Yakowitz S
Systems and Industrial Engineering Department, University of Arizona, Tucson 85721, USA.
Math Biosci. 1995 May;127(1):99-121. doi: 10.1016/0025-5564(94)00048-5.
AIDS models, and epidemiological models generally, are almost exclusively either differential equations or Markov processes. Of course, the phenomena are fundamentally random, so at best differential equations track the expectation of the process and variability is masked. There are few techniques in the literature for numerical analysis of Markov chains of any size, and so those wishing to analyze stochastic epidemics presently have little alternative to simulation. The contribution of the present paper is to propose numerical techniques capable of finding marginal probabilities of Markov chains having thousands and even millions of states. The ideas are illustrated by application to AIDS models in the literature which formerly had been investigated only through Monte Carlo. This introductory foray has not plumbed the depths of the computational methodology, which yet needs refinement and streamlining that comes through experience. Yet in its primitive form, it is shown herein to be adequate for a computation on the scale of a two-population partition of the San Francisco homosexual epidemic. The closing discussion compares the strengths and weaknesses of the present numerical techniques with the simulation approach to investigation of Markov epidemics.
艾滋病模型以及一般的流行病学模型几乎无一例外都是微分方程或马尔可夫过程。当然，这些现象本质上是随机的，所以微分方程充其量只能追踪过程的期望值，而变异性则被掩盖了。文献中几乎没有针对任何规模的马尔可夫链进行数值分析的技术，因此目前那些希望分析随机流行病的人除了模拟之外几乎没有其他选择。本文的贡献在于提出能够找到具有数千甚至数百万个状态的马尔可夫链的边际概率的数值技术。通过应用于文献中的艾滋病模型来说明这些想法，这些模型以前仅通过蒙特卡罗方法进行研究。这一初步尝试尚未深入探究计算方法的深度，该方法仍需要通过经验进行完善和简化。然而，就其原始形式而言，本文表明它足以用于对旧金山同性恋群体疫情的两群体划分规模进行计算。结尾的讨论比较了当前数值技术与模拟方法在研究马尔可夫流行病方面的优缺点。
