An algorithm for the direct estimation of the parameters of the SIR epidemic model from the I(t) dynamics

The discrete SIR (Susceptible-Infected-Recovered) model is used in many studies to model the evolution of epidemics. Here, we consider one of its dynamics-the exponential decrease in infected cases I(t). By considering only the I(t) dynamics, we extract three parameters: the exponent of the initial exponential increase $$ ; the maximum value $$ ; and the exponent of the final decrease $$ . From these three parameters, we show mathematically how to extract all relevant parameters of the SIR model. We test this procedure on numerical data and then apply the methodology to real data received from the COVID-19 situation in France. We conclude that, based on the hospitalized data and the ICU (Intensive Care Unit) cases, two exponentials are found, for the initial increase and the decrease in I(t). The parameters found are larger than reported in the literature, and they are associated with a susceptible population which is limited to a sub-sample of the total population. This may be due to the fact that the SIR model cannot be applied to the covid-19 case, due to its strong hypotheses such as mixing of all the population, or also to the fact that the parameters have changed over time, due to the political initiatives such as social distanciation and lockdown.