  Ordinary differential equation (ODE) models used in mathematical epidemiology
assume explicitly or implicitly large populations. For the study of infections
in a hospital this is an extremely restrictive assumption as typically a
hospital ward has a few dozen, or even fewer, patients. This work reframes a
well-known model used in the study of the spread of antibiotic-resistant
bacteria in hospitals, to consider the pathogen transmission dynamics in small
populations. In this vein, this paper proposes a Markov chain model to describe
the spread of a single bacterial species in a hospital ward where patients may
be free of bacteria or may carry bacterial strains that are either sensitive or
resistant to antimicrobial agents. We determine the probability law of the
\emph{exact} reproduction number ${\cal R}_{exact,0}$, which is here defined as
the random number of secondary infections generated by those patients who are
accommodated in a predetermined bed before a patient who is free of bacteria is
accommodated in this bed for the first time. Specifically, we decompose the
exact reproduction number ${\cal R}_{exact,0}$ into two contributions allowing
us to distinguish between infections due to the sensitive and the resistant
bacterial strains. Our methodology is mainly based on structured Markov chains
and the use of related matrix-analytic methods. This guarantees the
compatibility of the new, finite-population model, with large population models
present in the literature and takes full advantage, in its mathematical
analysis, of the intrinsic stochasticity.
