Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.

季节性变化会影响许多传染病的动态，包括流感、霍乱和疟疾。传染性个体首次被引入人群的时间对于预测是否发生重大疾病爆发至关重要。在这项调查中，我们对具有季节性周期性的霍乱流行应用时间非齐次随机过程和多类型分支过程近似，以获得爆发概率的分析估计。特别是，对疾病灭绝概率的分析估计满足常微分方程组，该方程组遵循后向 Kolmogorov 微分方程。均值的显式表达式（分别为 假设发生灭绝的第一次灭绝时间的方差）是基于灭绝概率的分析估计得出的。我们的结果表明，疾病爆发的概率以及疾病首次灭绝的平均和标准推导在时间上是周期性的，并且取决于传染性个体或自由生活病原体被引入的时间。然后使用一般霍乱模型的两个例子进行数值模拟以验证分析预测。最后，将发展的理论结果扩展到更一般的传染病模型。疾病灭绝第一时间的平均值和标准推导在时间上是周期性的，取决于传染性个体或自由生活病原体被引入的时间。然后使用一般霍乱模型的两个例子进行数值模拟以验证分析预测。最后，将发展的理论结果扩展到更一般的传染病模型。疾病灭绝第一时间的平均值和标准推导在时间上是周期性的，取决于传染性个体或自由生活病原体被引入的时间。然后使用一般霍乱模型的两个例子进行数值模拟以验证分析预测。最后，将发展的理论结果扩展到更一般的传染病模型。