Supplementary Information Impact of viral drift on vaccination dynamics and patterns of seasonal influenza

We assumed that the cross-protective immunity acquired through natural infection declines gradually in the absence of re-exposure to similar strains or vaccination [1]. We considered this decline in the model, and assumed that the pre-existing immunity (resulted from natural infection) becomes ineffective after m1 seasons (due to continual viral drift). Consistent with the observed data, we considered the recurrence of seasonal epidemics with a period of T = 1 year. Let a denote the recovery age (the time-period elapsed since last infection), and r(t, a) ∈ C (R× [0, m1T )→ R) represent the density of the individuals with the recovery age a at time t, as shown in Figure 1. Similar to the work of Metz and Diekmann [3], the dynamics of the density r(t, a) at nth season