Stability analysis of an atherosclerotic plaque formation model with time delay

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 $L L_*$. Moreover, we find that smaller plaque are associated with the presence of time delay.