Analysis and simulation of a meso-scale model of diffusive restistance of bacterial biofilms to penetration of antibiotics.

Most bacteria live in biofilm communities, which offer protection against harmful external impacts. This makes treatment of biofilm borne bacterial infections with antibiotics difficult. We discuss a dynamic mathematical model that focuses on the diffusive resistance that a growing biofilm exerts against penetration of antibiotics. This allows bacteria in the protected inner layers to grow while those in the outer rim are inactivated. The model consists of four parabolic partial differential equations for the dependent variables antibiotic concentration, oxygen concentration, active biomass fraction and inert biomass fraction. The equations for the last two variables show power law degeneracy (like the porous medium equation) as the dependent variable vanishes, and a power law singularity (like the fast diffusion equation) as the dependent variable approaches ist a priori known maximum value, and thus are highly non-linear. We show the existence of solutions to this model. This proof uses a positivity criterion, which is formulated and proved as a Lemma for more general nonlinear parabolic systems. Furthermore, a number of computer simulations are carried out to illustrate the behavior of the antibiotic isinfection model in dependence of the antibiotics added to the system