On a diffusive bacteriophage dynamical model for bacterial infections

Abstract

Bacteriophages or phages are viruses that infect bacteria and are increasingly used to control bacterial infections. We develop a reaction–diffusion model coupling the interactive dynamic of phages and bacteria with an epidemiological bacteria-borne disease model. For the submodel without phage absorption, the basic reproduction number [Formula: see text] is computed. The disease-free equilibrium (DFE) is shown to be globally asymptotically stable whenever [Formula: see text] is less than one, while a unique globally asymptotically endemic equilibrium is proven whenever [Formula: see text] exceeds one. In the presence of phage absorption, the above stated classical condition based on [Formula: see text], as the average number of secondary human infections produced by susceptible/lysogen bacteria during their entire lifespan, is no longer sufficient to guarantee the global stability of the DFE. We thus derive an additional threshold [Formula: see text], which is the average offspring number of lysogen bacteria produced by one infected human during the phage–bacteria interactions, and prove that the DFE is globally asymptotically stable whenever both [Formula: see text] and [Formula: see text] are under unity, and infections persist uniformly whenever [Formula: see text] is greater than one. Finally, the discrete counterpart of the continuous partial differential equation model is derived by constructing a nonstandard finite difference scheme which is dynamically consistent. This consistency is shown by constructing suitable discrete Lyapunov functionals thanks to which the global stability results for the continuous model are replicated. This scheme is implemented in MatLab platform and used to assess the impact of spatial distribution of phages, on the dynamic of bacterial infections.