Bistability and Bacterial Infections

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection

Sparse Grid and Extrapolation Methods for Parabolic Problems

Sparse grids are a recently introduced new technique for discretizing partial differential equations having a very favorable complexity in the number of unknowns for higher dimensional problems so that they are especially attractive for instationary equations when time is treated as an additional dimension. The paper will introduce the sparse grid finite element technique and the sparse grid combination technique which can be interpreted as a multivariate extrapolation method. The concepts are closely related to the multilevel principle so that multigrid methods and multilevel preconditioning strategies are the natural solvers and so that the overall solution process has optimal complexity. Furthermore, the combination technique is easily parallelizable and applicable to nonlinear problems, like the Richardson equation. Besides an introduction of the algorithms with their basic analysis we will present numerical tests for a suite of characteristic model problems