Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.

中文翻译：

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鸟类西尼罗河病毒动力学随机流行模型中疾病灭绝或爆发的概率

疾病灭绝阈值为预防和控制疾病提供了重要信息。在本文中，基于其类似确定性模型的假设，开发了用于西尼罗河病毒在鸟类中传播动力学的随机流行模型，即连续时间马尔可夫链。分支过程用于导出随机模型的灭绝阈值和疾病灭绝或持续存在的条件。从分支过程计算出的疾病灭绝概率与数值模拟近似的概率非常吻合。比较两种模型的疾病动态，以确定人口随机性对西尼罗河病毒动态的影响。分析和数值结果表明，随机模型和确定性模型之间的模型预测和渐近动力学存在差异，这对于预防疾病爆发至关重要。研究发现，如果疾病来自暴露的蚊子，而不是来自传染性蚊子和鸟类，则疾病灭绝的可能性很高。使用样本路径估计疾病灭绝的有限时间，并且表明如果疾病是由暴露的蚊子引入的，则流行持续时间最短。