Stability of monostable waves for a class of epidemic system with asymmetry and nonlocal effects

The stability of monostable waves for a class of epidemic system with nonlocal effects is considered in this study, where the system can be non-quasimonotone and the kernel functions in diffusion terms and nonlocal reactions can all be asymmetric. By combining the Fourier transform with the anti-weighted energy method, the global stability of monostable waves with the decay rate of the form an exponential function multiplied by an algebraic function for non-critical wave speed and possible critical wave speed is established and the stability investigation of monostable waves to nonlocal dispersal equations with quasi-monotonicity and symmetric kernels is extended to asymmetric and non-quasimonotone cases. Moreover, an concrete example and numerical simulations are included to confirm the theoretical conclusions.