Multinomial analysis of smoothed HIV back-calculation models incorporating uncertainty in the AIDS incidence

Back-calculation models, developed to reconstruct the past trend of human immunodeficiency virus (HIV) and to project future acquired immunodeficiency syndrome incidence (AIDS), are usually and unrealistically based on the assumption that the observed AIDS counts are independently distributed according to a Poisson process. In contrast, we argue that a multinomial framework is more suitable to this situation, leading to a natural covariance structure. The ill-conditioned nature of the problem is solved by modelling the HIV parameters according to a cubic spline function to reduce the dimensionality of the parameter space and obtain smoother parameter estimates. We applied a regression spline technique which yields to a computationally stable basis incorporating the incubation period in the new design matrix. We directly incorporate the reporting delay distribution in the AIDS incidence data, leading to a more complex formulation of the variance and covariance model that is adapted to the iteratively reweighted least square (IRLS) algorithm. In this case we obtain more accurate estimates of the standard error of the HIV incidence, especially in the most recent time. Our model, which uses a cubic spline reparameterization based on a multinomial probability distribution, is applied to the AIDS epidemic data in Italy.