A cross-infection model with diffusive environmental bacteria

In order to investigate the role that environmental bacteria played in the dynamics of hospital infections, we propose a cross-infection model with diffusive bacteria in the environment. Firstly, we prove the global existence, uniform boundedness and ultimate boundedness of solutions as well as the existence of a global attractor for the equivalent model. Secondly, we investigate a limiting system to establish the threshold dynamics for the model in terms of the basic reproduction number R₀ by using the theories of monotone dynamical systems and chain transitive sets. More precisely, we show that if R₀≤1, then the infection-free steady state is globally stable; and if R₀>1, then the system has a globally stable endemic steady state. Finally, we use the numerical method to explore the influence of different diffusion coefficients on R₀. In the case where the transmission rate is independent of diffusion coefficient, the numerical results indicate that R₀ is decreasing with respect to the diffusion rate. In the case where the transmission rate is a function of diffusion coefficient, we find that in a less polluted environment, R₀ is a decreasing function with respect to the diffusion rate, which implies that the diffusion of bacteria is beneficial for patients; while in a more polluted environment, R₀ may increase with increasing diffusion rate, which means increasing diffusion of bacteria is harmful for the elimination of disease.