Birth-death processes are discrete-state, continuous-time Markov jump processes with one-step jumps. In this work, first passage times in birth-death processes and their near-continuum (i.e., large system size) limit processes are investigated. Mean first passage time to the absorbing zero-state or mean extinction time is an exponentially large quantity with exponent that is proportional to the system size, provided that there is at least one stable state in the full-continuum, deterministic limit of the system. On the epidemiological SIS model, it is illustrated that the associated diffusion process, i.e., the near-continuum limit of the underlying Markov jump process, leads to an exponential mean extinction time, but with a different exponent, independent of the system size. Since the extinction is a rare event, large deviations principles are introduced and used in order to obtain the above-mentioned exponents bypassing the exact solutions; these solutions may not be available for multistep or multidimensional Markov jump processes. Another sample model, the Schlogl model of chemical kinetics is used as a benchmark one. The deterministic description of this model exhibits bistability, hence the sporadic, fluctuation-driven switches between two ‘stable’ states are rare events or large deviations. The main novel result of this work concerns multistable models. It is shown that the most likely path between two states is not necessarily the most dominant one for the mean first passage time. The latter may be dominated by the so-called trapped paths, i.e., the paths that are first trapped in a neighborhood of another stable state for a long time.