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Conditional distributions and waiting times in multitype branching processes. (English) Zbl 1276.92081
Summary: We present novel results for discrete-time and Markovian continuous-time multitype branching processes. As a population develops, we are interested in the waiting time until a particular type of interest (such as an escape mutant) appears, and in how the distribution of individuals depends on whether this type has yet appeared. Specifically, both forward and backward equations for the distributions of type-specific population sizes over time, conditioned on the non-appearance of one or more particular types, are derived. In tandem, equations for the probability that one or more particular types have not yet appeared are also derived. Brief examples illustrate numerical methods and potential applications of these results in evolutionary biology and epidemiology.

MSC:
92D15 Problems related to evolution
92D30 Epidemiology
60J85 Applications of branching processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
92-08 Computational methods for problems pertaining to biology
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