Journal of Mathematical Biology

, Volume 75, Issue 6–7, pp 1463–1485 | Cite as

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



Antibiotic-resistant bacteria have posed a grave threat to public health by causing a number of nosocomial infections in hospitals. Mathematical models have been used to study transmission dynamics of antibiotic-resistant bacteria within a hospital and the measures to control antibiotic resistance in nosocomial pathogens. Studies presented in Lipstich et al. (Proc Natl Acad Sci 97(4):1938–1943, 2000) and Lipstich and Bergstrom (Infection control in the ICU environment. Kluwer, Boston, 2002) have provided valuable insights 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 analyses of the models in broader parameter regions that are biologically feasible. Bifurcation analysis and identification of the global stability conditions can be very helpful for assessing interventions that are aimed at limiting nosocomial infections and stemming the spread of antibiotic-resistant bacteria. In this paper we study the global dynamics of the mathematical model of antibiotic resistance in hospitals considered in Lipstich et al. (2000) and Lipstich and Bergstrom (2002). The invasion reproduction number \({{\mathcal {R}}}_{ar}\) of antibiotic-resistant bacteria is derived, and the relationship between \({{\mathcal {R}}}_{ar}\) and two control reproduction numbers of sensitive bacteria and resistant bacteria (\({{\mathcal {R}}}_{sc}\) and \({{\mathcal {R}}}_{rc}\)) is established. 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 Lipstich and Bergstrom (2002). More specifically, there exists a new threshold \({{\mathcal {R}}}_{ar}^c\), such that if \({{\mathcal {R}}}_{ar}^c<{{\mathcal {R}}}_{ar}<1\), then the system can have two positive interior equilibria, which leads to an interesting bistable phenomenon. This may have critical implications for controlling the antibiotic-resistance in a hospital.


Antibiotic resistance Invasion reproduction number Backward bifurcation Bistable phenomenon Global dynamics 

Mathematics Subject Classification

92C50 34C23 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xiuli Cen
    • 1
  • Zhilan Feng
    • 2
  • Yiqiang Zheng
    • 2
  • Yulin Zhao
    • 3
  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA
  3. 3.School of Mathematics (Zhuhai)Sun Yat-sen UniversityZhuhaiPeople’s Republic of China

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