Li Jing, Zou Xingfu
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
J Math Biol. 2010 May;60(5):645-86. doi: 10.1007/s00285-009-0280-9. Epub 2009 Jul 1.
In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-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 latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R0 is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R0<1, and becomes unstable if R0>1. In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.
在本文中，假设人群中的一种传染病具有固定的潜伏期，且潜伏个体可能会扩散，我们针对生活在n斑块环境（城市、城镇或国家等）中的人群，构建了一个具有简单人口结构的SIR模型。该模型由一个延迟微分方程组给出，其中固定延迟用于考虑潜伏期，非局部项则由潜伏期间个体的流动性引起。假设感染相关类别的传播矩阵不可约，推导出基本再生数R0的表达式，并表明当R0<1时，无病平衡点是全局渐近稳定的，而当R0>1时则变得不稳定。在后一种情况下，至少存在一个地方病平衡点，且疾病将持续存在。当n = 2时，考虑了两个允许传播矩阵可约的特殊情况，以说明疾病潜伏期和人口流动性对疾病动态的联合影响。除了无病平衡点和内部地方病平衡点的存在性外，还讨论了这两个特殊情况下边界平衡点的存在性及其稳定性。
