Statistical distributions can be grouped into families or systems.  The T-X family is the generalized family of distributions, suggested by Alzaatreh, et al.(2013). which consists of several sub-families of distributions, which can be used to develop new distributions. In this paper, Odds Burr XII generalized family of distributions is introduced. The special cases based on the Odds Burr XII-generalized family of distributions are proposed. The two suggested OBXII-distributions referred to as Odds Burr XII-logistic and Odds Burr XII -beta II distributions are constructed. Some properties of the proposed distributions are obtained. The maximum likelihood method is used under Type-I censored samples for estimating the parameters. A Monte Carlo simulation study is given to investigate the precision of the maximum likelihood estimates of the two suggested distributions. An application using COVID-19 data sets is introduced to show how the theoretical results can be used in practice.