On the evolution of the COVID-19 epidemiological parameters using only the series of deceased. A study of the Spanish outbreak using Genetic Algorithms

We propose a methodology for estimating the evolution of the epidemiological parameters of a SIRD model (acronym of Susceptible, Infected, Recovered and Deceased individuals) which allows to evaluate the sanitary measures taken by the government, for the COVID-19 in the Spanish outbreak. In our methodology the only information required for estimating these parameters is the time series of deceased people; due to the number of asymptomatic people produced by the COVID-19, it is not possible to know the actual number of infected people at any given time. Therefore, among the different time series that quantify the pandemic we consider just the number of deceased people to minimize the square sum of errors. The time series of deaths considered runs from March to the end of September and is divided into four sub-periods reflecting the different isolation measures taken by the Spanish government. The parameters that we can estimate are the time from the beginning of the disease, the transmission rate, and the recovery rate; these last two ratios are estimated in each of the different sub-periods. In this way the model considered has 2x4+1=9 parameters that are estimated jointly over the whole period from the data of deceased. Given the complexity of the model, to estimate the parameters that minimize the square sum of errors, a Genetic Algorithm is used. Our methodology confirms the effectiveness of the sanitary measures taken by the Spanish government showing a dramatic reduction in the basic reproductive number $R_0$ during confinement; also, a further increase in $R_0$ after the end of the alarm state decreed by the government on June 21 was detected. Our results also point out that the Patient Zero in the COVID-19 Spanish outbreak emerged between the end of December and early January, at least four weeks before January 31st, that was the moment when the Spanish authorities reported the first positive case.