A Strain-Structured SIS Model with Mutation: Spectral Thresholds and Long-Time Dynamics

We study the global dynamics of a strain-structured SIS model with a nonlocal mutation kernel. We characterize the epidemic threshold in terms of the spectral bound of the linearized operator and relate it to a generalized basic reproduction number R 0 , a formulation that remains valid even when the recovery rate vanishes locally. We show that the sign of the spectral bound determines extinction and persistence, and characterize the critical dynamics in the regular case. In the regular supercritical regime, we also construct an example with a unique unstable endemic equilibrium, showing that convergence towards the endemic state fails in general. Under an additional balanced transmission condition, however, we prove convergence towards the endemic equilibrium.