= 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 is
$$
\mu_i(t)=\alpha_i-\log t+\mu_{0i}.
$$
Exponentiating this log-scale center gives
$$
\boxed{\frac{e^{\mu_i(t)}}{T_{0i}}=\frac{e^{\alpha_i}}t.}
$$
Thus 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 gives
$$
\frac{\mathbb E\{T_i(t)\mid\alpha_i,\mu_{0i},\sigma^2\}}{T_{0i}}
=\frac{e^{\alpha_i+\sigma^2/2}}t.
$$
The 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.
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