Upcoming Seminars
Upcoming Seminars
A coagulation toy model for silicosis
Fernando Manuel Pestana da Costa (Universidade Aberta)
Location: Seminar Room 311B, School of Mathematics & Statistics, University of Glasgow
Date: Tuesday, May 6 at 2:00 pm
Abstract: We present a system with a countable number of ordinary differential equations of coagulation type that can be considered a simple model for silicosis. Silicosis is a respiratory disease due to the ingestion of quartz dust and their accumulation in the lungs. The mathematical model consists in an ODE for the concentration of free quartz particles, an ODE for the concentration of macrophages without quartz (cells of the immune system that identify, capture and try to expel entities strange to the body), and a countable number of ODEs each one describing the concentration of macrophages with a number $i\in\mathbb{N}$ of captured quartz particles. We briefly describe basic results such as existence, uniqueness, regularity, and semigroup results of the set of solutions. Then we study the dynamics of the infinite dimensional system in the case of a particular class of rate coefficients that allows for the decoupling of the full infinite dimensional system into a finite dimensional one and a lower triangular infinite system. By the analysis of the finite dimensional subsystem, we conclude that it has a saddle-node bifurcation, possessing two equilibria when a bifurcation coefficient which is the ratio of the input rate of quartz to the rate of creation of empty macrophages is below a critical value, and no equilibria above that value. The stability properties of the bifurcating branches of the full infinite dimensional system are studied and we end with some comments about a possible interpretation of these results.
Note: based on works with P. Antunes, M. Drmota, M. Grinfeld, J.T. Pinto, R. Sasportes
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