstrate a more general result, which embraces and substantially extends that in Ref. [1], ̄ows from Stefanov's [2] results on ®nite-state birth± death processes. The relevance of such results to biological sciences is spelled out in Ref. [1] (see also the references therein). Furthermore, we provide a simple route for numerical evaluation of the variance of this integral together with a code for implementing such evaluation. In what follows, we will use similar notation to that adopted in Ref. [1], letting X …t† be a birth±death Markov process on the state space f0; 1; . . . ; g, with birth and death rates ki and li, respectively for state i. Further, denote by Z…k† the waiting time till reaching state 0 from state k, that is Z…k† ˆ infft > 0 : X …t† ˆ 0 jX …0† ˆ kg and de®ne