In this article we present a new INteger-valued AutoRegressive (INAR) model with the aim of extracting baseline patterns of cattle fallen stock registered over an 5-year period at a local scale. We introduce HINAR as a generalization of the classical Poisson-based INAR models whose innovations follow a Hermite distribution. In order to assess trends and seasonality in these time series, we fit different models with time-dependent parameters by specifying proper functions. Using real world examples, we illustrate how to estimate parameters by maximum likelihood and validate the fitted models. We also show a detailed method to forecast. Our proposed model supposes a good solution for studying discrete time series when the counts have many zeros, low counts and moderate overdispersion. This model has been applied to the analysis of fallen cattle registered at a local scale as part of the development of a veterinary syndromic surveillance system.