Large families of bivariate probability models based on Panjer-type relations are proposed. These relations are generated by differential equations of probability generating functions (p.g.fs), expressed by rational functions that incorporate parameters. In this setting, models are not explicit, but their p.g.fs may be obtained explicitly by solving such differential equations. Then, for parameter estimation, minimization of distances between p.g.fs and empirical p.g.fs is used. Numerical applications to real data sets are presented.