Exponential stability of traveling fronts in a diffusion epidemic system with delay

This paper is concerned with the stability of traveling wave fronts of a monostable reaction–diffusion epidemic system with delay. The existence and comparison theorem of solutions of the corresponding Cauchy problem in a weighted space are established for the system on RR by appealing to the theories of semigroup and abstract functional differential equations. When the initial perturbation around the traveling wave decays exponentially as x→−∞x→−∞ (but the initial perturbation can be arbitrarily large in other locations), we prove the exponential stability of all traveling wave fronts for the monostable epidemic system with delay. This is probably the first time the weighted-energy method combining comparison principle is applied to reaction–diffusion systems to solve the stability of traveling fronts in some appropriate exponentially weighted space.