Levy geometric graphs.

We present a family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including nonspatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.

Effective epidemic model for COVID-19 using accumulated deaths.

The severe acute respiratory syndrome COVID-19 has been in the center of the ongoing global health crisis in 2020. The high prevalence of mild cases facilitates sub-notification outside hospital environments and the number of those who are or have been infected remains largely unknown, leading to poor estimates of the crude mortality rate of the disease. Here we use a simple model to describe the number of accumulated deaths caused by COVID-19. The close connection between the proposed model and an approximate solution of the SIR model provides estimates of epidemiological parameters. We find values for the crude mortality between 10 - 4 and 10 - 3 which are lower than estimated numbers obtained from laboratory-confirmed patients. We also calculate quantities of practical interest such as the basic reproduction number and subsequent increment after relaxation of lockdown and other control measures.

Oedema-based model for diffuse low-grade gliomas: application to clinical cases under radiotherapy.

OBJECTIVES: Diffuse low-grade gliomas are characterized by slow growth. Despite appropriate treatment, they change inexorably into more aggressive forms, jeopardizing the patient's life. Optimizing treatments, for example with the use of mathematical modelling, could help to prevent tumour regrowth and anaplastic transformation. Here, we present a model of the effect of radiotherapy on such tumours. Our objective is to explain observed delay of tumour regrowth following radiotherapy and to predict its duration. MATERIALS AND METHODS: We have used a migration-proliferation model complemented by an equation describing appearance and draining of oedema. The model has been applied to clinical data of tumour radius over time, for a population of 28 patients. RESULTS: We were able to show that draining of oedema accounts for regrowth delay after radiotherapy and have been able to fit the clinical data in a robust way. The model predicts strong correlation between high proliferation coefficient and low progression-free gain of lifetime, due to radiotherapy among the patients, in agreement with clinical studies. We argue that, with reasonable assumptions, it is possible to predict (precision ~20%) regrowth delay after radiotherapy and the gain of lifetime due to radiotherapy. CONCLUSIONS: Our oedema-based model provides an early estimation of individual duration of tumour response to radiotherapy and thus, opens the door to the possibility of personalized medicine.

Modeling the role of water in Bacillus subtilis colonies.

We propose a simple cellular automaton model for the description of the evolution of a colony of Bacillus subtilis. The originality of our model lies in the fact that the bacteria can move in a pool of liquid. We assume that each migrating bacterium is surrounded by an individual pool, and the overlap of the latter gives rise to a collective pool with a higher water level. The bacteria migrate collectively when the level of water is high enough. When the bacteria are far enough from each other, the level of water becomes locally too low to allow migration, and the bacteria switch to a proliferating state. The proliferation-to-migration switch is triggered by high levels of a substance produced by proliferating bacteria. We show that it is possible to reproduce in a fairly satisfactory way the various forms that make up the experimentally observed morphological diagram of B. subtilis. We propose a phenomenological relation between the size of the water pool used in our model and the agar concentration of the substrate on which the bacteria migrate. We also compare experimental results from cutting the central part of the colony with the results of our simulations.

Improving the time-machine: estimating date of birth of grade II gliomas.

OBJECTIVES: Here we present a model aiming to provide an estimate of time from tumour genesis, for grade II gliomas. The model is based on a differential equation describing the diffusion-proliferation process. We have applied our model to situations where tumour diameter was shown to increase linearly with time, with characteristic diametric velocity. MATERIALS AND METHODS: We have performed numerical simulations to analyse data, on patients with grade II gliomas and to extract information concerning time of tumour biological onset, as well as radiology and distribution of model parameters. RESULTS AND CONCLUSIONS: We show that the estimate of tumour onset obtained from extrapolation using a constant velocity assumption, always underestimates biological tumour age, and that the correction one should add to this estimate is given roughly by 20/v (year), where v is the diametric velocity of expansion of the tumour (expressed in mm/year). Within the assumptions of the model, we have identified two types of tumour: the first corresponds to very slowly growing tumours that appear during adolescence, and the second type corresponds to slowly growing tumours that appear later, during early adulthood. That all these tumours become detectable around a mean patient age of 30 years could be interesting for formulation of strategies for early detection of tumours.

Modelling intercellular communication and its effects on tumour invasion.

We present a model aiming at the description of intercellular communication on the invasive character of gliomas. We start from a previous model of ours based on a cellular automaton and develop a new version of it in a three-dimensional geometry. Introducing the hydrodynamic limit of the automaton we obtain a macroscopic model involving a nonlinear diffusion equation. We show that this macroscopic model is quite adequate for the description of realistic situations. Comparison of the simulations with experimental results shows agreement with the finding that the inhibition of intercellular communication (through gap junctions) tends to decrease migration. As an application of our model we estimated the possible increase in life expectancy, due to reduced cell migration mediated by the inhibition of intercellular communication, on patients suffering from gliomas. We find that the obtained increase may amount to a 20% gain in the case of unresectable tumours.

A model for short- and long-range interactions of migrating tumour cell.

We examine the consequences of long-range effects on tumour cell migration. Our starting point are previous results of ours where we have shown that the migration patterns of glioma cells are best interpreted if one assumes attractive interactions between cells. Here we complement the cellular automaton model previously introduced by the assumption of the existence of a chemorepellent produced by the main bulk of large spheroids (in the hypoxic/necrotic areas). Visible effects due to the presence of such a substance can be found in the density profiles of cells migrating out of a single spheroid as well as in the angular distribution of cells coming from two close-lying spheroids. These effects depend crucially on the diffusion speed of the chemorepellent. A comparison of the simulation results to experimental data of Werbowetski et al. allows to draw (tentative) conclusions on the existence of a chemorepellent and its properties.

A model for glioma cell migration on collagen and astrocytes.

We present a model for the migration of glioma cells on substrates of collagen and astrocytes. The model is based on a cellular automaton where the various dynamical effects are introduced through adequate evolution rules. Using our model, we investigate the role of homotype and heterotype gap junction communication and show that it is possible to reproduce the corresponding experimental migration patterns. In particular, we confirm the experimental findings that inhibition of homotype gap junctions favours migration while heterotype inhibition hinders it. Moreover, the effect of heterotype gap junction inhibition dominates that of homotype inhibition.

A cellular automaton model for the migration of glioma cells.

We present a study of in vitro cell migration in two dimensions as a first step towards understanding the mechanisms governing the motility of glioma cells. Our study is based on a cellular automaton model which aims at reproducing the kinetics of a lump of glioma cells deposited on a substrate of collagen. The dynamical effects of cell attraction and motion inertia are introduced through adequate automaton rules. We compare the density profiles given by the model to those obtained experimentally. The result of the best fit indicates a substantial cell-cell attraction due to cell-cell communication through gap junctions (or chemotaxis) and negligible inertia effects during migration. Tracking of individual migrating cells indicates highly convoluted cell trajectories.