Modelling silicosis: existence, uniqueness and basic properties of solutions

TL;DR

This paper analyzes a mathematical model of silicosis disease progression, proving key properties like existence, uniqueness, and stability of solutions for an infinite differential equation system based on coagulation, fragmentation, and death processes.

Contribution

It establishes rigorous mathematical results for a complex infinite ODE model of silicosis, extending previous models with new proofs of solution properties.

Findings

Proved existence and uniqueness of solutions

Demonstrated continuous dependence on initial data

Established differentiability of solutions

Abstract

We present a model for the silicosis disease mechanism following the original proposal by Tran, Jones, and Donaldson (1995) [9], as modified recently by da Costa, Drmota, and Grinfeld (2020) [4]. The model consists in an infinite ordinary differential equation system of coagulation-fragmentation-death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.