Two-Type Branching Processes with Subexponential Life-Spans and SIR Epidemic Models

Abstract

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.

Keywords:

• Branching process
• Epidemic
• Extinction
• Incubation period
• Malthusian parameter
• Subexponential class

Mathematics Subject Classification:

• Primary 60J80
• Secondary 62M05

These results are part of the project No FT-2006/03 funded by KFUPM, Dhahran, Saudi Arabia. The author is indebted to King Fahd University of Petroleum and Minerals for excellent research facilities. He is also grateful to the referee and one of editors for valuable comments on the first version of the article.