This paper reports a broad study using epidemic-related counting data of COVID-19 disease caused by the novel coronavirus (SARS-CoV-2). The considered dataset refers to Brazil's daily and accumulated counts of reported cases and deaths in a fixed period (from January 22 to June 16, 2020). For the data analysis, it has been adopted a nonlinear rational polynomial function to model the mentioned counts assuming Gaussian errors. The least-squares method was applied to fit the proposed model. We have noticed that the curves are still increasing after June 16, with no evidence of peak being reached or decreasing behavior in the period for new reported cases and confirmed deaths by the disease. The obtained results are consistent and highlight the adopted model's capability to accurately predict the behavior of Brazil's COVID-19 growth curve in the observed time-frame.