In this paper, we use a linear birth and death process with immigration to model infectious disease propagation when contamination stems from both person-to-person contact and contact with the environment. Our aim is to estimate the parameters of the process. The main originality and difficulty comes from the observation scheme. Counts of infected population are hidden. The only data available are periodic cumulated new retired counts. Although very common in epidemiology, this observation scheme is mathematically challenging even for such a standard stochastic process. We first derive an analytic expression of the unknown parameters as functions of well-chosen discrete time transition probabilities. Second, we extend and adapt the standard Baum-Welch algorithm in order to estimate the said discrete time transition probabilities in our hidden data framework. The performance of our estimators is illustrated both on synthetic data and real data of typhoid fever in Mayotte.

翻译：本文采用含移民的线性生灭过程，对由人际接触和环境接触共同引发的传染病传播进行建模。我们的目标是估计该过程的参数。主要创新点与难点源于观测方案：受感染人口数量为隐藏变量，唯一可观测的数据是定期累积的退出计数。尽管该观测方案在流行病学中极为常见，但即使对于此类标准随机过程，其在数学上仍具挑战性。我们首先推导出未知参数与精心选取的离散时间转移概率之间的解析表达式；其次，扩展并改进标准Baum-Welch算法，以在隐藏数据框架下估计前述离散时间转移概率。通过合成数据及马约特岛伤寒实际数据验证了估计量的性能。