Stability of traveling wave solutions for a spatially discrete SIS epidemic model (Q2421788)

The paper deals with the spatially discrete SIS epidemic model of the form \[ \frac{d}{dt}y_{j,m}(t)=-\mu_m y_{j,m}(t)+(1-y_{j,m}(t))\sum_{r=-\infty}^{\infty}\sum_{n=1}^k\sigma_m\lambda_{mn}y_{j-r,m}(t)p_{mn}(r)\tag{*} \] (\(t\in\mathbb{R}\), \(j\in\mathbb{Z}\), \(m=1,\dots,k\)), where \(y_{j,m}(t)\) represents the proportion of individuals in the \(m\)-th population \(\sigma_m\) at the position \(j\) who are infectious in time \(t\). Here, the authors use the known fact [\textit{K. F. Zhang} and \textit{X.-Q. Zhao}, Nonlinearity 21, No. 1, 97--112 (2008; Zbl 1139.92023), p. 110] that there exists some \(c^*>0\) such that for any \(c\geq c^*\) equation \((^*)\) has a non-increasing traveling wave solution \(y_{j,m}(t)=\varphi_m(j-ct)\), \(t\in\mathbb{R}\) and \(m=1, 2, \dots, k\), connecting \(0\) with the unique (positive) globally asymptotically stable equilibrium \(\mathbf{E}\) in \([0,1]^k\setminus\{0\}\). It is shown that, if the perturbation around such a traveling wave solution belongs to a suitable weighted Banach space, then any solution of equation (*) converges exponentially to this traveling wave solution.