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Continuous-time Markov processes with a finite state space can be used to model countless real world phenomena. Therefore, researchers often encounter the problem of estimating the transition rates that govern the dynamics of such processes. Ideally, the estimation of transition rates would be based on observed transition times between the states in the model, i.e., on continuous-time observation of the process. However, in many practical applications only the current status of the process can be observed on a pre-defined set of time points (discrete-time observations). The estimation of transition rates is considerably more challenging when based on discrete-time data as compared to continuous observation. The difficulty arises from missing data due to the unknown evolution of the process between the actual observation times. To be able to estimate the rates reliably, additional constraints on how they vary in time will usually be necessary. A real world application considered in this thesis involves the asymptomatic carriage state (colonisation) with the bacterium Streptococcus pneumoniae (the pneumococcus)