The incidence of AIDS in Portugal adjusted for reporting delay and underreporting

The Human Immunodeficiency Virus can evolve to severe illness having a major impact on socio-demographic and economic features of the affected countries. Most countries rely on surveillance systems to monitor the status of the epidemic which are based on cases notification by physicians. It can take several months until the diagnosed cases are notified and there are even cases that are not reported at all. The purpose of this paper is to adjust the Portuguese notification data to these main possible biasing problems. For the reporting delay we will use the traditional conditional likelihood estimation for count data assuming a Poisson distribution. For the cases that are not reported at all, we will use a mixture of Poisson distributions, based on natural conjugate prior distributions, and estimate the unknown parameters through maximum likelihood. The Poisson model suggests that approximately 80% of the Portuguese AIDS cases were reported within one year after the diagnosis and that the majority of cases were notified in the first three months. The BB/NBD model suggests that the probability of a new AIDS case being notified gets higher once a large amount of AIDS cases has been notified.