Cen Xiuli, Feng Zhilan, Zheng Yiqiang, Zhao Yulin
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People's Republic of China.
Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA.
J Math Biol. 2017 Dec;75(6-7):1463-1485. doi: 10.1007/s00285-017-1128-3. Epub 2017 Apr 10.
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 [Formula: see text] of antibiotic-resistant bacteria is derived, and the relationship between [Formula: see text] and two control reproduction numbers of sensitive bacteria and resistant bacteria ([Formula: see text] and [Formula: see text]) is established. More importantly, we prove that a backward bifurcation may occur at [Formula: see text] when the model includes superinfection, which is not mentioned in Lipstich and Bergstrom (2002). More specifically, there exists a new threshold [Formula: see text], such that if [Formula: see text], 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.
抗生素耐药细菌通过在医院引发多种医院感染，对公众健康构成了严重威胁。数学模型已被用于研究医院内抗生素耐药细菌的传播动态以及控制医院病原体中抗生素耐药性的措施。Lipstich等人（《美国国家科学院院刊》97(4):1938 - 1943, 2000）和Lipstich与Bergstrom（《重症监护病房环境中的感染控制》。Kluwer出版社，波士顿，2002）发表的研究为理解医院中抗生素耐药细菌的传播提供了有价值的见解。然而，他们的结果仅限于对几种不同情况的数值模拟，而没有在更广泛的生物学可行参数区域对模型进行分析。分岔分析和全局稳定性条件的确定对于评估旨在限制医院感染和遏制抗生素耐药细菌传播的干预措施非常有帮助。在本文中，我们研究了Lipstich等人（2000）和Lipstich与Bergstrom（2002）中所考虑的医院抗生素耐药数学模型的全局动态。推导了抗生素耐药细菌的入侵繁殖数[公式：见正文]，并建立了[公式：见正文]与敏感细菌和耐药细菌的两个控制繁殖数（[公式：见正文]和[公式：见正文]）之间的关系。更重要的是，我们证明了当模型包括双重感染时，在[公式：见正文]处可能会发生向后分岔，这在Lipstich与Bergstrom（2002）中未提及。更具体地说，存在一个新的阈值[公式：见正文]，使得如果[公式：见正文]，那么系统可以有两个正的内部平衡点，这导致了一个有趣的双稳态现象。这可能对控制医院中的抗生素耐药性具有关键意义。
