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We introduce a new probabilistic model to estimate the real spread of thenovel SARS-CoV-2 virus along regions or countries. Our model simulates thebehavior of each individual in a population according to a probabilistic modelthrough an inverse problem; we estimate the real number of recovered andinfected people using mortality records. In addition, the model is dynamic inthe sense that it takes into account the policy measures introduced when wesolve the inverse problem. The results obtained in Spain have particularpractical relevance: the number of infected individuals can be $17$ timeshigher than the data provided by the Spanish government on April $26$ $th$ inthe worst-case scenario. Assuming that the number of fatalities reflected inthe statistics is correct, $9.8$ percent of the population may be contaminatedor have already been recovered from the virus in Madrid, one of the mostaffected regions in Spain. However, if we assume that the number of fatalitiesis twice as high as the official numbers, the number of infections could havereached $19.5\%$. In Galicia, one of the regions where the effect has been theleast, the number of infections does not reach $2.5 \%$ . Based on ourfindings, we can: i) estimate the risk of a new outbreak before Autumn if welift the quarantine; ii) may know the degree of immunization of the populationin each region; and iii) forecast or simulate the effect of the policies to beintroduced in the future based on the number of infected or recoveredindividuals in the population.

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