Analysis of the family of angiogenesis models with distributed time delays

TL;DR

This paper generalizes angiogenesis models with distributed delays, analyzing stability and bifurcations, and supports findings with numerical simulations based on real parameters.

Contribution

It introduces a family of models extending Hahnfeldt et al.'s work, providing new stability and bifurcation results for distributed delay systems.

Findings

Proved global existence and uniqueness of solutions.

Identified conditions for stability switches and Hopf bifurcations.

Numerical analysis confirms theoretical predictions with real parameter estimates.

Abstract

In the presented paper a family of angiogenesis models, that is a generalisation of Hahnfeldt et al. model is proposed. Considered family of models is a system of two differential equations with distributed time delays. The global existence and the uniqueness of the solutions are proved. Moreover, the stability of the unique positive steady state is examined in the case when delay distributions are Erlang or piecewise linear distributions. Theorems guaranteeing the existence of stability switches and occurrence of the Hopf bifurcation are proved. Theoretical results are illustrated by numerical analysis performed for parameters estimated by Hahnfeldt et al. (Cancer Res., 1999).