Distributions of prevalence and daily new cases in a stochastic linear SEIR model

Model parameters are typically estimated by calibrating the model to new case counts. This is important for understanding disease dynamics and guiding control measures. For parameter estimation, it is essential to identify the distribution of new cases and establish an appropriate likelihood function. This study employs a stochastic linear SEIR model to approximate the distributions of the number of infectious individuals and the number of daily new cases. We show that the probability-generating function (PGF) of the number of infectious individuals can be approximated as the product of PGFs of two birth-and-death processes. We theoretically derive formulas for the mean and variance of both the number of infectious individuals and daily new cases. Furthermore, we demonstrate that the distribution of the infectious population size can be approximated by a binomial or negative binomial distribution, depending on the relationship between its mean and variance. The distribution of daily new cases can also be well approximated by a binomial or negative binomial distribution, depending on the distribution of the infectious population. Specifically, if the number of infectious individuals follows a binomial distribution, the number of daily new cases is also binomial; if it follows a negative binomial distribution, the number of daily new cases is negative binomial as well. These findings provide a robust theoretical basis for parameter estimation and epidemic forecasting.