Logistic regression models integrating disease presence/absence data are widely used to identify risk factors for a given disease. However, when data arise from imperfect surveillance systems, the interpretation of results is confusing since explanatory variables can be related either to the occurrence of the disease or to the efficiency of the surveillance system. As an alternative, we present spatial and non-spatial zero-inflated Poisson (ZIP) regressions for modelling the number of highly pathogenic avian influenza (HPAI) H5N1 outbreaks that were reported at subdistrict level in Thailand during the second epidemic wave (July 3rd 2004 to May 5th 2005). The spatial ZIP model fitted the data more effectively than its non-spatial version. This model clarified the role of the different variables: for example, results suggested that human population density was not associated with the disease occurrence but was rather associated with the number of reported outbreaks given disease occurrence. In addition, these models allowed estimating that 902 (95% CI 881-922) subdistricts suffered at least one HPAI H5N1 outbreak in Thailand although only 779 were reported to veterinary authorities, leading to a general surveillance sensitivity of 86.4% (95% Cl 84.5-88.4). Finally, the outputs of the spatial ZIP model revealed the spatial distribution of the probability that a subdistrict could have been a false negative. The methodology presented here can easily be adapted to other animal health contexts.