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We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria.Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods.

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We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vectorq. In the case whereq = 1, we derive a global extinction criterion which holds under second moment conditions, and whenq < 1 we develop necessary and sufficient conditions for q =q.

We apply matrix analytic methods and branching processes theory to a comparison of female populations in different countries. We show how the same mathematical model allows us to determine characteristics about individual women, such as the distribution of her lifetime, the time until her first and her last daughter, and the number of daughters, as well as to analyze properties of the whole female family generated by a first woman, such as the extinction probability of the family, the distributions of the time until extinction, of the family size at any given time and of the total progeny.