Analysis of an Antimicrobial Resistance Transmission Model

Abstract

We present an analysis of a system of differential equations that models the transmission dynamics of pathogens with antimicrobial resistance (AMR) in an intensive care unit (ICU) studied by Austin and Anderson (1999). In Austin and Anderson’s four–dimensional compartmental model, patients and health care workers are viewed as hosts and vectors of the pathogens, respectively, and subdivided into uncolonized and colonized populations. In the analysis, we reduce the model to a two–dimensional non–autonomous system. Noting that the reduced system has an autonomous limiting system, we then apply the theory of asymptotically autonomous differential equations systems in the plane developed by Markus (1956) and extended by Thieme (1992, 1994), and later by Castillo–Chavez and Thieme (1995). We first present a stability analysis of the limiting system and prove the existence of a locally asymptotically stable equilibrium point under a set of constraints expressed in terms of reproductive numbers. We then proceed to an asymptotic analysis of the non–autonomous, two–dimensional system by applying a Poincaré–Bendixson type trichotomy result proved by Thieme (1992, 1994). In particular, we establish that any forward bounded trajectory of the non–autonomous system that starts within a defined rectangular region will converge toward the equilibrium point of the limiting system, provided that certain conditions given in terms of the reproductive numbers are satisfied.