 Solution

Acquired immunity means that the reinfection rate is no longer the common constant $a$: it should depend on infection history, usually decreasing immediately after recovery and perhaps rising as immunity wanes. The two observed states are then not Markov unless the state is enlarged to record time since infection, number of prior infections, or latent immune class. Monthly binary tests identify infection status only at visits, so infection and recovery times are interval-censored and past short episodes may be missed. The sparse histories contain little information for estimating several history-dependent transition intensities.