In this paper, with the assumption that infectious individuals, once recovered for a period of fixed length, will relapse back to the infectious class, we derive an epidemic model for a population living in a two-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the fixed constant relapse time and a non-local term caused by the mobility of the individuals during the recovered period. We explore the dynamics of the model under two scenarios: (i) assuming irreducibility for three travel rate matrices; (ii) allowing reducibility in some of the three matrices. For (i), we establish the global threshold dynamics in terms of the principal eigenvalue of a 2×2 matrix. For (ii), we consider three special cases so that we can obtain some explicit results, which allow us to explicitly explore the impact of the travel rates. We find that the role that the travel rate of recovered and infectious individuals differs from that of susceptible individuals. There is also an important difference between case (i) and (ii): under (ii), a boundary equilibrium is possible while under (i) it is impossible.

中文翻译：

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在两修补程序环境中复发的流行病模型的动力学

在本文中，假设传染性个体一旦恢复了固定的时间长度，便会复发并回到传染性阶层，我们推导了一个居住在两地环境中（城市，城镇或国家）的人群的流行模型。等）。该模型由一类延迟微分方程系统提供，该系统具有固定的延迟，该延迟考虑了固定的恒定重复时间和由恢复期间个体的活动性引起的非局部项。我们在两种情况下探索模型的动力学：（i）假设三个旅行率矩阵的不可约性；（ii）在这三个矩阵中的某些矩阵上具有可约性。对于（i），我们根据2×2矩阵的本征值建立全局阈值动力学。对于（ii），我们考虑三种特殊情况，以便获得一些明确的结果，这使我们可以明确地探索旅行费率的影响。我们发现，康复和感染者的出行率与易感者的出行率不同。情况（i）和（ii）之间还有一个重要区别：在（ii）下，边界平衡是可能的，而在（i）下，边界平衡是不可能的。