modeling ratio between test response and control response in a repeated measures design

#1
Hello everyone !

I am currently struggling with the analysis of a repeated measures model. Data are organized as follow: Each subject has received a particular vaccine. The blood of each subject is then tested with different viruses. Three quantitative measures are made for each couple subject/virus. Three control measures are also made for each virus.

The objective is to modelize the ratio between the subject measures and the control measures for each virus in order to see if the response to the virus depends on the vaccine. This ratio is in fact a proportion as the measures of a subject should only be a fraction of the control's measures

My first attempt was to perform a loglinear mixed model. After exponentialization, the difference between the marginal mean of each vaccine and the marginal mean of the control (at the link scale) should give the proportion I'm looking for. The problem of course is that as I used a gaussian distribution for the modelisation, the confidence interval of this ratio is not bounded between 0 and 1...

I then tried two more approach. In the first one I took the predicted values of each subject in the first model so to have only one measure per subject/control instead of three. For each subject I calculated the ratio between his measure and the control and then modelized a beta regression to achieve my goal.
In the second approach I first modelized the control response with a linear mixed model. I then computed the ratio for each of the subject's three measures with the corresponding predicted control's value. I finally ran a mixed beta regression to get the final estimates.

The two approaches give slightly different results. In both cases I am mainly concerned about proceeding to a regression on the basis of results predicted by another.
Is this approach acceptable according to you?