Ai Shangbing, Albashaireh Reem
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899 USA.
J Dyn Differ Equ. 2014;26(1):143-164. doi: 10.1007/s10884-014-9348-3. Epub 2014 Jan 28.
We study traveling wavefront solutions for two reaction-diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle's invariance principle, and their uniqueness by a geometric singular perturbation argument.
我们研究了两个反应扩散系统的行波前解，它们分别是作为两个非局部空间SIRS模型的扩散近似推导出来的。这些解刻画了潜在疫情波空间传播的进展和速度。对于第一个扩散系统，我们找到了波速的一个下界，并证明了对于所有大于此下界的速度，行波都存在。对于第二个扩散系统，我们找到了最小波速，并表明对于所有大于或等于最小速度的速度，行波都存在。我们进一步证明了对于足够大的波速，这些解（在平移意义下）的唯一性。这些解的存在性通过结合射击法和拉萨尔不变性原理来证明，而它们的唯一性则通过几何奇异摄动法来证明。
