We present a new method for studying the star formation history of late-type, cluster galaxies undergoing gas starvation or a ram-pressure stripping event by combining bidimensional multifrequency observations with multi-zones models of galactic chemical and spectrophotometricevolution. This method is applied to the Virgo cluster anemic galaxy NGC 4569. We extract radial profiles from recently obtained UV GALEX images at 1530 and 2310 A, from visible and near-IR narrow (Halpha) and broad band images at different wavelengths (u, B, g, V, r, i, z, J, H, K), from Spitzer IRAC and MIPS images and from atomic and molecular gas maps. The model in the absence of interaction (characterized by its rotation velocity and spin parameter) is constrained by the unperturbed H band light profile and by the Halpha rotation curve. We can reconstruct the observed total-gas radial-density profile and the light surface-brightness profiles at all wavelengths in a ram-pressure stripping scenario by making simple assumptions about the gas removal process and the orbit of NGC 4569 inside the cluster. The observed profiles cannot be reproduced by simply stopping gas infall, thus mimicing starvation. Gas removal is required, which is more efficient in the outer disk, inducing a radial quenching in the star formation activity, as observed and reproduced by the model. This observational result, consistent with theoretical predictions that a galaxy-cluster IGM interaction is able to modify structural disk parameters without gravitational perturbations, is discussed in the framework of the origin of lenticulars in clusters. ; Comment: Accepted for publication on ApJ