基于图模型推理和凸规划的流行病预测与预防
Prediction and Prevention of Pandemics via Graphical Model Inference and Convex Programming
摘要：
难以预测的COVID-19大流行爆发揭示了统计建模的重要性，这将解决地理区域的时空相关性，例如感染在人口普查区域粒度的城市中的传播。在本文中，我们为以下两个相互关联的公共卫生挑战提供了算法答案。(1)推理挑战：假设在城市中有$N$人口普查区（节点），并且在任何一组节点处给定初始感染，在感染爆发的传播稳定时，人口普查区的子集被感染的概率是多少？(2)预防挑战：什么是最低限度的控制行动，可以采取最小化的稳定状态足迹的感染部分？为了应对这些挑战，我们建立了一个有吸引力的伊辛（成对，二进制）类型的流行病的图形模型，其中每个节点代表一个普查轨迹，每个边缘因子代表一对节点之间的成对相互作用的强度。我们发现，几乎所有有吸引力的伊辛模型稠密图的结果在两种模式的最可能的状态：要么所有的节点，最初没有被感染的感染，或所有最初未感染的节点仍然未被感染。推理挑战的这种双模态解决方案允许我们将预防挑战重新表述为以下易于处理的凸规划：对于具有成对和偏置因子的裸伊辛模型，表示没有预防措施的系统，使得对于至少一个初始感染模式，MAP状态被完全感染，在$l_1$范数中找到最接近的，一组因素导致伊辛模型的所有MAP状态，应用最佳预防措施，变得安全。
英文摘要：
Hard-to-predict bursts of COVID-19 pandemic revealed significance of statistical modeling which would resolve spatio-temporal correlations over geographical areas, for example spread of the infection over a city with census tract granularity. In this manuscript, we provide algorithmic answers to the following two inter-related public health challenges. (1) Inference Challenge: assuming that there are $N$ census blocks (nodes) in the city, and given an initial infection at any set of nodes, what is the probability for a subset of census blocks to become infected by the time the spread of the infection burst is stabilized? (2) Prevention Challenge: What is the minimal control action one can take to minimize the infected part of the stabilized state footprint? To answer the challenges, we build a Graphical Model of pandemic of the attractive Ising (pair-wise, binary) type, where each node represents a census track and each edge factor represents the strength of the pairwise interaction between a pair of nodes. We show that almost all attractive Ising Models on dense graphs result in either of the two modes for the most probable state: either all nodes which were not infected initially became infected, or all the initially uninfected nodes remain uninfected. This bi-modal solution of the Inference Challenge allows us to re-state the Prevention Challenge as the following tractable convex programming: for the bare Ising Model with pair-wise and bias factors representing the system without prevention measures, such that the MAP state is fully infected for at least one of the initial infection patterns, find the closest, in $l_1$ norm, set of factors resulting in all the MAP states of the Ising model, with the optimal prevention measures applied, to become safe.
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