Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.009 ) Pub Date : 2020-09-15 , DOI: 10.1017/prm.2020.65
Wenzhang Huang; Chufen Wu

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.

down
wechat
bug