= Solution
Condition on each patient's initial observed state and assume <independent> patient trajectories and observation times that do not carry additional information about the unobserved path. The <Markov property> factors each trajectory's <likelihood function> into its successive one-year <transition probabilities>. Thus
$$
\boxed{\ell(\lambda,\mu)=\sum_{r,s=1}^2n_{rs}\log p_{rs}(1;\lambda,\mu).}
$$
More explicitly, with $q=\lambda+\mu>0$ and $E=e^{-q}$,
$$
\ell=n_{11}\log\frac{\mu+\lambda E}{q}+n_{12}\log\frac{\lambda(1-E)}q+n_{21}\log\frac{\mu(1-E)}q+n_{22}\log\frac{\lambda+\mu E}q.
$$
Successive transitions within one patient need not be <independent> unconditionally: this is a product of conditional factors justified by the <Markov property>. If initial-state <probabilities> are modeled rather than conditioned on, their <likelihood contributions> must also be included. Zero-count terms contribute zero; a positive count attached to a zero model <probability> makes the <log-likelihood> equal to $-\infty$.
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