Hopf bifurcation analysis on a delayed reaction-diffusion system modelling the spatial spread of bacterial and viral diseases

•A diffusive system modelling spread of bacterial and viral diseases is considered.•The stability of steady states and Hopf Bifurcation driven by delay are discussed.•The normal form is derived by applying the method of Faria and Magalháes.•The diffusion-driven instability of the homogeneous periodic solution is studied.•Two examples are given and stable periodic solutions are simulated.