Coexistence conditions for strains of influenza with immune cross-reaction.

The accumulation of cross-immunity in the host population is an important factor driving the antigenic evolution of viruses such as influenza A. Mathematical models have shown that the strength of temporary non-specific cross-immunity and the basic reproductive number are both key determinants for evolutionary branching of the antigenic phenotype. Here we develop deterministic and stochastic versions of one such model. We examine how the time of emergence or introduction of a novel strain affects co-existence with existing strains and hence the initial establishment of a new evolutionary branch. We also clarify the roles of cross-immunity and the basic reproductive number in this process. We show that the basic reproductive number is important because it affects the frequency of infection, which influences the long term immune profile of the host population. The time at which a new strain appears relative to the epidemic peak of an existing strain is important because it determines the environment the emergent mutant experiences in terms of the short term immune profile of the host population. Strains are more likely to coexist, and hence to establish a new clade in the viral phylogeny, when there is a significant time overlap between their epidemics. It follows that the majority of antigenic drift in influenza is expected to occur in the earlier part of each transmission season and this is likely to be a key surveillance period for detecting emerging antigenic novelty.