 Solution

Let $T_{0i}=e^{{\mu }_{0i}}$ denote the child's latent true baseline titre. With the constrained slopes, the conditional mean of log follow-up titre isExponentiating this log-scale center givesThus the conditional median, or geometric-mean titre, expressed as a fraction of true baseline titre, decays inversely with elapsed time; the child-specific intercept supplies the proportionality constant. It is not an exponential decay in time, and the formula is a follow-up model for $t>0$, not an extrapolation to the immunisation instant $t=0$.

For the arithmetic conditional mean, the lognormal distribution correction givesThe correction changes the constant but not the inverse-time dependence when the residual variance is constant. Averaging over random intercepts similarly preserves the time factor whenever the required expectation is finite.