Influenza, often referred to as the flu, is an extremely contagious respiratory illness caused by influenza viruses, impacting populations globally with significant health consequences annually. A hallmark of influenza is its seasonal patterns, influenced by a mix of geographic, evolutionary, immunological, and environmental factors. Understanding these seasonal trends is crucial for informing public health decisions, including the planning of vaccination campaigns and their formulation. In our study, we introduce a genotype-structured infectious disease model for influenza transmission, immunity, and evolution. In this model, the population of infected individuals is structured according to the virus they harbor. It considers a symmetrical fitness landscape where the influenza A and B variants are considered. The model incorporates the effects of population immunity, climate, and epidemic heterogeneity, which makes it suitable for investigating influenza seasonal dynamics. We parameterize the model to the genomic surveillance data of flu in the US and use numerical simulations to elucidate the scenarios that result in the alternating or consecutive prevalence of flu variants. We show that the speed of virus evolution determines the alternation and co-circulation patterns of seasonal influenza. Our simulations indicate that slow immune waning reduces how often variants change, while cross-immunity regulates the co-circulation of variants. The framework can be used to predict the composition of future influenza outbreaks and guide the development of cocktail vaccines and antivirals that mitigate influenza in both the short and long term.