Critical traveling waves in a diffusive disease model (Q2633865)

This paper considers a diffusive disease model \[ S_t=d_1 S_{xx}-\beta \frac{SI}{S+I}, \] \[ I_t=d_2 I_{xx}+ \beta \frac{SI}{S+I}-\gamma I \] and resolves a question left open in [\textit{X.-S. Wang} et al., Discrete Contin. Dyn. Syst. 32, No. 9, 3303--3324 (2012; Zbl 1241.92069)], whos proved the existence of travelling waves for the this system with velocities $c>c_*=2\sqrt{d_2(\beta-\gamma)}$. Here the existence of a travelling wave with the critical speed $c=c_*$ is proved. As the authors note, a special feature of this system is that the reaction system involved has infinitely many equilibria ($I=0$, $S$ arbitrary). The proof applies the Schauder fixed point theorem to an appropriately constructed operator.