Propagation direction of traveling waves for a class of nonlocal dispersal bistable epidemic models

Abstract

This work is devoted to studying the propagation direction of the following nonlocal dispersal epidemic model (0.1)$\left(\begin{array}{cccc}\frac{\partial u}{\partial t}& =d_1\left({\int }_{R}J(y-x)u(y,t)dy-u\right)-u+\alpha v,& & x\in R,t>0,\\ \frac{\partial v}{\partial t}& =d_2\left({\int }_{R}J(y-x)v(y,t)dy-v\right)-\beta v+g\left(u\right),& & x\in R,t>0,\end{array}\right)$where $d_1,d_2,\alpha ,\beta >0$. By discussing the case $c=0$ and using the monotone dependence of the wave speed of traveling wave solutions on parameters, we state the sufficient conditions for the speed $c>0$ and $c<0$ under some calculations and analysis. Compared to the known works for classical diffusive epidemic models, we have to overcome difficulties due to the appearance of nonlocal dispersal operators in the current paper.