Dynamics of spatially heterogeneous viral model with time delay (Q2332295)

The authors study the viral model \[ \begin{cases} \partial_t S(x,t) = \lambda (x) - dS(x,t) - f(x,S(x,t))v(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \\ \partial_t I(x,t) = e^{-n\tau} f(x,S(x, t-\tau)) v(x, t-\tau) - a I(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \\ \partial_t v(x,t) = D \Delta v(x,t) + k(x) I(x,t) - u v(x,t), & \quad (x,t) \in \Omega \times (0,\infty), \end{cases} \] endowed with homogeneous Neumann boundary conditions for \(v\) and continuous and nonnegative initial data. Here, \(\Omega \subset \mathbb{R}^N\) is a bounded domain with smooth boundary, \(a\), \(d\), \(D\), \(n\), \(u\), and \(\tau\) are positive constants, \(k, \lambda \in C( \overline{\Omega})\) satisfy \(k>0\) and \(\lambda \ge 0\) in \(\overline{\Omega}\), and \(f(x, \cdot) \in C([0,\infty)) \cap C^2 ((0,\infty))\) satisfies \(f(x,0) = 0\), \(f^\prime (x,S) >0\) and \(f^{\prime \prime} (x,S) \le 0\) for all \((x,S) \in \overline{\Omega} \times (0,\infty)\). In this viral model, \(S\), \(I\), and \(v\) denote the uninfected target cells, infected target cells producing the virus, and free virus particles, respectively. The model includes spatial heterogeneity, diffusion of the virus, nonconstant production rates \(\lambda\) and \(k\) as well as a time delay \(\tau\). A similar model without time delay has been analysed e.g. in [\textit{F.-B. Wang} et al., Appl. Anal. 93, No. 11, 2312--2329 (2014; Zbl 1307.35054)]. The authors determine the basic reproduction number \(R_0\) of the model and prove that in case of \(R_0 <1\) the disease-free equilibrium is globally attractive, while in case of \(R_0 >1\) the disease is uniformly persistent. Moreover, if \(k\) and \(\lambda\) are positive and constant, then it is proved that the endemic equilibrium is globally attractive. Important ingredients of the proofs are comparison arguments and results from the theory of dynamical systems.