Allen L J, Burgin A M
Department of Mathematics and Statistics, Texas Tech University, Lubbock 79409-1042, USA.
Math Biosci. 2000 Jan;163(1):1-33. doi: 10.1016/s0025-5564(99)00047-4.
The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.
分析并比较了确定性和随机性离散时间流行病模型的动态特性。离散时间随机模型是马尔可夫链，是连续时间模型的近似。分析了具有恒定种群规模和一般感染强度的SIS和SIR类型模型，然后分析了一个更一般的具有可变种群规模的SIS模型。在确定性模型中，基本再生数R0的值决定了疾病的持续存在或灭绝。如果R0 < 1，疾病被消除，而如果R0 > 1，疾病在种群中持续存在。由于本文考虑的所有随机模型都具有至少一个吸收状态的有限状态空间，无论R0的值如何，疾病最终灭绝是确定的。然而，在某些情况下，直到疾病灭绝的时间可能非常长。在这些情况下，如果概率分布以非灭绝为条件，那么当R0 > 1时，存在一个准平稳概率分布，其均值与确定性地方病平衡一致。对疫情的预期持续时间进行了数值研究。
