We propose several spatial-temporal epidemiological mathematical models to study their suitability to approximate the dynamics of the early phase of the COVID-19 pandemic in Chile. The model considers the population density of susceptible, infected, and recovered individuals. The models are based on a system of partial differential equations. The first model considers a space-invariant transmission rate, and the second modeling approach is based on different space-variant transmission rates. The third modeling approach, which is more complex, uses a transmission rate that varies with space and time. One main aim of this study is to present the advantages and drawbacks of the mathematical approaches proposed to describe the COVID-19 pandemic in Chile. We show that the calibration of the models is challenging. The results of the model's calibration suggest that the spread of SARS-CoV-2 in the regions of Chile was different. Moreover, this study provides additional insight since few studies have explored similar mathematical modeling approaches with real-world data.