In this study we propose a deformation-based framework to jointly model the influence of aging and Alzheimer's disease (AD) on the brain morphological evolution. Our approach combines a spatio-temporal description of both processes into a generative model. A reference morphology is deformed along specific trajectories to match subject specific morphologies. It is used to define two imaging progression markers: 1) a morphological age and 2) a disease score. These markers can be computed regionally in any brain region. The approach is evaluated on brain structural magnetic resonance images (MRI) from the ADNI database. The model is first estimated on a control population using longitudinal data, then, for each testing subject, the markers are computed cross-sectionally for each acquisition. The longitudinal evolution of these markers is then studied in relation with the clinical diagnosis of the subjects and used to generate possible morphological evolutions. In the model, the morphological changes associated with normal aging are mainly found around the ventricles, while the Alzheimer's disease specific changes are located in the temporal lobe and the hippocampal area. The statistical analysis of these markers highlights differences between clinical conditions even though the inter-subject variability is quite high. The model is also generative since it can be used to simulate plausible morphological trajectories associated with the disease. Our method quantifies two interpretable scalar imaging biomarkers assessing respectively the effects of aging and disease on brain morphology, at the individual and population level. These markers confirm the presence of an accelerated apparent aging component in Alzheimer's patients but they also highlight specific morphological changes that can help discriminate clinical conditions even in prodromal stages. More generally, the joint modeling of normal and pathological evolutions shows promising results to describe age-related brain diseases over long time scales.