Description
We study an individual based SIR-model where the pathogen acquires immune-escape mutations at rate $N^{\alpha}$ per capita, where $N$ is the population size and alpha in $(0,1)$.
In this regime, even when the population is closed and immunity to a strain is permanent, the epidemic can persist thanks to the accumulation of sufficiently many mutations. We derive conditions for persistence of the pathogen in the large population limit, and observe emergence of interesting phenomena such as the presence of shadow variants, infecting only a negligible proportion of individuals, but that are necessary for the epidemic to overcome herd immunity and persist. Upon persistence, the epidemic sometimes reaches a stationary traveling wave dynamics, which, in simple cases, can be explicitly described.