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 the environment. Our aim is to estimate the parameters of the process. The main originality and difficulty comes from the observation framework. Indeed, counts of infected population are hidden. The only data available are periodic cumulated new infected counts. 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：本文利用带有移民的线性生灭过程来模拟传染病传播，其中污染源于人际接触和环境两方面。我们的目标是估计该过程的参数。主要创新点和难点在于观测框架：感染人群的计数是隐藏的，仅有周期性累积的新增感染人数数据可用。我们首先推导出未知参数的解析表达式，将其表示为精心选择的离散时间转移概率的函数。其次，我们扩展并改进了标准Baum-Welch算法，以在隐藏数据框架下估计这些离散时间转移概率。通过合成数据及马约特岛伤寒真实数据验证了估计量的性能。