Complex Patterns in a Space-Time Discrete Mathematical Model of Antibiotic Resistance in Hospitals with Self-Diffusion.

In this work, we extend an antibiotic resistance mathematical model in hospitals spatially on a two-dimensional coupled map lattice. The complex dynamics of the resultant space–time discrete model are investigated. An invasion reproduction number Rar and two control parameters Rsc and Rrc for sensitive bacteria and resistant bacteria, respectively, are defined. The three numbers play an important role in the existence and stability of fixed points. According to the normal form theorem and the center manifold theory, we show that the discrete model exhibits both the flip and the Turing bifurcations. We prove that the existence of the interior fixed point excludes the possibility of the occurrence of a Neimark–Sacker bifurcation. Numerical simulations, including phase diagrams, Lyapunov exponents, bifurcation diagrams, and complex patterns, are enrolled to verify the analyses. The complex patterns induced here are either a result of the pure Turing instability, or the flip instability (non-Turing), while no Turing-flip patterns are formed. The results of the current study show that the space–time discrete version of the model of antibiotic resistance in hospitals captures complex and rich nonlinear dynamics and adds to our understanding of the complicated pattern formation of the model.