Congestion is a phenomenon that impacts most cities in the world. Due to car emissions, it is a significant source of pollution. Even though mobility restrictions can reduce congestion and emissions, essential activities still need cars. With lockdown measures during the global pandemic of Covid-19, measuring essential traffic data has been made possible. This paper concerns analysis and modelling of such essential traffic. It appears that congestion dynamics of essential traffic exhibits dynamics than can be represented with a linear model. This paper introduces such a model and provide a method to jointly estimate the parameters and the model input. The model is validated with data collected in Johannesburg, South Africa.