论文标题
传染病的随机建模，第一部分：了解负二项式分布并更可靠地预测流行病
Stochastic Modeling of an Infectious Disease, Part I: Understand the Negative Binomial Distribution and Predict an Epidemic More Reliably
论文作者
论文摘要
为什么在不同城市或人口，医疗基础设施和人们的行为中相似的不同城市或国家中，Covid-19的流行模式如此不同？为什么关于被感染或死亡的人的数量，所谓的专家的预测或预测通常是严重错误的？这项研究的目的是更好地了解流行病的随机性，并回答上述问题。关于传染病的大部分工作都是基于“确定性模型”的（Kermack和McKendrick：1927。）我们将探索可以捕捉传染病行为的本质的随机模型。随机模型以表述，考虑了传染病的随机性质。 我们在这里研究的随机模型基于“移民的出生和死亡过程”（简称BDI），这是在研究人口生长或某些生物物种灭绝的研究中提出的。但是，流行病学社区尚未对BDI过程模型进行调查。 BDI过程是一些出生和死亡过程之一，我们可以通过分析解决。它的时间依赖性概率分布函数是“负二项式分布”，其参数$ r $少于$ 1 $。该过程的“变化系数”大于$ \ sqrt {1/r}> 1 $。此外，它具有像Zeta分布一样长的尾巴。这些特性解释了为什么感染模式显示出极大的变化。确定性模型预测的感染数量远大于分布的中位数。这就解释了为什么基于确定性模型的任何预测都会更频繁地失败。
Why are the epidemic patterns of COVID-19 so different among different cities or countries which are similar in their populations, medical infrastructures, and people's behavior? Why are forecasts or predictions made by so-called experts often grossly wrong, concerning the numbers of people who get infected or die? The purpose of this study is to better understand the stochastic nature of an epidemic disease, and answer the above questions. Much of the work on infectious diseases has been based on "SIR deterministic models," (Kermack and McKendrick:1927.) We will explore stochastic models that can capture the essence of the seemingly erratic behavior of an infectious disease. A stochastic model, in its formulation, takes into account the random nature of an infectious disease. The stochastic model we study here is based on the "birth-and-death process with immigration" (BDI for short), which was proposed in the study of population growth or extinction of some biological species. The BDI process model ,however, has not been investigated by the epidemiology community. The BDI process is one of a few birth-and-death processes, which we can solve analytically. Its time-dependent probability distribution function is a "negative binomial distribution" with its parameter $r$ less than $1$. The "coefficient of variation" of the process is larger than $\sqrt{1/r} > 1$. Furthermore, it has a long tail like the zeta distribution. These properties explain why infection patterns exhibit enormously large variations. The number of infected predicted by a deterministic model is much greater than the median of the distribution. This explains why any forecast based on a deterministic model will fail more often than not.
