Shaping the phylogenetic tree of influenza by cross-immunity

Cross-immunity among related strains can account for the selection producing the slender phylogenetic tree of influenza A and B in humans. Using a model of seasonal influenza epidemics with drift (Andreasen, 2003. Dynamics of annual influenza A epidemics with immuno-selection. J. Math. Biol. 46, 504-536), and assuming that two mutants arrive in the host population sequentially, we determine the threshold condition for the establishment of the second mutant in the presence of partial cross-protection caused by the first mutant and their common ancestors. For fixed levels of cross-protection, the chance that the second mutant establishes increases with rho the basic reproduction ratio and some temporary immunity may be necessary to explain the slenderness of flu's phylogenetic tree. In the presence of moderate levels of temporary immunity, an asymmetric situation can arise in the season after the two mutants were introduced and established: if the offspring of the new mutant arrives before the offspring of the resident type, then the mutant-line may produce a massive epidemic suppressing the original lineage. However, if the original lineage arrives first then both strains may establish and the phylogenetic tree may bifurcate.