论文标题
均匀树上的离散的先生模型及其连续极限
Discrete SIR model on a homogeneous tree and its continuous limit
论文作者
论文摘要
我们研究了传染病在同质树上传播的易感感染的（SIR）模型，以及该模型的极限行为在树顶级倾向于无穷大。我们获得了易感顶点所需的时间分布，以在模型参数上广泛的假设下，根据非线性积分方程的解决方案感染。也就是说，假定感染率是时间依赖性的，并且恢复时间由具有相当任意分布的随机变量给出。然后，当树顶级倾向于无穷大时，我们研究模型的行为，并适当地缩放感染率。我们表明，在此限制中，离散模型的积分方程意味着易感人群室的方程式。从某种意义上说，这是一个主方程，即传染性和回收的隔间都可以根据其解决方案明确表示。换句话说，主方程意味着所有三个种群隔室的联合时间演变的连续SIR模型。
We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends to infinity. We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters. Namely, infection rates are assumed to be time-dependent, and recovery times are given by random variables with a fairly arbitrary distribution. We then study the behavior of the model in the limit when the tree vertex degree tends to infinity, and infection rates are appropriately scaled. We show that in this limit the integral equation of the discrete model implies an equation for the susceptible population compartment. This is a master equation in the sense that both the infectious and the recovered compartments can be explicitly expressed in terms of its solution. In other words, the master equation implies a continuous SIR model for the joint time evolution of all three population compartments.
