Abstract
Atherosclerosis is a chronic inflammatory disease that poses a serious
threat to human health. It starts with the buildup of plaque in the
artery wall, which results from the accumulation of pro-inflammatory
factors and other substances. In this paper, we propose a mathematical
model of early atherosclerosis with a free boundary and time delay. The
time delay represents the transformation of macrophages into foam cells.
We obtain an explicit solution and analyze the stability of the model
and the effect of the time delay on plaque size. We show that for
non-radial symmetric perturbations, when $n = 0$ or $1$, the
steady-state solution $(M_*,p_*,r_*)$ is linearly stable; when $n
\ge 2$, there exists a critical parameter $L_*$ such
that the steady-state solution is linearly stable for $L <
L_*$ and unstable for $L > L_*$. Moreover, we find
that smaller plaque are associated with the presence of time delay.