Fig. 2: Components of the mathematical model.

a Bacteria grow and spread as a wave on the outside surface of the catheter. b Behaviour of Eq.(1) (‘Methods’) subject to open boundaries and an initial inoculation of 100 bacteria per mm² at the bottom of the catheter(x =  0), with all parameters as given in Table 1. Each curve is a snapshot of the surface density profile (bacterial abundance on the catheter surface at different distances along the catheter) at 24 (pink), 36 (light brown), and 48 (dark brown) h, respectively. c Physical parameters controlling outer surface model behaviour. d Bacteria grow in residual urine within the bladder. e Behaviour of Eq. (2), with an initial inoculation of 10³ mL⁻¹, subject to different dilution rates k_D. Each curve shows a computer simulation of the bacterial abundance (in cells per mL) in the bladder, for dilution rates of 0 (darkest green), 0.36 (dark green), 0.72 (mid green), and 1.08 (light green) h⁻¹, respectively. f Physical parameters controlling bladder model behaviour. g Urine transports bacteria downwards through the catheter. h Numerical solution of Eqs. (3) and (4) gives the spatial distribution of bacterial abundance within the catheter. At the top of the catheter, bacteria are uniformly distributed within the urine with abundance 10⁹ mL⁻¹. However, further down the catheter, the urine close to the surface becomes depleted of bacteria, due to bacterial deposition on the luminal (inside) surface. i Physical parameters controlling luminal flow model behaviour.