Bifurcation analysis and global dynamics of a mathematical model of antibiotic resistance in hospitals

Abstract

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.