The main objective of this paper is to describe and interpret an SIR (Susceptible-Infectious-Recovered) epidemic model though a logistic equation, which is parameterized by a Malthusian parameter and a carrying capacity parameter, both being time-varying, in general, and then to apply the model to the COVID-19 pandemic by using some recorded data. In particular, the Malthusian parameter is related to the growth rate of the infection solution while the carrying capacity is related to its maximum reachable value. The quotient of the absolute value of the Malthusian parameter and the carrying capacity fixes the transmission rate of the disease in the simplest version of the epidemic model. Therefore, the logistic version of the epidemics' description is attractive since it offers an easy interpretation of the data evolution especially when the pandemic outbreaks. The SIR model includes recruitment, demography, and mortality parameters, and the total population minus the recovered population is not constant though time. This makes the current logistic equation to be time-varying. An estimation algorithm, which estimates the transmission rate through time from the discrete-time estimation of the parameters of the logistic equation, is proposed. The data are picked up at a set of samples which are either selected by the adaptive sampling law or allocated at constant intervals between consecutive samples. Numerical simulated examples are also discussed.