Traveling waves in the Kermack–McKendrick epidemic model with latent period

Abstract

We study traveling waves for a diffusive susceptible–infected–recovery model, due to Kermack and McKendrick, of an epidemic with standard incidence and latent period included. In contrast to the classical case where the mass action incidence is employed, the total population is varied in the present model. It turns out that the governing equation for the recovery species cannot be decoupled from the other two equations for the susceptible and the infected species, and hence that the present model cannot be reduced to a two-component system as the classical one does. The existence of traveling waves of the model in this study can be completely characterized by the basic reproduction number of the system of ordinary differential equations associated with the present model. The model admits a continuum of traveling waves parameterized by wave speed c when waves do exist. Our approach is based on the fixed point theory and a delicately designed pair of super-/sub-solutions. This set of super-/sub-solutions also allows us to completely answer two unsolved questions in the existing literatures where the latent period is zero: (i) the existence of the minimal-speed wave which is believed to play a key role in the evolution of epidemic diseases and (ii) the existence of traveling waves does not depend on the relative ratio of the diffusivity of the infected species to the one of the recovery species.