Spectral Stability of Time-Periodic Traveling Fronts in Seasonal Epidemic Models

Abstract We prove spectral stability of the time-periodic traveling invasion fronts of aseasonally forced diffusive SIR epidemic model. These fronts are waves that carry theinfection into a disease-free population under seasonally varying transmission andrecovery.Existence of these fronts is known down to the minimal invasion speed $\cstar$; we study their stability here. The linearization is a $T$-periodic (Floquet) operator, whoseessential spectrum, for a pulled front, is confined to the left half-plane by anexponentially weighted space for every speed $c\ge\cstar$, established in a companionpaper. What remains is the point spectrum, usually accessed through an Evansfunction. We show instead that, in the weighted space, a direct energy estimatesettles the entire spectrum at once: a $T$-periodic Lyapunov transformation (a gauge)replaces the time-dependent potential by its average, and a pointwise property of the front profilemakes the symmetric part of the operator negative definite under an explicitinequality relating the wave speed, the diffusion rates, and the transmissioncoupling. This gives a threshold $\ce\ge\cstar$ above which every front is spectrallystable with a quantitative gap, and a sharper resolvent estimate, exploiting thelocalization of the infected pulse, lowers the threshold to a $\ce'<\ce$. On theremaining narrow band $c\in[\cstar,\ce')$ we use a periodic Evans function: a parityargument from the sign of the Evans function on the real axis excludes the genericreal instability, and an argument-principle winding count returns winding zerothroughout, showing that $\ce$ marks a limitation of the energy method, anon-normality gap, rather than a genuine stability transition. MSC 2020: 35B35, 35K57, 35P05, 35B10, 92D30.