Discrete distributions have their important in modeling count data in several applied fields such as epidemiology, public health, sociology, medicine, and agriculture. This paper discusses the estimation of the parameters of two discrete models called discrete Poisson-Lindley and discrete Lindley distributions, using several frequentist estimation methods. Parameter estimation can provide a guideline for choosing the best method of estimation for the model parameters, which would be very important to reliability engineers and applied statisticians. The finite sample properties of the estimates are explored using extensive simulation results. The ordering performance of the proposed estimators is determined by the partial and overall ranks of different parametric values. We also derived two important actuarial measures of the two discrete models. A computational study for the two risk measures is conducted. Finally, applications of the two discrete distributions have been examined and compared with other discrete distributions via three data sets from the medicine field including two COVID-19 data sets.