Non-monotone waves of a stage-structured SLIRM epidemic model with latent period

Abstract

We propose and investigate a stage-structured SLIRM epidemic model with latent period in a spatially continuous habitat. We first show the existence of semi-travelling waves that connect the unstable disease-free equilibrium as the wave coordinate goes to − ∞, provided that the basic reproduction number$\mathcal {R}_0 > 1$and$c > c_*$for some positive number$c_*$. We then use a combination of asymptotic estimates, Laplace transform and Cauchy's integral theorem to show the persistence of semi-travelling waves. Based on the persistent property, we construct a Lyapunov functional to prove the convergence of the semi-travelling wave to an endemic (positive) equilibrium as the wave coordinate goes to + ∞. In addition, by the Laplace transform technique, the non-existence of bounded semi-travelling wave is also proved when$\mathcal {R}_0 > 1$and$0 < c < c_*$. This indicates that$c_*$is indeed the minimum wave speed. Finally simulations are given to illustrate the evolution of profiles.