传染病模型平均场方程收敛性的初等证明
An elementary proof of convergence to the mean-field equations for an epidemic model
摘要：
通常使用由常微分方程（ODE）系统描述的隔室流体模型来对疾病传播进行建模。除了简单之外，这些模型也是接触网络上疾病传播的更精确随机模型的平均场近似。对于一个完整网络上随机的易感染-易感染-易受感染（SIS）过程（带恢复的感染）的最简单的情况，已经证明，随着节点数量的增加，感染节点的比例收敛到平均场常微分方程。然而，证明并不简单，需要复杂的概率，偏微分方程（PDE），或无穷系统的常微分方程。在这种情况下，我们提供了一个简短的证明均方收敛在有限的时间间隔上使用两个常微分方程和矩不等式的系统，也获得了第一个下界的预期分数感染节点。
英文摘要：
It is common to use a compartmental, fluid model described by a system of ordinary differential equations (ODEs) to model disease spread. In addition to their simplicity, these models are also the mean-field approximations of more accurate stochastic models of disease spread on contact networks. For the simplest case of a stochastic susceptible-infected-susceptible (SIS) process (infection with recovery) on a complete network, it has been shown that the fraction of infected nodes converges to the mean-field ODE as the number of nodes increases. However the proofs are not simple, requiring sophisticated probability, partial differential equations (PDE), or infinite systems of ODEs. We provide a short proof in this case for convergence in mean-square on finite time-intervals using a system of two ODEs and a moment inequality and also obtain the first lower bound on the expected fraction of infected nodes.
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