8 Branching Processes Branching processes are used as a tool for modeling in genetics, biomolecular reproduction, population growth, genealogy, disease spread, photomultiplier cascades, nuclear fission, earthquake triggering, queueing models, viral phenomena, social networks, neuroscience, etc. This chapter mainly deals with the computation of probabilities of extinction and explosion in finite time for branching processes. 8.1 Construction and Examples 310 8.2 Probability Generating Functions 314 8.3 Extinction Probabilities 318 Exercises . . 332 8.1 Construction and Examples Consider a time-dependent population made of a number X, of individuals at generation n > 0. In the branching process model, each of these X, individuals may have a random number of descendants born at time n + 1. For each k = 1, 2, . .., Xn we let Yx denote the number of descendants of individual no k. That means, we have Xo = 1, X1 = Y1, and at time n + 1, the new population size Xn+1 will be given by Xn X,HEYit .. + Ym EY k, (8.1.1) k=1 where the (Yk) k21 form a sequence of independent, identically distributed, nonnegative integer valued random variables which are assumed to be almost surely finite, i.e. P(Yk U satisfying (:0 + c1 = 1. Then the extinction probability of the branching process is equal to 1. ms No