Identification of a time-varying SIR Model for Covid-19

Throughout human history, epidemics have been a constant presence. Understanding their dynamics is essential to predict scenarios and make substantiated decisions. Mathematical models are powerful tools to describe an epidemic behavior. Among the most used, the compartmental ones stand out, dividing population into classes with well-defined characteristics. One of the most known is the $SIR$ model, based on a set of differential equations describing the rates of change of three categories over time. These equations take into account parameters such as the disease transmission rate and the recovery rate, which both change over time. However, classical models use constant parameters and can not describe the behavior of a disease over long periods. In this work, it is proposed a $SIR$ model with time-varying transmission rate parameter with a method to estimate this parameter based on an optimization problem, which minimizes the sum of the squares of the errors between the model and historical data. Additionally, based on the infection rates determined by the algorithm, the model's ability to predict disease activity in future scenarios was also investigated. Epidemic data released by the government of the State of Rio Grande do Sul in Brazil was used to evaluate the models, where the models shown a very good forecasting ability, resulting in errors for predicting the total number of accumulated infected persons of 0.13% for 7 days ahead and 0.6 for 14 days ahead.