We discuss predictive density for Poisson sequence models under sparsity constraints. Sparsity in count data implies situations where there exists an overabundance of zeros or near-zero counts. We investigate the exact asymptotic minimax Kullback--Leibler risks in sparse and quasi-sparse Poisson sequence models. We also construct a class of Bayes predictive densities that attain exact asymptotic minimaxity without the knowledge of true sparsity level. Our construction involves the following techniques: (i) using spike-and-slab prior with an improper prior; (ii) calibrating the scaling of improper priors from the predictive viewpoint; (iii) plugging a convenient estimator into the hyperparameter. For application, we also discuss the performance of the proposed Bayes predictive densities in settings where current observations are missing completely at random. The simulation studies as well as applications to real data demonstrate the efficiency of the proposed Bayes predictive densities. This talk is based on the joint work with Fumiyasu Komaki (University of Tokyo) and Ryoya Kaneko (University of Tokyo).