Diffusion coefficients

[trvrb]

I found the section fitting diffusion coefficients (in different branches) to the antigenic maps confusing -- if only because strict diffusion (in one or two dimensions) has no advection term and therefore no tendency to move in one direction or another; whereas the data clearly show an advective tendency (moving to the "right" over time in antigenic dimension 1). I presume the authors attempt to deal with this by imposing a prior (eq 12) for the location of a virus based on its location in a molecular phylogeny. But this would seem to conflate antigenic drift, which the authors wish to quantify using a diffusion coefficient, with genetic mutations (many of which are driven by selection for antigenic escape). In other words, this hybrid approach of assuming a prior based on a strain's genotype, and then fitting a diffusion coefficient to summarize strength of drift, does not seem to to be a pure measure of drift. What exactly the fitted diffusion coefficient means is extremely unclear to me, in any formal sense, as the procedure is so ad-hoc.

[trvrb]

We agree that applying the standard measure of diffusion coefficient D based on displacement vs time, without accounting for advection was problematic. We have substantial revised the text to focus on direct parameter inference of drift parameter μ and diffusion volatility parameter σ_x, properly separating advection and volatility in the diffusion process. The section "Antigenic evolution across influenza lineages" most reflects these changes, including the addition of a new table 2.