  Antibiotic-resistant bacteria has posed a grave threat to public health by
causing a number of nosocomial infections in hospitals. Mathematical models
have been used to study the transmission of antibiotic-resistant bacteria
within a hospital and the measures to control antibiotic resistance in
nosocomial pathogens. Studies presented in \cite{LBL,LB} have shown great value
in understanding the transmission of antibiotic-resistant bacteria in a
hospital. However, their results are limited to numerical simulations of a few
different scenarios without analytical analysis of the models in all
biologically feasible parameter regions. Bifurcation analysis and
identification of the global stability conditions are necessary to assess the
interventions which are proposed to limit nosocomial infection and stem the
spread of antibiotic-resistant bacteria. In this paper we study the global
dynamics of the mathematical model of antibiotic resistance in hospitals in
\cite{LBL,LB}. The invasion reproduction number $\mathcal R_{ar}$ of
antibiotic-resistant bacteria is introduced. We give the relationship of
$\mathcal R_{ar}$ and two control reproduction numbers of sensitive bacteria
and resistant bacteria ($\mathcal R_{sc}$ and $\mathcal R_{rc}$). More
importantly, we prove that a backward bifurcation may occur at $\mathcal
R_{ar}=1$ when the model includes superinfection which is not mentioned in
\cite{LB}. That is, there exists a new threshold $\mathcal R_{ar}^c$, and if
$\mathcal R_{ar}^c<\mathcal R_{ar}<1$, then the system can have two interior
equilibria and it supports an interesting bistable phenomenon. This provides
critical information on controlling the antibiotic-resistance in a hospital.
