By analysing the evolution of the COVID-19 epidemic in the state of Minas Gerais, Brazil, we showed the importance of considering the sub-notification not only of deaths but also of infected cases. It was shown that the largely used criteria of a historical all-deaths baseline is not approachable in this case, where most of the deaths are associated with causes that should decrease due to social distancing and reduction of economic activities. A quite simple and intuitive model based on the Gompertz function was applied to estimate excess deaths and excess of infected cases. It fits well the data and predicts the evolution of the epidemic adequately. Based on these analyses, an excess of 21.638 deaths and 557.216 infected cases is predicted until the end of 2020, with an upper bound of the case fatality rate of around 2.4 % and a prevalence of 2.6 %. The geographical distribution of cases and deaths and its ethnic correlation are also presented. This study points out the necessity of governmental and private organizations working together to improve public awareness and stimulate social distancing to curb the viral infection, especially in critical places with high poverty.