Understanding the interaction between epidemiological and evolutionary dynamics for antigenically variable pathogens remains a challenge, particularly if analytical insight is wanted. In particular, while a variety of relatively complex simulation models have reproduced the evolutionary dynamics of influenza, simpler models have given less satisfying descriptions of the patterns seen in data. Here, we develop a set of relatively simple deterministic models of the transmission dynamics of multi-strain pathogens which give increased biological realism compared with past work. We allow the intensity of cross-immunity generated against one strain given exposure to a different strain to depend on the extent of genetic difference between the strains. We show that the dynamics of this model are determined by the interplay of parameters defining the cross-immune response function and can include fully symmetric equilibria, self-organized strain structures, regular periodic and chaotic regimes. We then extend the model by incorporating transient strain-transcending immunity that acts as a density-dependent mechanism to lower overall infection prevalence and thus pathogen diversity. We conclude that while some aspects of the evolution of influenza can be captured by deterministic models, overall, the description obtainable using a purely deterministic framework is unsatisfactory, implying that stochasticity of strain generation (via mutation) and extinction needs to be captured to appropriately capture influenza dynamics.