COVID-19: Estimating spread in Spain solving an inverse problem with a probabilistic model

We introduce a new probabilistic model to estimate the real spread of the novel SARS-CoV-2 virus along regions or countries. Our model simulates the behavior of each individual in a population according to a probabilistic model through an inverse problem; we estimate the real number of recovered and infected people using mortality records. In addition, the model is dynamic in the sense that it takes into account the policy measures introduced when we solve the inverse problem. The results obtained in Spain have particular practical relevance: the number of infected individuals can be $17$ times higher than the data provided by the Spanish government on April $26$ $th$ in the worst-case scenario. Assuming that the number of fatalities reflected in the statistics is correct, $8$ percent of the population may be contaminated or have already been recovered from the virus in Madrid, one of the most affected regions in Spain. However, if we assume that the number of fatalities is twice as high as the official numbers, the number of infections could have reached $19.5$\%. In Galicia, one of the regions where the effect has been the least, the number of infections does not reach $2.5$ \%. Based on our findings, we can: i) estimate the risk of a new outbreak before Autumn if we lift the quarantine; ii) may know the degree of immunization of the population in each region; and iii) forecast or simulate the effect of the policies to be introduced in the future based on the number of infected or recovered individuals in the population.