An evolutionary model of influenza A with drift and shift

Influenza A virus evolves through two types of evolutionary mechanisms - drift and shift. These two evolutionary mechanisms allow the pathogen to infect us repeatedly, as well as occasionally create pandemics with large morbidity and mortality. Here we introduce a novel model that incorporates both evolutionary mechanisms. This necessitates the modelling of three types of strains - seasonal human strains, bird-to-human transmittable H5N1 strains and evolved pandemic H5N1 strain. We define reproduction and invasion reproduction numbers and use them to establish the presence of dominant and coexistence equilibria. We find that the amino acid substitution structure of human influenza can destabilize the human influenza equilibrium and sustained oscillations are possible. We find that for low levels of infection in domestic birds, these oscillations persist, inducing oscillations in the number of humans infected with the avian flu strain. The oscillations have a period of 365 days, similar to the one that can be observed in the cumulative number of human H5N1 cases reported by the World Health Organization (WHO). Furthermore, we establish some partial global results on the competition of the strains.