Analysis of a mathematical model of immune response to fungal infection
The authors considered the PDE system \begin{align*} \frac{\partial F}{\partial t}-\delta_F\nabla^2F&=-\lambda_FF^2K^{-1}+F(\lambda_F-\eta N-\zeta T)\\ \frac{\partial N}{\partial t}-\delta_N\nabla^2N&=\lambda_FF-d_N(N-N_0)\\ \frac{\partial T}{\partial t}-\delta_T\nabla^2N&=\lambda_TF\lambda_{TN}(N-N_0)-d_T(T-T_0) \end{align*} where denote the fungi density by $F$, the neutrophils density by $N$, and the density of CD8+ T cells by $T$, the density of $F$ is dispersing, with a diffusion coefficient $\delta F$, that $F$ has a logistic growth with rate coefficient $\lambda F$ and carrying capacity $K$, and that it is being killed by $N$ at rate $\eta$, and by T cells at rate $\zeta$, as a mathematical model for the spread of fungal infections invasive Candida albicans The boundary condition at the free boundary $\partial\Omega(t)$ are taken to be $F=0$, $N=N_0$, $T=T_0=\frac{\lambda_{TN}}{d_T}N_0$ where $F,N,T,\Omega(t)=\{0\leq r\leq R(t)\}$ are radially symmetric, \[ \frac{dR}{dt}(t)=h\bigg(-\frac{\partial R}{\partial r}(r,t)\bigg|_{r=R(t)}\bigg) \] The main results of the work are statements \begin{itemize} \item[1.] If $\eta N_0+\zeta T_0>\lambda_F$, then $\lim\limits_{t\to\infty} R(t)<\infty$ and $\lim\limits_{t\to\infty}\lim\limits_{0\leq r\leq r(t)}\sup F(r,t)=0$ \item[2.] $\eta N_0+\zeta T_0<\lambda_F$, and $R(0)$ is the large enough, then $\lim\limits_{t\to\infty} R(t)=\infty$, \end{itemize}
scientific article; zbMATH DE number 7371056