Swansea University, Singleton Park, Swansea SA2 8PP, UK.
Math Biosci. 2011 Dec;234(2):108-17. doi: 10.1016/j.mbs.2011.09.002. Epub 2011 Sep 24.
The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the randomized SI (susceptible/infected) model. We obtain the probability density function for the early behavior of the epidemic. This provides an appropriate distribution for the initial conditions with which to describe the subsequent deterministic evolution of the system. We apply the method of matching asymptotic expansions to link the two stages. This allows us to estimate the standard deviation of the number of infectives in the developed outbreak, and the statistical characteristics of the outbreak time. The potential trajectories caused by the stochastic nature of the contamination stage show greatest divergence at the initial and fade-out stages and coincide most tightly just after the peak of the epidemic. The time to the peak of the outbreak is not strongly dependent on the initial trajectory.
一个描述初始污染的随机 Markov 过程和一个带有随机初始条件的关联确定动力系统，用于爆发的持续发展。初始污染阶段可以很好地用随机化 SI（易感/感染）模型来近似。我们得到了传染病早期行为的概率密度函数。这为描述系统后续确定性演化的初始条件提供了合适的分布。我们应用匹配渐近展开的方法来连接两个阶段。这使我们能够估计在已发展的爆发中感染人数的标准差，以及爆发时间的统计特征。污染阶段的随机性引起的潜在轨迹在初始和消退阶段分歧最大，而在疫情高峰期后则最为吻合。疫情高峰期的时间与初始轨迹没有很强的相关性。
