Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA.
J Biol Dyn. 2012;6:590-611. doi: 10.1080/17513758.2012.665502.
The basic reproduction number, ℛ(0), one of the most well-known thresholds in deterministic epidemic theory, predicts a disease outbreak if ℛ(0)>1. In stochastic epidemic theory, there are also thresholds that predict a major outbreak. In the case of a single infectious group, if ℛ(0)>1 and i infectious individuals are introduced into a susceptible population, then the probability of a major outbreak is approximately 1-(1/ℛ(0))( i ). With multiple infectious groups from which the disease could emerge, this result no longer holds. Stochastic thresholds for multiple groups depend on the number of individuals within each group, i ( j ), j=1, …, n, and on the probability of disease extinction for each group, q ( j ). It follows from multitype branching processes that the probability of a major outbreak is approximately [Formula: see text]. In this investigation, we summarize some of the deterministic and stochastic threshold theory, illustrate how to calculate the stochastic thresholds, and derive some new relationships between the deterministic and stochastic thresholds.
基本繁殖数 ℛ(0) 是确定性传染病理论中最著名的阈值之一，如果 ℛ(0)>1 ，则预示着疾病的爆发。在随机传染病理论中，也存在着预测大爆发的阈值。在单个感染组的情况下，如果 ℛ(0)>1 且有 i 个感染者被引入易感人群，则大爆发的概率约为 1-(1/ℛ(0))( i )。如果有多个可能引发疾病的感染组，那么这个结果就不再成立。多组的随机阈值取决于每个组内的个体数量 i ( j )， j=1 ，…， n ，以及每个组的疾病灭绝概率 q ( j )。从多类型分支过程中可以得出，大爆发的概率约为[公式：见文本]。在本研究中，我们总结了一些确定性和随机性阈值理论，说明了如何计算随机性阈值，并推导出了确定性和随机性阈值之间的一些新关系。
