A stochastic model for immune response with mutations and evolution: the non-spatial setting

We consider a stochastic model for a pathogen population in the presence of an immune response, in which pathogen types are partially ordered by ancestry and the immune system must eliminate ancestor types before it can eliminate their descendants. In this model, pathogens reproduce independently at rate $\lambda >0$ and, at each birth, a mutation occurs with probability $r\in (0,1]$ , producing a novel type that is antigenically distinct and whose elimination by the immune system is delayed relative to its ancestors. We provide an explicit characterization of the survival-extinction phase transition and compute the expected total progeny in the subcritical regime. We then extend the model by allowing mutations to produce two possible outcomes: conditional on mutation, with probability $p\in (0,1]$ the newborn is a viable immune-escape mutant, while with probability $1-p$ it is a sterile mutant that produces no offspring. For this extension, we obtain an explicit survival criterion in terms of $(\lambda ,r,p)$ and identify parameter regimes in which survival is possible only for an intermediate range of mutation probabilities, reflecting the balance between immune escape and mutational load.