A Multitype Branching Processes Approach to the Evolutionary Dynamics of Escape

Abstract: Evolutionary dynamics of escape is a recent development in theoretical biology. It is an attempt to predict possible patterns of population dynamics for a certain strain of viruses placed in a hostile environment. The only way to escape extinction for the virus is to find a new form better adapted to the new environment. This is usually achieved by mutations in certain positions of the genome. In this thesis we use multitype Galton-Watson branching processes to model the evolution of such virus populations and provide answers to some of the most relevant questions arising in them. We determine the asymptotic probability of escape for a population stemming from a single progenitor. The calculations are obtained assuming mutations are rare events and generalize results previously known for particular reproduction laws. We also give a description of the random path to escape, that is the chain of mutations leading to the escape form of the virus. Using this description, we also study the waiting time to escape, i.e., the time it takes to produce the escape form of the virus. We start by deriving results for simple populations allowing for two-types of individuals and simple mutation schemes. Later we perform asymptotic analysis, again assuming mutations are rare, for populations with quite general reproduction and mutation schemes.