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The main concern of this paper is providing a flexible discrete model that captures every kind of dispersion (equi-, over- and under-dispersion). Based on the balanced discretization method, a new discrete version of Burr–Hatke distribution is introduced with the partial moment-preserving property. Some statistical properties of the new distribution are introduced, and the applicability of proposed model is evaluated by considering counting series. A new integer-valued autoregressive (INAR) process based on the mixing Pegram and binomial thinning operators with discrete Burr–Hatke innovations is introduced, which can model contagious data properly. The different estimation approaches of parameters of the new process are provided and compared through the Monte Carlo simulation scheme. The performance of the proposed process is evaluated by four data sets of the daily death counts of the COVID-19 in Austria, Switzerland, Nigeria and Slovenia in comparison with some competitor INAR(1) models, along with the Pearson residual analysis of the assessing model. The goodness of fit measures affirm the adequacy of the proposed process in modeling all COVID-19 data sets. The fundamental prediction procedures are considered for new process by classic, modified Sieve bootstrap and Bayesian forecasting methods for all COVID-19 data sets, which is concluded that the Bayesian forecasting approach provides more reliable results. [ABSTRACT FROM AUTHOR]