A stochastic model for the evolution of the influenza virus

Abstract

Consider a birth and death chain to model the number of types of a given virus. Each type gives birth to a new type at rate $\lambda$ and dies at rate 1. Each type is also assigned a fitness. When a death occurs either the least fit type dies (with probability $1-r$) or we kill a type at random (with probability $r$). We show that this random killing has a large effect (for any $r>0$) on the behavior of the model when $\lambda<1$. The behavior of the model with $r>0$ and $\lambda<1$ is consistent with features of the phylogenetic tree of influenza.