Application of a Semi-empirical Dynamic Model to Forecast the Propagation of the COVID-19 Epidemics in Spain
Authors: Mora Juan C. , Pérez Sandra , Rodríguez Ignacio , Núñez Asunción , Dvorzhak Alla 
Published: April 2020

A semi-empirical model, based on the logistic map approach, was developed and applied to forecast the different phases of the evolution of the COVID-19 epidemic. This model can be used to make predictions of the propagation of the SARS-CoV-2 virus in different spatial scales: from a world scale to a country or even a smaller scale. Predictions on persons hospitalized, number of ventilators needed at ICUs and potential numbers of deaths were successfully carried out in different countries using this approach. This paper shows the mathematical basis for the model together with a proposal for its calibration on the different phases of the epidemic.

Specific results are shown for the COVID-19 epidemic in Spain. For predicting the evolution of the epidemic four phases were considered: non-controlled evolution since the 20th of February; total lock-down from the 15th of March; partial easing of the lock-down from the 13th of April; and a phased lock-down easing from the 1st of May.

In a first phase, if no control is established, the model predicted in Spain 12 millions of infected people of a total of 46.6 millions inhabitants. From those infected nearly 1 million people would need intensive care and around 700,000 deaths would be directly produced by the disease. However, as these numbers would occur in a brief period (few months), the number of deaths would have been higher due to the saturation of the health system.

For a second phase, considering a total lock-down of the whole country from the 15th of March, the model predicted for the 17th of April 194,000 symptomatic infected cases, 85,700 hospitalized, nearly 8,600 patients with needs of an ICU and 19,500 deaths. The model also predicted the peak to be produced between the 29th of March and the 3rd of April. Although the data are still under revision, the accuracy in all the predictions was very good, as the reported values by that day were 197,142 infected, 7,548 inpatients needing an ICU and 20,043 deaths. The peak was produced between the 31st of May and the 2nd of April.

For the third phase, the ease of the lock-down which began the 13th of April, early predictions were made by the beginning of April [Mora et al., 2020]. Assuming conservatively an infection daily rate of a 3% (r = 1.03) the model predicted 400,000 infections and 46, 000 ± 15,000 deaths by the end of May. The predictions overestimated the real values, due to a stricter reduction of the infection daily rate which lead to values of r < 1% and a revision of the whole series of data by the health authorities carried out along the month of May. A new prediction performed with updated parameters at the beginning of May provided a prediction of 250,000 infected and 29,000 ± 15,000 deaths. The reported values by the end of May were 282,870 infected and 28,552 deaths.

After the total easing of the lock-down many uncertainties appear, but the model predicts that the health system would not saturate if the daily rate of infections r is kept below 1.02 (2% of daily increase in the number of symptomatic infected).

This simple model provides a system to predict the evolution of epidemics with a good accuracy, even during epidemics development, where other systems have difficulties in their calibration. As the parameters involved in the model are based in empirical values of the different quantities (e.g. number of inpatients or deaths, related with the number of infected persons) it can be dynamically adjusted and adapted to sudden changes in the statistics.

As other models, the results provided by this model can be used by the authorities to support decision making in order to optimize resources and to minimize the consequences of epidemics, including the future outbreaks of the COVID-19 which will occur.