Mathematical Models and Methods in Applied Sciences

Recent experiments for swarming of the bacteria Bacillus subtilis on nutrient-rich media show that these cells are able to proliferate and spread out in colonies exhibiting complex patterns as dendritic ramifications. Is it possible to explain this process with a model that does not use local nutrient depletion?

We present a new class of models which is compatible with the experimental observations and which predict branching instabilities and does not use nutrient limitation. These conclusions are based on numerical simulations. The most complex of these models is also the biologically most accurate but the essential effects can also be obtained in simplified versions which are amenable to analysis. An example of instability mechanism is the transition from a shock wave to a rarefaction wave in a reduced two by two hyperbolic system.

The paper presents a model of virus mutations and evolution of epidemics in a system of interacting individuals, where the intensity of the pathology, described by a real discrete positive variable, is heterogeneously distributed, and the virus is in competition with the immune system or therapeutical actions. The model is developed within the framework of the Kinetic Theory of Active Particles. The paper also presents a qualitative analysis developed to study the well-posedness of the mathematical problem associated to the general framework. Finally, simulations show the ability of the model to predict some interesting emerging phenomena, such as the mutation to a subsequent virus stage, the heterogeneous evolution of the pathology with the co-presence of individual carriers of the virus at different levels of progression, and the presence of oscillating time phases with either virus prevalence or immune system control.

In this paper, we construct a model to describe the spatial motion of a monolayer of cells occupying a two-dimensional dish. By taking care of nonlocal contact inhibition, quiescence phenomenon, and the cell cycle, we derive porous media-like equation with nonlocal reaction terms. The first part of this paper is devoted to the construction of the model. In the second part we study the well-posedness of the model. We conclude the paper by presenting some numerical simulations of the model and we observe the formation of colonies.

In this paper we study some qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens in biological systems. Questions related to the existence of steady states, the finite speed of propagating fronts or the regularization in the interior of the support are studied from analytical and numerical points of view.

Experiments have shown that wild type P. aeruginosa swarms much faster than rhlAB mutants on 0.4% agar concentration surface. These observations imply that development of a liquid thin film is an important component of the self-organized swarming process. A multiscale model is presented in this paper for studying interplay of key hydrodynamical and biological mechanisms involved in the swarming process of P. aeruginosa. This model combines a liquid thin film equation, convection–reaction–diffusion equations and a cell-based stochastic discrete model. Simulations demonstrate how self-organized swarming process based on the microscopic individual bacterial behavior results in complicated fractal type patterns at macroscopic level. It is also shown that quorum sensing mechanism causing rhamnolipid synthesis and resulting liquid extraction from the substrate lead to the fast swarm expansion. Simulations also demonstrate formation of fingers (tendrils) at the edge of a swarm which have been earlier observed in experiments.

The simulation of the creation and evolution of biological forms requires the development of computational methods that are capable of resolving their hierarchical, spatial and temporal complexity. Computations based on interacting particles, provide a unique computational tool for discrete and continuous descriptions of morphogenesis of systems ranging from the molecular to the organismal level. The capabilities of particle methods hinge on the simplicity of their formulation which enables the formulation of a unifying computational framework encompassing deterministic and stochastic models. In this paper, we discuss recent advances in particle methods for the simulation of biological systems at the mesoscopic and the macroscale level. We present results from applications of particle methods including reaction–diffusion on deforming surfaces, deterministic and stochastic descriptions of tumor growth and angiogenesis and discuss successes and challenges of this approach.