Two-type branching processes with subexponential life-spans and SIR epidemic models

We consider a model of age-dependent branching stochastic process that takes into account the incubation period of the life of individuals. We demonstrate that such processes may be treated as a two-type branching process with a periodic mean matrix. In the case when the Malthusian parameter does not exist study of the process requires additional restrictions on the life and incubation time distributions which define so called subexponential family (Athreya, K. 1972. Branching Processes, Springer, New York). We obtain certain new properties of subexponential distributions, in particular, describe a subclass, which is closed with respect to convolution. Using these results we derive asymptotic behavior of the first and second moments and of the probability of nonextinction. We also prove a limit theorem for the process conditioned on nonextinction.