School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China.
Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7, Canada.
Math Biosci Eng. 2020 Sep 14;17(5):6098-6127. doi: 10.3934/mbe.2020324.
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.
在本文中，我们假设感染个体一旦在固定长度的恢复期后康复，将重新回到感染类，从而为生活在双斑块环境（城市、城镇或国家等）中的人群建立了一个传染病模型。该模型由一个时滞微分方程组系统给出，其中固定时滞用于表示固定的恒定复发时间，非局部项则由个体在恢复期内的移动性引起。我们在两种情况下探索了模型的动力学：（i）假设三个旅行率矩阵不可约；（ii）允许其中三个矩阵中的一些可约。对于（i），我们根据一个 2×2 矩阵的主特征值来建立全局阈值动力学。对于（ii），我们考虑了三个特殊情况，以便能够得到一些显式结果，从而能够明确地探讨旅行率的影响。我们发现，康复和感染个体的旅行率的作用与易感染个体的旅行率的作用不同。在情况（ii）中，边界平衡点是可能的，而在情况（i）中则不可能。
