Modeling escape from extinction with decomposable multi-type Sevastyanov branching processes

Abstract

Biological populations under stress often face certain extinction unless they adapt toward unfavorable circumstances. In some scenarios such adaptation can assume the form of mutations within the genome of the population (e.g., cancer cells resisting chemotherapy, viruses developing resistance toward a vaccine), while in other scenarios adaptation can be in the form of movement toward some physical location (e.g., spreading of cancer cells to parts of the organism unaffected by treatment, animal populations fleeing polluted areas or areas struck by disaster). Regardless of the particular situation, it is often the case that cells/individuals with different levels of adaptation (which we may group into types) emerge among the cells/individuals of a stressed population. We propose a decomposable multi-type Sevastyanov branching process (possibly with multiple supercritical types) for modeling relevant aspects of the dynamics of such populations. The branching process developed within this paper is a generalization of the decomposable multi-type age-dependent branching process with a single supercritical type considered in Slavtchova-Bojkova and Vitanov. With respect to Slavtchova-Bojkova and Vitanov, we introduce additional, possibly supercritical, types into the interaction scheme between types, further, we incorporate possible dependence of the reproductive capabilities of cells/individuals from their age. We obtain a system of integral equations for the probability generating function of the new process and accordingly expand previous results from Slavtchova-Bojkova and Vitanov concerning probabilities of extinction, number of occurred mutations, waiting time to escape mutant, and immediate risk of escaping extinction. We also provide a general numerical scheme for calculating obtained systems of integral equations.