In this paper, an epidemic compartmental model with saturated type treatment function is presented to investigate the transmission dynamics of COVID-19 with a case study of Spain (in Europe). We obtain the basic reproduction number of the model which plays a very important role in disease spreading. We show that if the basic reproduction number is less than unity then the disease-free equilibrium point is locally asymptotically stable, but making the basic reproduction number less than unity is not sufficient to eradicate COVID-19 infection which is shown through backward bifurcation. The model is validated with the real COVID-19 data of Spain (in Europe), Algeria (in Africa), and India (in Asia) and also estimated important model parameters in all cases. The effect of an important model parameter for controlling the disease spreading is also investigated for the infection scenario of Spain only. We establish that the asymptomatic class plays a very important role for spreading this pandemic disease. The effective reproduction number has been estimated which varies in time in Spain. Finally, the model is reformulated as an optimal control problem which shows that the social distancing due to adapting a partial lockdown by some countries is highly effective for controlling COVID-19.