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.

中文翻译：

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具有潜伏期的阶段结构 SLIRM 流行病模型的非单调波

我们提出并研究了在空间连续生境中具有潜伏期的阶段结构 SLIRM 流行病模型。我们首先展示了当波坐标变为−∞时连接不稳定无病平衡的半行波的存在，前提是基本再生数$\mathcal {R}_0 > 1$和$c > c_*$对于一些正数$c_*$. 然后我们使用渐近估计、拉普拉斯变换和柯西积分定理的组合来显示半行波的持续性。基于持久性，我们构造了一个 Lyapunov 泛函，以证明当波坐标变为 + ∞ 时，半行波收敛到特有（正）平衡。此外，通过拉普拉斯变换技术，也证明了有界半行波的不存在$\mathcal {R}_0 > 1$和$0 < c < c_*$. 这表明$c_*$确实是最小波速。最后给出了模拟来说明轮廓的演变。