This paper introduces a composition of two single parameter generalized family of distributions: the alpha power transform and type II Topp-Leone-G families of distributions. Some basic mathematical treatments of the family of distributions are studied. The parameter estimates of the proposed family of distributions are derived via maximum likelihood estimation method and a Monte Carlo simulation study was conducted to examine the asymptotic behaviour of the parameter estimates of sub-model belonging to the proposed family of distributions. To illustrate the applicability of the proposed family of distributions in real world data fittings, two data sets consisting of the daily recovery and mortality rates of Covid-19 patients in Nigeria, from May 1 to June 30, 2020, was employed. The APTIITLK distribution arising from the proposed family of distributions, alongside with some bounded non-nested distributions was used to fit the two data sets and results obtained from the analysis clearly revealed that the APTIITLK distribution outperformed all the non-nested distributions used in fitting the two data sets. Some informative graphical plots for goodness of fit test were investigated to further validate the flexibility of the APTIITLK distribution over the competing distributions.